DAHLGREN DIVISION 

NAVAL SURFACE WARFARE CENTER 

DAHLGREN, VIRGINIA 22448-5100 

NSWCDD/TR-03/81 

APPLICATION OF THE LAMBERT PROBLEM 

TO INVERSE-SQUARE GRAVITY 

BY ISAAC SOFAIR 

STRATEGIC AND STRIKE SYSTEMS DEPARTMENT 

December 2003 

Approved for public release; distribution is unlimited 

i/ii

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 December 2003 

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS 

Application of the Lambert Problem to Inverse-Square Gravity 

6. AUTHOR(s) 

Isaac Sofair 

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Commander 

Dahlgren Division, Naval Surface Warfare Center 

Attn: K41 

17320 Dahlgren Road 

Dahlgren, VA 22448-5100 

NSWCDD/TR-03/81 

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Approved for public release; distribution is unlimited. 

13. ABSTRACT (Maximum 200 words) 

The Lambert Problem is to determine vc

 , given 1r

 , 2r , and f t . vc

 can be determined by employing an 

iterative procedure that utilizes the Newton-Raphson method. Three candidates for the iteration variable are 

considered. Equations for an elliptical trajectory in terms of each of the iteration variables are derived. Equations 

for hyperbolic and parabolic trajectories are then deduced, and it is demonstrated how the type of trajectory can be 

determined from the given parameters. 

14. SUBJECT TERMS 15. NUMBER OF PAGES 

114 

16. PRICE CODE 

Lambert, trajectory, correlated velocity, null-miss, time of flight, eccentric anomaly, 

reentry, target, release, impact, elliptical, hyperbolic, parabolic 

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 OF REPORT 

18. SECURITY CLASSIFICATION

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ABSTRACT 

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL 

NSWCDD/TR-03/81 

iii/iv

FOREWORD 

The computations described in this report were performed in the Fire Control Formulation 

Branch (K41), Submarine-Launched Ballistic Missile (SLBM) Research and Analysis Division, 

Strategic and Strike Systems Department. This report is a compilation of all the important 

mathematical results and their derivations pertaining to the application of the Lambert Problem 

to inverse-square gravity. In particular, expressions for the correlated velocity and the null-miss 

vector in terms of known parameters are derived. The treatment is carried out in detail for 

elliptical, parabolic, and hyperbolic trajectories. The appendices include background material 

needed for the main report, alternative derivations of or expressions for some key variables, and 

a derivation of an expression for the rate of change of the null-miss vector. 

It is intended that this report will serve as a comprehensive reference guide for all those 

who desire easy access to the Keplerian trajectory equations simulating the trajectory of a freefall reentry vehicle in an inverse-square gravitational field. 

The author would like to acknowledge the assistance of Davis Owen and William Davis in 

providing important feedback. He is also indebted to the Documentation Management System 

(DMS) group whose patient assistance in editing the report to conform with accepted format was 

crucial to its completion, and to the Technical Information Division for proofreading the report 

to ensure that the correct format was adhered to. 

Approved by:

SHEILA D. YOUNG, Head 

Strategic and Strike Systems Department 

NSWCDD/TR-03/81 

v

CONTENTS 

Page

INTRODUCTION ................................................................................................................................................1

TIME OF FLIGHT, CORRELATED VELOCITY, VELOCITY AT IMPACT ..................................................2

NULL-MISS DIRECTION...................................................................................................................................6

REENTRY FLIGHT PATH ANGLE, ANGLE OF ATTACK, AND YA ...........................................................9

TIME OF FLIGHT IN TERMS OF cθ v ............................................................................................................11

TIME OF FLIGHT IN TERMS OF p ..............................................................................................................14

EQUATIONS FOR A HYPERBOLIC TRAJECTORY.....................................................................................15

EQUATIONS FOR A PARABOLIC TRAJECTORY .......................................................................................18

DETERMINATION OF TYPE OF TRAJECTORY..........................................................................................21

REFERENCES ...................................................................................................................................................23

BIBLIOGRAPHY...............................................................................................................................................24

APPENDIXES 

A - TIME OF FLIGHT BETWEEN TWO POINTS OF AN ELLIPTICAL TRAJECTORY...................A-1 

B - AN ALTERNATIVE DERIVATION OF THE TIME OF FLIGHT................................................... B-1 

C - EXPRESSIONS FOR AND IN TERMS OF AND 1 2 f , , , r, r , gf g p   .......................................... C-1 

D - TWO DIFFERENT SETS OF INITIAL CONDITIONS ....................................................................D-1 

E - DERIVATION AND COMPUTATION OF THE ELEMENTS 

ARISING IN THE Q-GUIDANCE MATRIX ............................................................................ E-1 

F - DERIVATION OF AN ANALYTIC EXPRESSION FOR 

f R t

c v

















∂

∂

θ

θ ...............................................F-1 

G - A GENERAL EXPRESSION FOR THE NULL-MISS DIRECTION................................................G-1 

H - DETERMINATION OF FOOTPRINT ACHIEVABILITY IN CONNECTION 

WITH TIME OF FLIGHT AND CANTING ANGLE................................................................H-1 

I - COMPUTATION OF THE RATE OF CHANGE OF THE NULL-MISS VECTOR ........................... I-1 

J - DERIVATION OF SECOND PARTIALS OF AND c cr v v θ ................................................................ J-1 

K - DERIVATION OF , , , AND R R

R f b f Rf bf

ddd d

dt dt dt dt

θ θ β β

θβθ β

∂∂∂ ∂

∂∂∂ ∂

..................................................K-1 

DISTRIBUTION ............................................................................................................................................... (1) 
NSWCDD/TR-03/81 

vi 

ILLUSTRATIONS 

Figure Page

1 PARABOLA WHOSE FOCUS IS AT THE ORIGIN AND WHOSE VERTEX IS AT ) ( ,0 0r ..............19 

A-1 POSITION AND VELOCITY OF AN OBJECT UNDERGOING ELLIPTICAL MOTION 

WITH THE CENTER OF THE EARTH AS THE FOCUS OF THE ELLIPSE......................................A-3 

A-2 ECCENTRIC ANOMALY VERSUS TRUE ANOMALY ......................................................................A-4 

G-1 REPRESENTATION OF A SMALL PERTURBATION IN THE TARGET VECTOR.........................G-4 

H-1 CANTING ANGLE AS A MEASURE OF THE DEVIATION 

OF 2r′

 FROM THE DOWNRANGE ARC, R θ ........................................................................................H-3 
NSWCDD/TR-03/81 

1 

INTRODUCTION 

The correlated velocity, vc

 , is that velocity which will carry a reentry body on a free-fall 

trajectory from the initial position (at release), 1r

 , to the target position, 2r , in a specified time, 

f t . A guidance computer steers the missile in a direction which will cause the difference 

between the correlated velocity and the true velocity to approach zero. When this difference, 

which is called the velocity to be gained, is zero, the reentry body is released. 

The Lambert Problem is to determine vc

 , given 1r

 , 2r , and f t . It is well-known that there 

is no closed-form expression for vc

 in terms of the given parameters. However, vc can be 

determined by employing an iterative procedure that utilizes the Newton-Raphson method. 

In this report, three candidates for the iteration variable will be considered. They are: 

(i) 2

x , where 

2

E

x

∆ = , ∆E being the change in the eccentric anomaly; 

(ii) cθ v , which is the tangential component of vc

 ; 

(iii) p, known as the semi-latus rectum, where 

2

1 , c

h p h

μ

= =× r v

  , and μ = GM G, 

being the universal gravitational constant and M the mass of the earth. 

It will be shown that , x cθ v , and p are related by the equations 

2

2 cos cos

2

2 sin

1 2 1 2

2

1 2 2 2

1

R

R

c

r r r r x

r r

r v

p θ

θ

μ

θ

+ −

= = , 

where θ R is the range angle defined by 

1 2

1 2

1 2

arccos arccos( ), 0 ˆ ˆ R R

rr

θ θπ ⋅ = = ⋅ <<

r r

r r

 

. 

Using the above relations, we will derive the equations for an elliptical trajectory in terms 

of each of the above iteration variables. We will then deduce the equations for hyperbolic and 

parabolic trajectories, and demonstrate how the type of trajectory can be determined from the 

given parameters. Since we will be using results that are derived in Appendix A, it is advisable 

to read Appendix A before delving into the derivations that follow. Note that all the equations in 

this report are based on the hypothesis of an inverse-square gravitational field. 
NSWCDD/TR-03/81 

2 

TIME OF FLIGHT, CORRELATED VELOCITY, 

VELOCITY AT IMPACT 

It is shown in Appendix A that the time of flight for an elliptical trajectory is given by 

{ } ( )

3

1 2

21 2 1 sin sin 2 sin cos

2 2 fe

ab a E E E t E Ee E E E e

h μ

 ∆ + = − − − = ∆−    

{ } ( )

3

1 2 sin cos , 2

a E xe x xE x

μ

∆

= − += . (1) 

Now 

rr a e E a e E a e E E a e x xE 12 1 2 1 2 1 += − + − = − + = − + ( )( ) ( ) 1 cos 1 cos 2 cos cos 2 1 cos cos { } { ( )}

( ) 1 2

1 cos 1 sec

2

r r e xE x

a

 +

 + =−     . (2) 

( ) ( ) ( ) ( )

( ) ( )

{ } ( )

21 1 2 1 2

1 2 12

1 2 12

1 2 12

1 2

21 12

1

12 12

cos cos cos cos sin sin

2 2 2 2 22

1 1 11

cos cos sin sin

2 2 22

1 cos cos 1 sin sin

2 2 22

cos cos cos cos .

2 2

R

ae ae ae ae E E EE

r r rr

a E E EE

e e

rr

a EE EE a e xe xE

rr rr

θ θθ θ θ θ θ − == +

− − ++

= +

 = − ++    

 − +

= − = −+    

2 2

1 2 12 1 2 12

1 2 cos cos 2 sin csc 2 cos cos

22 2

R R r r rr x a x a x r r rr x

θ θ  ∴ +− =  = +−     . 

For the sake of brevity, put 2

2 cos cos 1 2 1 2

R S r r r r x

θ = + − , as this expression will be used 

frequently. Then 

a S x

2 csc

2

1 = . (3) 
NSWCDD/TR-03/81 

3 

Substituting Equations (2) and (3) into Equation (1), we obtain 

( )

3

3 2 1 2

1 2 12

2

2

1 2 12

csc tan 1 sin

2

2 cos cos csc

2

csc

2

sec cos 2 cos cos

2

fe

R

R

S rr

t xx x x

S

r r rr x x x

S

x

x r r x rr x

μ

θ

μ θ

   + = −−       	 



      + −   	 
 =  −+−      

( )( ) ( ) 2

1 2 12 csc csc cos 2 cos 1 cot

2 2

R S

x r r x x x rr x x

θ

μ

 = + −+ −     . (4) 

Differentiating Equation (4) with respect to , x we obtain 

() ( ) { }

{ } ( )

( ) { } ( )( )

2 2

1 2

22 2

1 2

2 2

12 1 2

3csc 1 cot 1

1 csc 2 cos 6csc 5 3 cot 2csc 1

2 2

cos 3 csc 4csc 3 12csc 1 cos

2

fe R

R

rr x x x

t

x rr x x x x

x S

rr r r x x x x x

θ

μ

θ

 

  + −−

∂

= + −− −

∂

+ + −− −  

. (5) 

The expressions (4) and (5) are analytic in the region 2 0 ≤ x < π. x is updated by the 

standard Newton-Raphson method as follows: 

22 2 ( ) { ( )}

1

2

2

n n

f fe n n f fe n

nn n

fe fe

xx xx

t tx x t tx

xx x

t t

x x

+

= =

− − =+ =+ ∂ ∂

    ∂ ∂

, 

where a good initial guess for 2

x , according to Reference 1, is 

{ ( )} 2 4 7 11

1 1.333 10 4.5 10 8.333 10 f ff xt t t −− − = × + × −⋅ × . 

The iteration continues until fe t is infinitesimally close to f t , at which point we obtain the 

correct value of x . Usually, four iterations are sufficient for convergence. 
NSWCDD/TR-03/81 

4 

Proposition: The correlated velocity is given by 

2 1

c

f

g

− = r r

v

   , 

where 

2 2 12

1 1

2 2 cos cos , cos 2 2

R R r r rrS f xg

r r

θ θ

μ

=− + = . 

Proof: Since 1r

 , 2r , and vc

 are coplanar vectors, each of them may be written as a linear 

combination of the other two. Let 2 1 c r rv = + f g    . Then 

2 21 1 12 1 , cc c r v rr rv rr rv × =×= × = ×= × = f fh g gh        . 

Let 

1 1

1 11 1 , 

0 0

x x

y y

    = =         

r r

   , 

with similar expressions for 2r

 and 2r



 . Then 

x11 1 1 11 1 1 = = −= = r a Eey r b E cos cos , sin sin θ θ ( ) , 

with similar expressions for 2 x and 2 y . Using the fact that br

h E = , we obtain 

1 1 1 11 1 1 1

1 1

sin sin , cos cos ah hx aE E E y bE E E br r =− =− = =     , 

with similar expressions for 2 x and 2 y. Hence, 

{ } ( )

{ } ( ) ( )

{ } ( )

2 1 21 12

1 2 12

1

1

21 1 21

1 1

2 2 2

2 1

1 1 11 1

cos cos sin sin

cos cos cos 1

2 1 1 cos 1 sin 1 2 cos cos , 2

R

xy xy a f E Ee E E h hr

a ar EE e E EE

r ra

a a Sr r EE x x

r r rr r

θ

× − = = = −+

 = − − = − −+    

= − − − = − = − =− +

r r

 NSWCDD/TR-03/81 

5 

{ } ( )( )

{ } ( )( ) { } ( )

1 2 12 21

1 22 1

3 3

21 2 1 1

3

1 2 12 12

cos sin cos sin

sin sin sin 2 sin cos cos

2 2 sin cos 2 sin cos cos .

2 22

R RR

x y x y ab g Ee E Ee E h hh

a a E E e E E x xe xE

a arr rr S rr

x x

a

μ μ

θ θθ

μ μμ

× − == = − −−

= −− − = − +

= ==

r r

 

The magnitude of the correlated velocity is given by 















 = − 













 = − r a r a

vc

2 1 2 1

1 1

μ μ , 

where a is given by Equation (3). 

Proposition: The velocity at impact is given by 

t c 1 v rv = + f g    , 

where 

1 1

12 2 2

cos 2 , 2 cos cos 2

R x rr f Sg x

rr r r

θ

=− =− + μ   . 

First proof: 

( ) ( ) 2 1 21 12

2 1

12 12 12

1 1 2 2 1 1

2 22 2

cos sin sin 2 2 ,

2 1 sin 1 2 cos cos .

2

t c

t R

x y x y ah x a f EE x S

h h h brr rr rr

a Sr r

g xx h h r rr r

μ μ

θ

× × − = = = =− − =− =−

× ×

= = = − = − =− +

v v r r

r v r r

   



   



Second proof: By considering the reverse trajectory, we have 

1 2

t

f

g

− ′ = − ′r r

v

   , 

where 

1 2

2

2 1 , cos 2

R S rrS f g g

r

θ

μ ′ ′ =− = = . 
NSWCDD/TR-03/81 

6 

Now 

( ) 1 1 21 1 2

1 , t cg fg g fg g f fg f f f f

g g g g ggg

 ′ =+ =+ − =− +  − =− =   ′ ′  

v rv r rr r r

        . 

The second equation yields immediately g = f ′. The first equation yields 

( ) 1 2

1 2 12 1 2

1 2

1 11 2 1 1 1 cos cos

2

cos 2 .

R fg ff S S S S f Srr x

g g g r r grr g rr

x S

rr

θ

μ

− −′       = = = − − − = − − =−    	 	    
 
 
 

= −

 

Third proof: The expressions for f

 and g may also be obtained from 

2 2

2 2

, f g f Eg E E E

∂ ∂ = = ∂ ∂

    , 

where 

{ } ( )( )

3

2 2 1

21 2 1 2

1 2

2 1 1 sin , sin sin sin , 2a EE a f g EE e E E E

r ra

μ

μ

− =− = − − − =  . 

The magnitude of the velocity at impact is given by 















 = − 













 = − r a r a

vt

2 1 2 1

2 2

μ μ . 

NULL-MISS DIRECTION 

The null-miss direction is the direction along which a velocity error, vc

 ∆ , at release can 

exist and the reentry body will still hit a target on a rotating earth. It is assumed that the 

atmosphere is moving with the earth and no winds exist. The null-miss direction is defined by 

the unit vector PDK

ˆ , where 

ˆ , 1

c

f c

DK f

c c

f

t

t

t

∆ − ∆ ∆ = = ∆ =−

∆ ∆

∆

v

v P

v v





  . 
NSWCDD/TR-03/81 

7 

Since ( ) fe fe R t = t x,θ , we have 

R

R

fe fe

f

t

x

x

t

t θ

θ

∆

∂

∂

∆ +

∂

∂

∆ = , 

where 

1 2

1 2

1 2

ˆ ˆˆ ˆ ˆ , csc Rf R θ θ t

× ∆ = ⋅ ∆ =− ⋅ = = ×

×

r r Ω n Ω nn r r

r r

      , 

and Ω



 is the vector representing the earth rotation around itself. 

x

t

t

x

t

t

t

x

fe

R

R

fe

fe

R

R

fe

f

∂

∂

∆

∂

∂

+

= −

∂

∂

∆

∂

∂

∆ −

∴ ∆ =

θ

θ

θ

θ

1

, 

where 

x

t fe

∂

∂

 is given by Equation (5), and 

() ( ) { }

( )

2

1 2

1 2

1 2

3csc 1 cot 1

1 sin

22 2 6 cos csc cot 1 cot

2

fe R

R

R

rr x x x

t rr

S rr x x x x

θ

θ θ μ

 + −− ∂   = −   ∂ − −    

. (6) 

Similarly, by taking differentials of both sides of the equation 2 1

c

f

g

− = r r

v

   , we obtain 

c c g

g

g

f

g

r v

r

v     ∆ − ∆ − ∆ ∆ = 1

2 , 

where 

22 2 , , f RR

R R

f f gg t fx gx

x x

θ θ

θ θ

∂∂ ∂∂ ∆ = × ∆ =− × ∆ = ∆ + ∆ ∆ = ∆ + ∆

∂∂ ∂∂

r Ω r Ω r     . 

Now 

2 2

1 1

2 1 2

12 12 1 2

2 sin cos , cos sin , 2 2

2 sin cos , sin 3 cos cos . 2 22 2

R R

R

R RR

R

f r fr x x

xr r

g g rr rr x rr x r r

xS S

θ θ

θ

θ θθ

μ θμ

∂ ∂ =− =− ∂ ∂

∂ ∂  = = −−   ∂ ∂  
NSWCDD/TR-03/81 

8 

Hence, 

2

1

2 1 2

12 12 1 2

2 sin cos cos sin , 2 22

2 sin cos sin 3 cos cos

222 2

R RR

RR R

R

r f x xx

r

rr

g rr x x rr x r r S S

θ θθ

θθ θ θ

μ μ

 ∆ ∆ =− ∆ +    

 ∆ = ∆+ − − ∆    

2 1 2

12 12 1 2

1 2

1 2

12 1 2

1 2

2 sin cos sin 3 cos cos

222 2

2

cos

2

1 sin cos tan 3 cos cos

222 2

sin cos cos sin tan .

2 22 22

RR R

R

R

RR R

R

R RR RR

rr

rr x x rr x r r

g S S

g rrS

r r

x x rr x r r

S S

r r

x xx

S

θθ θ θ

μ μ

θ

μ

θθ θ θ

θ θθ θθ

 ∆+ − − ∆   ∆    =

 = ∆+ − − ∆    

 ∆ ∆ = ∆+ −    

Putting 2 2

cos sin

2

sin cos R R R z x x x

θ θ ∆θ = ∆ + , we obtain 

2 2 1 2

1

1

2

tan

2 2

R R

c c

z r rr

z

g gr S

× ∆  θ θ ∆ =− + + −  

 

Ω r

vr v

     . (7) 

Equation (7) may be written in terms of 1 1 ˆ rˆ , θ , and nˆ which form a right-handed orthogonal 

coordinate system. We have 

( ) ( ) ( ) 11 1 1 ˆ ˆ ˆˆ ˆˆ ∆ =∆ ⋅ +∆ ⋅ +∆ ⋅ cc c c v v rr v θ θ v nn    . 

Now 

21 2 1 2 1 2 2 2

2 1

21 2 1 2 2 1 2 2 2

ˆ ˆ ˆˆ ˆ sin sin ,

ˆ ˆ cos ˆ ˆ ˆˆ ˆ cot cos cos .

sin

R RR

R

R R RR

R

rrr

r rr

θ θθ

θ θ θ θθ

θ

× ⋅ = ⋅ × =− ⋅ × =− ⋅ = ∆

− × ⋅ = ⋅ × = ⋅ × = ⋅ × = ⋅ =− ∆

Ω r r Ω r r Ω r r Ω n

r r Ω r θ Ω r θ Ω r Ω r r Ω n

    

    

( ) ( ) ( )

( )

2 211 2 1 1 2

2 12 1 2

ˆ ˆ ˆˆ ˆˆ

ˆ sin cos . ˆ ˆˆ RR RR r r θθ θθ

∴ ×= ×⋅ + ×⋅ + ×⋅

= ∆ − ∆ + ×⋅

Ω r Ω rrr Ω r θθ Ω r nn

r θ Ω r nn

     

 
NSWCDD/TR-03/81 

9 

2 1 1 11 ( ) ( ) 2 1 2 11 2 1

2 2 12 1 2 1

2

2 12 1 2 1

1 2

2

1

ˆ ˆ ˆ cos sin ˆ cos sin ˆ

ˆ cos 2 cos cos sin ˆ 2

ˆ 2 cos 2 cos cos 2 sin cos ˆ 2 2 22

2

cos

2

2

cos c

2

R R R R

c

R

R R

R R RR

R

R

f r fr r fr r

gg g

r r rr x r

g

r rr x r

rrS

r

S r

θ θ θ θ

θ θ θ

θ θ θθ

θ

μ

μ θ

− + − − + == =

   +− +   =

   − +   =

= −

r r r θ r r θ

v

r θ

r θ

  

2

1 1

1

2 ˆ os sin . ˆ 2

R r

x

r S

 μ θ   +

 

r θ

Substituting the expressions for 2 Ω r

 

× and vc

 into Equation (7), we obtain 

{ } 2 1 2 1 2 12 1 ( )

1 2 2 2

1 1

1 1

2

3 1 2 12

2

3 1

1

1 2 ˆ sin cos ˆ ˆˆ ˆ

2 2 ˆ tan cos cos sin ˆ 22 2 2

2 sin sin cos cos

2 2

sin s

2 2

c RR RR

RR R R

R R

R

z

r r rr

g g

rr r r

z x

S S r rS

x r r rr x x

S

r

r

r S

θθ θθ

θθ μ θ μ θ

μθ θ

μ θ

∆ =− ∆ − ∆ + × ⋅ +

  ∆     

+ − −+    	   
     
 
 

    +− ∆ 
  =

−

v r θ Ω r nn r

r θ

  

( )

( )

1

2

2 12

2 2

3 3 2 1 2 12 1

1

2

ˆ

in cos cos

2

ˆ sin sin cos cos 1 cos

2 22 2

1 ˆ ˆ.

R

R

R R

R R

x r rr x

r

r x x r r rr x

S rS

g

θ θ

μ μθ θ θ θ

 

 

    +− ∆ 
   

     +− ∆+ + − + ∆    	    
 
   

− ×⋅

r

θ

Ω r nn  

 (8) 

REENTRY FLIGHT PATH ANGLE, ANGLE OF 

ATTACK, AND YA 

These three angles are computed at reentry, i.e., at 400,000 ft. Let 2r′

 be the position vector 

corresponding to that altitude. Then 
NSWCDD/TR-03/81 

10 

( ) { } ( )

{ } ()( )

( )

2

2 21 2 1

2

2 12 1

1

2

1 1

12 2

1 1 2 arccos , , 2 sin sin , 2

2 1 sin , sin 2 sin sin , ,

2 sin 2 , 1 sin , , arccos ˆ ˆ

fe

c

t cR

r

ab a E x E E t xe E E

e h

a ab f xg x e E E f g r h

ah a f xg x f g brr r

π

θ

′ −

′ ′′ ′ ′ ′ =− = − = − −

′ ′′ ′ ′ ′ ′ ′ =− = − − = +

′ ′ ′ ′′ ′ ′ ′ ′ =− = − = + = ⋅ ′ ′

rrv

v r v rr

  

   ( ) 2 .

The velocity at reentry takes into account the earth rotation. It is given by 

2 v′ = v′ − Ω × r′

    r t . 

The reentry flight path angle is the angle between vr

′

 and the plane normal to the 

instantaneous local geodetic vertical, hˆ (Reference 2). It is given by 

( ) r r = − h ⋅ vˆ′ ˆ arccos

2

π

γ . 

The null-miss angle of attack is the angle between PDK

ˆ at release and vr

′

 . It is given by 

( ) ˆ arccos ˆ α DK r = ⋅ P v′ . 

If vr

 is the analog of vr

′

 at impact, then 

2 v v Ω r

    r = t − ×

and the zero-angle of attack is given by 

( ) vr vr = arccos ˆ ⋅ ˆ′ α0 . 

Let ( ) ˆ , , ˆ ˆ r n uv ′ ′ be a right-handed orthogonal reference frame, where 

r

r

r v

r v

n ′ × ′

′ × ′ ′ =  

 

2

2 ˆ ; 

then ˆ ˆ ˆ . r uv n = ×′ ′ Let YA be the angle between uˆ and the projection of PDK

ˆ on the nˆ′uˆ plane, to 

be measured clockwise from uˆ in the nˆ′uˆ plane. It can easily be shown that 

P u

P n

ˆ ˆ

ˆ ˆ arctan ⋅

⋅ ′ =

DK

DK YA . 
NSWCDD/TR-03/81 

11 

TIME OF FLIGHT IN TERMS OF cθ v

From the expression for vc

 on page 9, we have 

2

1

2 sin

2

R

c

r

v

r S θ

μ θ =

2

1

2

2

1 2 1 2

2

2 sin

2

2 cos cos

θ

θ

μ θ

c

R

R

r v

r

 S = r + r − r r x = (9) 

( ) 2 2

11 2 2

2

1 12

2 sin

2 cos

2 cos

2

R

c

R

c

rr rv r

x

r rr v

θ

θ

θ

μ

θ

+ −

 = (10) 

( )

( ) ( )

2

34 2 2 2

12 1 1 2 2

2

2

1 12

24 2 2 2 2 22 4

1 1 2 12 12 1 2 2

2

1 12

2 2

2 1 1 2 12

4 cos 2 sin

2 2 sin 1 cos

2 cos

2

2 cos 4 sin 4 sin

2 2

2 cos

2

2 sin 2 cos

2 2

R R

c c

R

c

R R

c Rc

R

c

R R

c

r rv r r r v r

x x

r rr v

r v r r rr rr r r v r

r rr v

r rv r r rr

θ θ

θ

θ θ

θ

θ

θ θ

μ

θ

θ θ θμ μ

θ

θ θ

μ

 −+−      =− =

− +− + + −

=

 	 − +− 


 =

2 2

1 1 2 12 2

2

1 12

2 cos 2 sin

2 2 .

2 cos

2

R R

c

R

c

rv r r rr r

r rr v

θ

θ

θ θ

μ

θ

   	    
  ++ −

   
 


Put 

22 2 2

2 1 1 2 12 1 1 2 12 2 2 sin 2 cos 2 cos 2 sin

2 2 22

R R RR F r rv r r rr rv r r rr r c c θ θ

θ θ θθ

μ μ

     = − +− ++ −          	 
	 
 . 

Then 

2

1 12

sin

2 cos

2

R

c

F

x

r rr v θ

θ = , (11) 
NSWCDD/TR-03/81 

12 

( ) ( ) 22 22

11 2 2 11 2 2

tan arctan

2 sin 2 sin

2 2

R R

c c

F F

x x

rr rv r rr rv r θ θ

θ θ

μ μ

=  =

+− +−

. (12) 

Substituting Equations (9), (10), (11), and (12) into Equation (4), we obtain 

( ) ( )

( )

2 22 2

2 1 2 1 1 2 12

1 2 222 2

1 2

2 2

11 2 2

2 sin 2 cos

2 sin

2 sin

arctan

2 sin

2

R

c R

c R

fe c R

R

c

r r r rv r r rr

rrv

t rrv F F

F rr rv r

θ

θ

θ

θ

θ μ θ

θ

μ θ

θ

μ

 + − +−  

=  

  +

+ −  

. (13) 

Differentiating Equation (13) with respect to c v θ , we obtain 

( )

( )

24 2 2 22 4

1 1 2 12 2

332 3

1 2

2 2

11 2 2

2342 3 2

1 2

2

1 2 cos 4 sin .

2

4 sin

arctan

2 sin

2

16 sin sin

2 .

fe R

c R

c

fe c R

c R

c

R

c R

t

r v r r rr r

v F

t rrv F

v F F rr rv r

rrv

F

θ

θ

θ

θ

θ

θ

θ θ μ

μ θ

θ

μ

θ μ θ

∂  = +− −   ∂  

 

    +

  + −  

−

 (14) 

Note that 

2 2

2 2

2

1

1 1 2 12

2 2

2 2

2

1 1 2 12 1 1 2 12

2 sin 2 sin

2 2

2 cos cos

2

2 sin 2 sin

2 2 .

2 cos 2 cos

2 2

R R

c

R

R R

c

R R

r r

v

r S

r r r rr x

r r

v

r r r rr r r r rr

θ

θ

θ θ

μ μ

θ

θ θ

μ μ

θ θ

= =    + −  

 < <     ++ +−   

Observing the expression for F, we see immediately that the above inequality implies that F is 

positive for elliptical trajectories. 
NSWCDD/TR-03/81 

13 

c v θ is updated by the Newton-Raphson method, where a good initial estimate, according to 

Reference 3, is 

2

2

2

1 1 2 12

2 sin sin 2

2 cos

2

R

R

c

R f

r

r

v

t

r r r rr

θ

θ

μ θ

θ = +    + +  

which, by the above inequality, holds provided that 

















+ − + + +

+ − >

2

2 cos

2

2 cos

2

2 cos

2

1

1 2 1 2 1 2 1 2

1 2

2

2

2

1 R R R

f r r r r r r r r

r r r r

t

θ θ

μ

θ . 

cr v , the radial component of vc

 , may be expressed in terms of cθ v . From the expression 

for vc

 on page 9, we have 

2

1

2

cos cos

2

R

cr

r

v x

S r

μ θ  = −  

 

. 

Substituting Equations (9) and (10) into the above expression, we obtain 

2

cot csc tan

2 1

1 R

c

cr R R c r v

v

r

r

v μ θ θ θ

θ

θ + 













 = − . 

Hence, 1 1 ˆ ˆ c cr c v v = + θ v r θ  , where θ R θ R ˆ ˆ ˆ csc ˆ cot ˆ

1 1 2 1 θ = n × r = r − r . By considering the 

reverse trajectory and using the fact that cθ tθ h r v r v = 1 = 2 , we obtain for the velocity at impact 

2 2 ˆ ˆ t tr t v v = + θ v r θ  , 

where 

( )

1

2

2 1

1 2 21

22 1 1 1 1 2 1

,

csc cot tan csc cot tan , 2 2

ˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ cos sin cos sin cot csc .

t c

R R

tr R R t R R c

t c

R R RR R R

r

v v

r

r r

vv v

r rv r rv

θ θ

θ θ

θ θ

μ θ μ θ θθ θ θ

θ θ θθ θθ

=

= − − =− −    

θ =× =× + = − = − nr n r θ θ rrr
NSWCDD/TR-03/81 

14 

TIME OF FLIGHT IN TERMS OF p

We have 

1 1

c

h p v

r r

θ

μ = = . 

Substituting this value of cθ v into Equations (13) and (14), we obtain 

( ) ( )

( )

2 22

12 1 2 1 2 12

1 2 22 2

1 2

2

1 2 12

2 sin 2 cos

2 sin

2 sin

arctan

2 sin

2

R

R

R

fe R

R

rr r r p r r rr

p rr t pr r F F

F r r p rr

θ θ

θ

θ μ θ

 + − +−  

=   ′ ′   + ′ + −  

, (15) 

( )

( )

1

2 2 2 22 4

1 2 12 1 2

33 3

1 2

2

1 2 12

44 3 2

1 2

2

1

2

1 2 cos 4 sin .

2

2 sin

arctan

2 2 sin

2

8 sin sin

2 ,

fe fe fe c

c c

R

R

fe R

R

R

R

tt t v

p v p r pv

p r r rr r r F

t p rr F

p F F r r p rr

r r

p

F

θ

θ θ

μ

θ θ

θ

μ θ

θ θ

μ

∂∂ ∂ ∂ = = ∂ ∂∂ ∂

 = +− −   ′  

 

  ′   + ′ ′   + −  

− ′

 (16) 

where 

2 2

12 1 2 12 1 2 12 12

2

1

2

2 sin 2 cos 2 cos 2 sin

2 2 22

.

R R RR F rr p r r rr p r r rr rr

r F

θ θ θθ

μ

     ′ = − +− ++ −          	 
	 


=

p is updated by the Newton-Raphson method, and a good initial estimate is 

μ

θ

2 2

1 c r v

p = , 

where cθ v is initialized as on page 13.
NSWCDD/TR-03/81 

15 

EQUATIONS FOR A HYPERBOLIC TRAJECTORY 

From Appendix A, we have 

2 cos 1 sin cos , sin 1 cos 1 cos

e e E E

e e

θ θ

θ θ

+ − = = + + . 

From the expression for , sin E we observe that since e > 1 for a hyperbola, E is imaginary in 

this case, i.e., 2 E is negative. Since 2

E

x

∆ = , it follows that x is imaginary, and hence 2

x is 

negative. Also, ( )

2

2 0.

1

h

a

μ e

= < −

Let 2 2 H =− =− Ey x , ; then E == = = iH x iy E i H x y , , sin sinh , cos cosh ,

2 2 sin sinh , tan tanh , csc csch , cot coth , csc csch x = = = = =− i y x i yx x y yx x y y x y ; also, 

arctan ( ) arctanh 1 1 ln

2 1

iu u u

iu u u u

+ = = − for u <1. Using the fact that 





 



 + −

>





 



 + −

=

2

2 cos

2

2 sin

2

2 cosh cos

2

2 sin

1 1 2 1 2

2

2

1 1 2 1 2

2

2

2

R

R

R

R

c

r r r r r

r

r r r r r y

r

v

θ

θ

μ

θ

θ

μ

θ , 

we see immediately that F and F′ are negative. Putting 

( ) ( ) 22 2

1 1 2 2 1 2 12 2 sin 2 sin

2 2

R R

c

F F

u

r r r v r r r p rr θ

θ θ

μ

− − ′ = =

+ − +−

and noting that ( ) 2 2

11 2 2 2 sin 0, 2

R

c rr rv r θ

θ

+− > μ it is easy to verify that u <1. We have 

( )

( ) ( )

( )

2

2 2 34 2

1 1 2 2 12

22 22

11 2 2 11 2 2

2 2

11 2 2

2 sin 4 cos 0

2 2

2 sin 2 sin 0

2 2

2 sin 1.

2

R R

c c

R R

c c

R

c

F r r r v r r rv

F rr rv r F rr rv r

F rr rv r u

θ θ

θ θ

θ

θ θ

μ

θ θ

μ μ

θ

μ

 − − + − =− <    

      −+ + − −− + + <   

 −< + −  <
NSWCDD/TR-03/81 

16 

Putting 2

2 cosh cos 1 2 1 2

R S r r r r y

θ ′ = + − , we obtain the following expressions for the 

hyperbolic time of flight: 

{ } ( )

( )( ) ( )

( ) ( )

( )

21 2 1

2

1 2 12

222 2

1 2 2 22 2 1 2

2 1 2 1 1 2 12

1 2

12 1 2

sinh sinh

csch cosh csch 2 cos coth 1

2 2

sin sin 1 2 sin 2 cos ln

2 1

sin 2 sin

fh

R

c R R c R

c R

R

a t a H He H H

S

y r r y y y rr y y

rrv r r v u

r r r rv r r rr

F F u

p r r

rr r r

F

θ θ

θ

μ

θ

μ

θ μθ θ μ θ

θ

μ

− = −− −

′  = + −+ −    

 +

= + − +− +     − −

= + ′ ( )

22 2

2 22 1 2

1 2 12

sin 1 2 cos ln .

2 1

R R

R

pr r u p r r rr

F u

θ θ θ  +   − +− +

  − ′ −

Also, 

() ( ) { }

{ } ( )

( ) { } ( ) ()

2 2

1 2

2 22

1 2

2 2

12 1 2

3csch coth 1 1

1 csch 2 cos 3 coth 2csch 1 6csch 5

2 2

cos 12csch 1 cosh 3 csch 4csch 3

2

fh R

R

r r yy y

t

y rr y y y y

y S

rr r r y y y y y

θ

μ

θ

 

  + −−

∂

= + +− −

∂ ′

++ + − +  

, 

( )

332 3

24 2 2 22 4 1 2

1 1 2 12 2

2342 3 2

1 2

2

1 1 2 sin 2 cos 4 sin ln

2 1

16 sin sin

2 ,

fh R fh c R

c R

c c

R

c R

t t rrv u

r v r r rr r

vF v u F F

rrv

F

θ

θ

θ θ

θ

θ μ θ θ μ

θ μ θ

∂   +

= +− − +    ∂ 	 
  − −

−

( )

33 3

2 2 2 22 4 1 2

1 2 12 1 2

44 3 2

1 2

2

1 1 sin 2 cos 4 sin ln

22 1

8 sin sin

2 ,

fh R R fh

R

R

R

t t p r r u

p r r rr r r

p F pu F F

r r

p

F

θ θ θ

μ

θ θ

μ

∂   +

= +− − +    ∂ − ′ 	 
  ′ ′ −

− ′

() ( ) { }

( )

2

1 2

1 2

1 2

3csch coth 1 1

1 sin

22 2 6 cos csch coth coth 1

2

fh R

R

R

r r yy y t rr

S rr y y y y

θ

θ θ μ

 + −− ∂   = −   ∂ ′ − −    

. 
NSWCDD/TR-03/81 

17 

2

, c y v θ , and p may be updated by the Newton-Raphson method as follows: 

{ ( )} () () {( ) }

( )

( ) 2 2

1 1 1

2 , , 

n c c n n

n f fh n f fh c n f fh n

nn c c nn n n fh fh fh

y y c v v p p

yt t y tt v ttp

yy v v pp t t t

y p v θ θ

θ

θ θ

θ

+ + +

= = =

− − − =+ = + =+  ∂ ∂ ∂

      ∂ ∂ ∂ 

, 

where the usual initial guesses may be used. 

The correlated velocity and velocity at impact are given by 

2 1

1 , hc t h hc

h

f f g

g

− = =+

r r

v vrv

     , 

where 

2 2 12 1 1

1 1 12 2 2

2 cosh 2 cosh cos , cos , 2 , 2 cosh cos . 22 2

R R R

h hh h

r r rr S r r y f y g f Sg y

r r rr r r

θθ θ

μ

μ

′ =− + = =− =− + ′  

Alternatively, 11 22 ˆ ˆ ˆ ˆ , c cr c t tr t vv vv =+ =+ θ θ v r θ v r θ   , where , , , and cr c tr t vvv v θ θ are given on 

page 13. Also, 

11 2 2

21 2 1 21 2 1 , c t v v

ra r a ra r a

μμ μμ

  

= −= + = −= +            . 

The null-miss vector is given by 

2 2 1 2

1

1

2

3 1 2 12

1

2 2

3 1 2 12

1

3 2

2

tan

2 2

2 sinh sin cosh cos

2 2 ˆ

sin sinh cosh cos

22 2

 sinh sin

2

R R

c c

h h

R R

R R

R

R

z r rr

z

g gr S

y r r rr y y S

r

r y r rr y r S

r

ry y S

θ θ

μθ θ

μθ θ θ

μ μ θ

× ∆ ′  

∆ =− + + −  ′ ′  

   	 
   +− ∆ ′   = −

 + − +− ∆   ′    

+ ∆+ ′

Ω r

vr v

r

    

( )

( )

2 2

3 1 2 12 1

1

2

ˆ cos cosh 1 cos

22 2

1 , ˆ ˆ

R R

R

h

r r rr y r S

g

θ θ θ  
    	   +− + ∆   
 ′      

− ×⋅

θ

Ω r nn  

where 

1

. , , sinh cos cosh sin ˆ 2 22

fh

R

R R RR

R

fh

t

y z y yy t

y

θ

θ θ θθ θ

∂

+ ∆

∂ ∆ ∆ =− ∆ =− =− ∆ + ′ ∂

∂

Ω n  . 
NSWCDD/TR-03/81 

18 

EQUATIONS FOR A PARABOLIC TRAJECTORY 

Since the time of flight is a well-behaved function of 2x , it follows from Equation (4) that 

the time of flight for a parabolic trajectory is given by 

( )( ) ( )

( )

2

1 2 12 0 0

2 4 24

0 2

2 1 2 12 0

lim lim csc csc cos 2 cos 1 cot

2 2

11 2 lim ... ... 2 cos ...

2 3 15 3 45 2 3 45

R

fp fe x x

R

x

S

t t x r r x x x rr x x

S x x xx

r r x rr

x

θ

μ

θ

μ

→ →

→

   = = + −+ −       	 


 
 
  
      = ++ + + − + + + +         

         	 


0

1 2 12 0 1 2 12

1 2 cos , where 2 cos 32 2

R R S r r rr S r r rr θ θ

μ

 = ++ =+−     . (17) 

Also, from Equations (5) and (6), 

( ) ( )

( ) ( )

2 2

2 1 2 12 12 1 2 0

0

2 2 0 2

1 2 12 12 1 2 0 0

2

2

2

0 0

2 1 2

1 lim cos cos ,

5 2 2 2

2 lim lim cos cos , 5 sin 2 2

1 lim lim

5 sin sin

fe R R

x

fe

fe R R

x x c c R

fe

fe

x x R

R

t

r r rr rr r r

x S

t

t S x r r rr rr r r

v r v

x

t

t x

p p rr

x

θ θ

θ θ

μ

θ θ

μ θ

θ θ

→

→ →

→ →

∂  = +− + +   ∂  

∂

∂ ∂  = =− + − + +   ∂ ∂  

∂

∂

∂ ∂ = =− ∂ ∂

∂

( ) ( )

3

2 0 2

1 2 12 12 1 2

1 2

1 2

0

0

cos cos , 2 22

2

sin lim .

2 2

R R

fe R

x R

S

r r rr rr r r

rr

t rr

S

θ θ

μ

θ

→ θ μ

   +− + +  

∂ = ∂

The correlated velocity and velocity at impact are given by 

2 01

01 0

0

, c tc

f f g

g

− = =+

r r

v vrv

     , 

where 

0 12 0 0 0

00 0 0

1 12 2

2 2

1 , cos , , 1 2

R S rrS S S

fg f g

r rr r

θ μ

μ = − = =− = −   . 

Since ( )

2

2 1

h

a

μ e = − and e = 1 for a parabola, it follows that a = ∞ for a parabola. Hence, 

1 1

2 1 2 lim

r a r

v a c

μ μ = 













 = − →∞

, which is known as the escape velocity. Similarly, 

2

2

r

vt

μ = . 
NSWCDD/TR-03/81 

19 

The null-miss vector is given by 

( )

2 02 1 2

1 0

0 01 0

2 2 2

3 3 1 2 12 2 12 1

0 10

2 2 2

3 3 2 1 2 12

0 10

2

tan

2 2

sin cos sin cos ˆ 2 2 2 22 2

1 sin cos 1 cos

22 2 2 2

R R

c c

RR R R

R

R R

R

z r rr

z

g gr S

r

r r rr x r rr

S rS

r

r x r r rr

S rS

θ θ

μ θ θ μθ θ θ

μ μθ θ θ

× ∆  

∆ =− + + −  

 

   = + − ∆− − ∆ 	 
        

 +− ∆ + + − + 

Ω r

vr v

r

    

( )

1

2

0

ˆ

1 , ˆ ˆ

R

g

θ   
 	 
     ∆

   

− ×⋅

θ

Ω r nn  

where 

0 2 2

0

2 0

1 lim

1 , cos sin 22 2 lim

fe

R x R R R

R

fe

x

t

x zx t

x

θ

θ θ θ θ →

→

∂

+ ∆

∂  ∆ =− = ∆ + ∆   ∂  

∂

. 

The equation for the time of flight for a parabolic trajectory may be derived directly from 

the equation of a parabola in polar coordinates, namely, 2

sec2

0

θ

r = r , obtained by setting e = 1

in the conic equation ( )

1 cosθ

1 0

e

r e

r

+

+ = (Figure 1). 

Figure 1. Parabola whose focus is at the origin and 

whose vertex is at ( ,0) 0

r

x

y

( ) r0 ,0

v0



( ) 0,0
NSWCDD/TR-03/81 

20 

We have 

2 2 2 2

1 1 1 1

2 2 2

1 1 1

4

2 2

2 22 0 0

2 2

0 0 2 22 3

2

0 21 2 1 3 3

1 sec sec 1 tan

2 22

2 sec sec tan 2 tan tan

2 2 2 23 2

2 1

tan tan tan tan

2 23 2 2

fp

d r r t rd d d

hh h

r r d d

h h

r

h

θ θ θ θ

θ θ θ θ

θ θ θ

θ θ θ

θ θ θθ θθ θ

θ

θ θθ θ θ θ θ

θθ θ θ

 == = = +    

  = + =+        

   = −+ − 	 
      

  

 

 

1 2 , 0 . ≤<< θθ π

Now 

2 1

2 1 1 2 21

1 2

12 2 1 12

21 2 1 3 3

2 1

tan tan

2 2 tan tan sec sec sin , tan tan 2 2 222 2 2 1 tan tan

2 2

1 tan tan tan tan cot sec sec cos

22 2 2 2 22 2

1

tan tan tan tan

2 23 2 2

1 tan tan 1

223

R R

R R

θ θ

θ θ θ θ θ θ θθ

θ θ

θθ θ θ θ θθ θ

θθ θ θ

θ θ

− − −= = =

+

  + =− =    

  −+ −    

 =− +    

2 2 1 2 12

12 12 1 2 2 2

12 1 2 12 2 2

1 2

2 1 2 12

0

tan tan tan tan

2 2 22

1 sec sec sin 1 tan tan sec sec

32 2 2 2 2 2 2

1 sec sec sin sec sec sec sec cos

32 2 2 2 2 2 2 2

sin cos

32 2

R

R R

R R

fp

rr

r r rr t

r

θ θ θθ

θθθ θθ θ θ

θθθ θ θ θθ θ

θ θ

 	  
    + +

 
  

 = + ++    

 = ++    

 = ++    =  

1 2

1 2 12

2

sin cos .

32 2

R R rr

r r rr

h

θ θ    + +  

Also, 

2 2 1 2 1 2 21

0 1 2 12 0

22 2 2 1 2 1 2 1 2 21

0

2

1 2

222 222 1 2 21 1 2

0 0

0

2 cos sec sec 2sec sec cos

2 2 2 22 2

sec sec cos cos 2cos cos cos

2 2 2 2 22 2

sin

2 sec sec sin sec sec sin .

22 2 222

R

R

R

S r r rr r

r

rr

r r

r

θ θ θ θ θ θθ

θ θ θ θ θ θ θθ

θ

θ θ θθ θ θ θ

 − =+− = + −    

 − = +−    

− = ==
NSWCDD/TR-03/81 

21 

12 0

0 0 0 0 1 2 12

0 0

2 2 12 2 sin cos

23 2

R R

fp

rr S h rv r r t r r rr

r S

μ μ θ θ

μ

μ

 ∴= = = =  = ++    

which is Equation (17). 

DETERMINATION OF TYPE OF TRAJECTORY 

Proposition: If 1 2 r and r   are given, then fh fp fe t < t < t . 

Proof: First consider Equation (4), namely, 

( )( ) ( )









 = x r + r x x − x + r r − x x

S

t R

fe 1 cot

2

csc csc cos 2 cos

2 1 2 1 2

2 θ

μ . 

It is clear that 

1 2 12 1 2 12 1 2 12 2 cos cos 2 cos 2 cosh cos

22 2

RR R r r rr x r r rr r r rr y

θθ θ

+− >+− >+−

0  SS S > > ′. (18) 

Now 

( ) 3

1 ... 3 45 ... 3 151 1

csc 1 cot

2 2 4

2

2 >















+ + 













 − = + + +

x x x

x

x x x (19) 

{ } ( ) { } ( ) 2 2 csc csc cos csc 3csc 1 cot 1 0, 0 d

xx x x x x x x x

dx

 − = − −> << π . 

Hence, csc x( ) x csc x cos x 2 − increases monotonically in the interval [0 . Since ,π )

{ } ( ) 2

0

2 lim csc csc cos

x 3

xx x x →− = , it follows that 

( ) 2 2 csc csc cos , 0 3

xx x x x − > << π . (20) 

Observing Equations (18), (19), and (20) in conjunction with Equation (4), we conclude that 

fe fp t > t . Now consider the expression for fh t derived on page 15, namely, 

( )( ) ( ) 2

1 2 12 csch cosh csch 2 cos coth 1

2 2

R

fh

S

t y r r y y y rr y y

θ

μ

′  = + −+ −     . (21) 
NSWCDD/TR-03/81 

22 

We have 

{ } ( ) { } ( )

( )

{ } ( ) { } ( ) ( )

{ } () ()

( ) ( )

2 2

42 2

4

4

3 2

4

csch coth 1 csch 3coth coth 1

csch sinh 3 cosh 3cosh sinh

1 csch cosh 2 1 3 cosh 2 1 3sinh 2

2

1 csch 3sinh 2 2 cosh 2 4

2

1 2 2

csch 3 2 ... 2 1 ...

2 3! 2!

d

yy y yy yy y dy

yy y y y y y

yy y y y y

y yy yy

y y

yy y

−= − −

= −+

= −− ++   

= −−

      = + +− + +  	 	

   
 
 

( ) ( ) ( )

( ) ( )

2 1 4

2

2 3 4

1

4

1 31 csch 2

2 2 1! 2 !

csch 2 0.

2 3!

n

n

n

n

y

y y

n n

n

y y

n

∞ +

=

∞ +

=

 

 
 −

 

    = −  	   + 
 

=− <

+





Hence, csch ( ) coth 1 2 y y y − decreases monotonically in the interval [0 . Since ,∞)

{ } ( ) 2

0

1 lim csch coth 1

y 3 yy y →

− = , it follows that 

( ) 2 1 csch coth 1 , 0. 3 yy y y −< > (22) 

{ } ( ) { } ( ) 2 2 csch cosh csch csch 3csch coth 1 1 0. d

y y y y y yy y dy∴ − = −−<

Hence, ( ) 2 csch cosh csch y yy y − decreases monotonically in the interval [0 . Since ,∞)

{ } ( ) 2

0

2 lim csch cosh csch 

y 3 y yy y →

− = , it follows that 

( ) 2 2 csch cosh csch , 0. 3 y yy y y − <> (23) 

Observing Equations (18), (22), and (23) in conjunction with Equation (21), we conclude that 

fh fp t < t . Hence, 

fh fp fe t < t < t . 

Therefore, given 1 2 , , and f r r t   , one can immediately determine the type of trajectory by 

computing fp t from Equation (17) and comparing it with f t . 
NSWCDD/TR-03/81 

23 

REFERENCES 

 1. Hand, George L., The Guidance Problem – Solution for All Types of Trajectories with 

Oblate Gravity (Addendum), CSDL-C-5560, Charles Stark Draper Laboratory, 

Cambridge, MA, Mar 1984. 

 2. Owen, Davis L., Geometry of the Ellipsoid, K40 Training Guide 4084.3, Naval 

Surface Weapons Center,*

 Dahlgren, VA, Dec 1984. 

 3. Davis, W. L., An Efficient Algorithm for Computing the Q-Guidance Matrix, 

NSWC/DL TR-3844, Naval Surface Weapons Center,∗

 Dahlgren, VA, Jan 1979. 

 

∗

 Currently Naval Surface Warfare Center, Dahlgren Division 
NSWCDD/TR-03/81 

24 

BIBLIOGRAPHY 

 Bate, Roger R.; Mueller, Donald D.; and White, Jerry E., Fundamentals of Astrodynamics, 

Dover Publications, Inc., New York, NY, 1971. 

 Cohen, C. J. and Lyddane, R. H., The Q-Guidance Matrix and Its Symmetry, NPG-TM-K-59/006, 

Naval Proving Ground,*

 Dahlgren, VA, Jun 1959. 

 Davis, W. L., An Efficient Algorithm for Computing the Q-Guidance Matrix, NSWC/DL TR-3844, 

Naval Surface Weapons Center,∗

 Dahlgren, VA, Jan 1979. 

 Davis, W. L., Computation of * DK

d and

dt R

∂

∂

P P

  , Naval Surface Weapons Center,* Strategic 

Systems Department, Dahlgren, VA (unpublished). 

 Fowles, Grant R., Analytical Mechanics, Third Edition, Holt, Rinehart and Winston, NY, 

1977. 

 Hand, George L., The Guidance Problem – Solution for All Types of Trajectories with 

Oblate Gravity, and Addendum, CSDL-C-5560, Charles Stark Draper Laboratory, 

Cambridge, MA, Oct 1982 and Mar 1984. 

 Owen, Davis L., Geometry of the Ellipsoid, K40 Training Guide 4084.3, Naval Surface 

Weapons Center,*

 Dahlgren, VA, Dec 1984. 

 

∗

 Currently Naval Surface Warfare Center, Dahlgren Division 
NSWCDD/TR-03/81 

A-1/A-2 

APPENDIX A 

TIME OF FLIGHT BETWEEN TWO POINTS OF AN 

ELLIPTICAL TRAJECTORY 

NSWCDD/TR-03/81 

A-3

For an inverse-square gravitational field, given 1 2 , , and c rv r

! ! ! at arbitrary points 1 2 P and P , 

it is possible to derive a closed-form expression for the time of flight between 1 2 P and P , namely, 

{ } ( ) 2 1 2 1 E E e sin E sin E

h

ab

t fe = − − − , 

where 

( )

1 2 2

1

1

1 2

1 2

1

2 , , , 1, ,

1 arctan , 2 arccos 1 .

c

c

c

hv p h p a e b ap

r a

b r E E

ha r e a

μ μ

π

−

 

=× = = − = − =    

⋅     = =− −     −    

r v

r v

! !

! !

It is a well-known fact that the focus of the elliptical trajectory is the center of the earth, and 

this can be taken as the origin of the rθ coordinate system (Figure A-1). 

Figure A-1. Position and velocity of an object 

undergoing elliptical motion with the 

center of the earth as the focus of 

the ellipse 

The derivation is facilitated by introducing the so-called eccentric anomaly, E. Choose the 

center of the ellipse to be at the origin of the xy coordinate system, and draw the circle 

circumscribing the ellipse (Figure A-2). The radius of this circle is clearly , a where a is the 

semimajor axis of the ellipse. If r is the magnitude of the radius vector at P and θ the angle 

between r

!

 and the x axis, then ( )2 2 2r x ae y =− + , where e is the eccentricity of the ellipse. If 

the perpendicular from P to the x axis cuts the circle in ( ) x, , then since the equation of the y′

ellipse is 1 2

2

2

2

+ = b

y

a

x , where b is the semiminor axis of the ellipse, and the equation of the 

circle is 2 22 x + = y a ′ , we have, for the same value of , x

2 2

2 2 1 1 y y yb

a b ya

′ − += ⇒ =

′ . 

v

!

r

!

θ
NSWCDD/TR-03/81 

A-4

Figure A-2. Eccentric anomaly versus 

true anomaly 

From Figure A-2, it is clear that cos , sin . bx a Ey y b E

a = ==′ Hence, 

() ( ) ( ) ( )

( ) () ()

22 2 2 2 22 2 2 2 2

2 2 22 2

cos sin cos 1 sin

1 2 cos cos 1 cos 1 cos .

r x ae y a E ae b E a E e a e E

a e Ee E a e E ra e E

=− += − + = − + −

=− + =− ⇒ = −

Now 

( )

( )( ) ( )( )

( ) ( )

( )( )

2

2 2

cos 1 sin cos cos , sin sin , 1 cos 1 cos

cos 1 1 cos

1 cos 1 1 1 cos

1 cos 1 cos

2 1

2cos 1 cos , or cos cos , 2 22 2

cos 1 1 cos

1 cos 1

1 cos 1 cos

x ae a E e y b e E Ee E

r r e E rr e E

Ee a e E

e E

eE eE r

aE E a e

e

r r

E e e E

eE eE

θ θ

θ

θ θ

θ

−− − = = −= == = − −

− − +

+ =+ = = − + − −

− ⇒ =− =

− + − − =− = = − − ( )( )

( ) 2 2 ( )

1 1 cos

2 1

2sin 1 sin , or sin sin . 2 22 2

1

tan tan .

21 2

a e E

r

aE E a e

e

r r

e E

e

θ θ

θ

+ −

+

⇒ =+ =

+

∴ = −

( ) a,0

( ) x, y′

E

( ) 0,0 ( ) x,0 ( ) ae,0

θ

r

!

P( ) x, y
NSWCDD/TR-03/81 

A-5

Differentiating the expression for cosθ with respect to time, we obtain 

( ) ()

( )

( )

( )

2 2

2 2 2

2

2

sin 1 cos sin cos 1 sin

sin sin

1 cos 1 cos

sin sin .

E E e E eE E E e eE E b E E

eE eE r

bb r E EE E

rr b

θ θ

θ θ

− −− − − − = =− =− − −

⇒ = ⇒ =

" " " " "

" " " "

Now, conservation of angular momentum implies that r = constant = h 2θ

" . 

( )

( ) { } ( ) 2

1

21 2 1

1 cos

1 cos sin sin .

E

fe

E

h h E

br ab e E

ab ab t e E dE E E e E E

h h

∴ == −

⇒ = − = −− − ∫

"

Now 

( ) ( ) ( )

( )

2

2

2

2

cos cos 1 cos cos , 1 cos 1 cos 1 cos 1 cos

1

1 cos , where 1 . 1 cos 1 cos

sin sin sin sin

1 cos

arcsin , arcsin .

Ee e e E eE

eE e e

a e p ra e E pa e

e e

aeh E pe eh r aeE E

br p e

brr pr E

aeh eh

θ θ

θ θ

θ θ

θθ θ

θ

θ

−+ − = ⇒ = −= −+ +

− ⇒ =− = = =− + +

=== = +

⇒ = =

" " "

" "

Since ˆ r rv r = = + ⋅= r r r rr ˆ ˆ , , it follows that θθ r v ! ! ! ! " " " . Hence, 

( )

1 arcsin arccos 1 arctan ,

1 arcsin arccos 1 arctan .

1

b rb E

aeh e a h a r

p p

reh e r r h

p

θ

⋅ ⋅     = = −=         −

⋅ ⋅     = = −=           −    

rv rv

rv rv

! ! ! !

! ! ! !

The energy equation is 

ε μ− = r

m

mv

2

2

1 , 

where ε is the total energy. 

( ) 1 2 22

2

m

mr r

r

μ ∴ + −= θ ε

" " . 
NSWCDD/TR-03/81 

A-6

Now 

2 2

2 22 2

22 4 , . h dr h dr h dr

r rr r

r d rd r d

θθ θ

θθ θ

   = == ⇒ += +        

"" " " "

Putting 

u

r

1 = , we have 

2

2 22 2 2

2

dr du du 1 rr h u

d ud d

θ

θθ θ

   = − ⇒ += +        

" " . 

Thus, the energy equation becomes 

( )

( )

2

22 2

2 2

2

0 2

2 2

2 2

2 2

2 2 0 2

2 0

1 22

2

arcsin

2 2 2

2 1 1 sin .

2 1 1 sin

du du u mh u mu u

d d mh h

du h u

u h

u

mh h m

h h

u r

h m h

m

ε μ μ ε θ θ

μ θ θ

ε μ ε

μ

μ ε θ θ

μ ε μ θθ

μ

          +− = ⇒ = +−

     

− ⇒ −= =

+ − +

    ⇒ = ++ −   ⇒ =          ++ − 

   

∫

Putting 2 0

π θ = − , we obtain the polar equation of the orbit, namely, 

2

2

2

2 1 1 cos

h

r

h

m

ε

μ θ

μ

=      + + 

   

. 

Comparing the above equation with the equation for r on page A-5, namely, 

1 e cosθ

p r

+ = , 

we have 

( )

2 2

2

1 2 22 2 2

2

2

1 1

2 , 1

22 2 1 , 1 . c c

h h

p e

m

h a h ah v v p ae a e

mr r a

ε

μ μ

ε

μμ μ μ μ

−

= =+

 

⇒ − = =− = −   ⇒ = − =−

 
NSWCDD/TR-03/81 

A-7

This completes our derivation. One can determine whether the orbiting object is before or 

past apogee by testing whether r v ! !

⋅ is positive or negative. If it is positive, the object is before 

apogee, and if it is negative, the object is past apogee. To prove this, we have r v⋅ = rr ! ! ". Since 

r is increasing in the region 0 ≤ θ ≤ π and decreasing in the region 2 , πθ π ≤ ≤ it follows that 

r" is positive in the first region and negative in the second region, which proves the proposition. 

As a relevant mathematical digression, the author derived the condition that a circle of 

radius R intersects an ellipse whose focus is the center of the circle, as in Figure A-1. The 

required condition is 

2

2

1 1, where 1 , 1 R b

e

ea a

η η   < = − =−     , in which case the overlapping 

area is 

( ) 2 2 arccos arccos 1

1

e R ab e

e

η η η

η

− + −− − . 
NSWCDD/TR-03/81 

A-8

THIS PAGE INTENTIONALLY LEFT BLANK 
NSWCDD/TR-03/81 

B-1/B-2 

APPENDIX B 

AN ALTERNATIVE DERIVATION OF THE TIME OF FLIGHT 

NSWCDD/TR-03/81 

B-3

First, the following compact relation will be derived: 

( ) D R D R

p r p r p r

θ θ θ −θ 













 = − 





 





 − − 













 − sin 1 1 sin 1 1 sin 1 1

2 1

, (B-1) 

where () ( ) 1 12 arccos , arccos ˆˆ ˆˆ θ θ D R = ⋅= ⋅ rr rr . From Appendix A, 

cos 1, sin r

p pr p ee v

r h

θ θ

μ =− = =! . 

Put θ = θ1 +θ D ; then 

( ) 111

1

2

1 1

2 1

1

cos cos cos cos sin sin 1 cos sin

, 

1 1 cos sin 1 1 cos sin

1 csc 1 cot

1 1 cos 1

D D D D cr D

D cr D R cr R

cr R R

D

p p ee e e v

r

p p r r

pp pp v v

r r

pp p

v

r r

p r

p

r

θ θθ θ θ θ θ θ θ

μ

θθ θθ

μ μ

θ θ

μ

θ

 

= += − =− −    

⇒ = =  

+− − +− −    

 

⇒ =− −−    

⇒ =  

−− + −     1 2

cot 1 csc sin R RD

p p

r r

θ θθ         − −

   

which yields Equation (B-1). Now, 

2 2 h p r pdt r d = = μ θ ⇒ μ = θ !

( )

( )

2

0

3

2

0

1 12

2

0

1

1 1 cos 1 cot 1 csc sin

, cos sin

R

R

R

f DD

D

D R RD

t rd

p

p d

p pp

r rr

d D

AB C

θ

θ

θ

θ θ

μ

θ

μ

θ θ θθ

θ

θ θ

 

⇒ =

=          −− + − −−     

      

= + +

∫

∫

∫

 

 

where 

3

11 2

1, 1, 1 cot 1 csc , R R

p pp p AB C D

rr r

θ θ

μ

  = =− =− −− =    

. 
NSWCDD/TR-03/81 

B-4

It can be shown after some algebra that 

( ) ( )

( )

222

222 222

2 22

12 1 2 1 2 12

1 2 22 2

1 2

2

1 2 12

tan sin cos 2 2 arctan

cos sin tan

2

2 sin 2 cos

2 sin

2 sin

arctan

2 sin

2

R

R R

f

R R R

R

R

R

R

R

ABC D BC C A

t

A B C AB C AB ABC ABC

rr r r p r r rr

p rr

pr r F F

F r r p rr

θ

θ θ

θ θ θ

θ θ

θ

θ μ θ

  − −   − +

= −+

−− + + + − − + +  

 + − +− 

 =  ′ ′ + ′  + − 

,















where 

2 2

12 1 2 12 1 2 12 12 2 sin 2 cos 2 cos 2 sin

2 2 22

R R RR F rr p r r rr p r r rr rr    θ θ θθ    ′ = − +− ++ −             . 

This is Equation (15) of the main text. 
NSWCDD/TR-03/81 

C-1/C-2 

APPENDIX C 

EXPRESSIONS FOR 1 2 f , , , , , gf g p AND IN TERMS OF r r AND   NSWCDD/TR-03/81 

C-3

Proposition: 

2 12 2 1 2

1 2

2 1 cos 1 1 2 1 sin , sin , tan , 1 sin , 22 2

R RR R

R

r rr r f gf g

p p p rr p p

θ μ θθ θ θ

μ

 − =− = = − − =−    

 

where ( ) 1 2 = arccos rˆ ⋅ rˆ θ R . 

Proof: From Appendix A, 

( ) ( ) 1 1( ) sin sin , cos cos , sin sin , cos cos 2 22 2

rr E E re re

E Ee

bp p p

θ θ θ θ − +

= =+ = =

( ) ( ) ( ) ( )

21 2 1 1 2

2 1 12 21 1 2

sin sin sin cos sin cos

2 22 22

11 11

sin cos sin cos

22 22

EE E E E E

x

re re re re

pp pp

θθ θθ

−  == −

−+ −+ = −

( ) 2

1 2 2 1 1 2

1

sin sin , 2 2

R

rr e rr

p b

− θθ θ − = = (C-1) 

() ( )

( ) ( ) { } ( )( )

21 2 1 1 2

2 1 1 2 12

2 1 1 2 2 1 21

12 2 1 1 2 2 1 12 1 2

sin 2 sin sin cos sin cos

sin cos sin cos sin sin sin

2 2 sin cos cos sin cos cos .

2 2 2 22 2

R R

x EE E E E E

r r r r rr

e ee

b p b p bp

rr rr

e e

bp bp

θ θ θ θ θ θ θθ

θθ θθ θθ θ θ θθ

= −= −

= + − + = −+−

−+− +   = += +      

Now 

( )

( )( )

1 . 1 1

2 2

sec

2

cos

2

cos

2

1 cos

2

1 cos 1 cos

2 cos cos

1 cos 1 cos

1 2

1 2

1 2 1 2

1 2

1 2

1 2

1 2











 −





	









 = +

+








	 




 + = +

+ +

+ + = +

+

+

+ =

r r

p e

e

p

rr

e e

e

p

e

p

e

p r r

R

R

θ θ θ

θ θ θ

θ θ

θ θ

θ θ

Substituting this into the formula for sin , we obtain ( ) 2x

( ) 









 − + 





	









 = +





































+

−





	











+

= 21 cos

1 1

2 2

tan

2

2

cos

2

cos

1 1 1

2

2

sin 2 sin 2 2

1 2

1 2 1 2 R 1 2 R R

R

R

r r

p

bp

r r rr

p

bp

rr

x

θ θ θ

θ

θ

12 12 2

12 12

1 1 2 1 1 1 cos tan sin tan

2 22

R RR R rr rr

b rr p b rr p

θ θθ θ   − = +− = +−      

. (C-2) 
NSWCDD/TR-03/81 

C-4

It has already been shown (pages 4 and 5 of the main text) that 

( ) 2 1 2 2

1 12 2

2 2 1 sin , 2 sin cos , sin 2 , 1 sin 2

R a arr a a f xg x f x g x

r rr r

θ μ

μ = − = =− = −   . 

Using Equations (C-1) and (C-2), we obtain 

2 2 2 2

2

12 12

12 12

1 1 2 2

2

2 2 1 sin 1 sin , 2 2

2 sin cos sin , 2 2

1 1 1 cos 1 cos 1 1

tan tan , 2 2

2 2 1 sin 1 sin , 2 2

R R

R R

R

R R RR

R R

ar r f b p

a rr rr

g b p

a f b rr p p p rr

ar r

g b p

θ θ

θ θ θ

μ μ

μ θ θ μθ θ

θ θ

=− =−

= =

  − − =− + − = − −      

=− =−





which are the desired results. 
NSWCDD/TR-03/81 

D-1/D-2 

APPENDIX D 

TWO DIFFERENT SETS OF INITIAL CONDITIONS 

NSWCDD/TR-03/81 

D-3

Set Ι. Given vc r r    1, 2 , and , determine the type of trajectory and its solution. 

We have 

1 2

1

2 −















 = − μ

c v

r

a . 

If 0, a > the trajectory is elliptical; if 0, a < it is hyperbolic; and if , a = ∞ it is parabolic. 

In the elliptical case, we have 

( )

( )

2 2

1 2 12

2

12 1 2

2

12 12 1 2

2 cos cos 2 sin 2 1 cos

2

2 cos 2 cos cos 2 0

2

1 arccos cos cos 2 2 , 22 2

R

R

R R

S r r rr x a x a x

a x rr x r r a

x rr rr a r r a

a

θ

θ

θ θ

=+− = = −

 − ++− =

    

 = ± − +−   	 



     

where ( ) 1 2 arccos ˆ ˆ θ R = ⋅ r r . Thus, the elliptical case is ambiguous. 

In the hyperbolic case, since 0, a < it follows that 

( ) ( ) 2 2 2 0 cos 2 2 cos 1 2 12 1 2 12 2 2

R R a r r a rr a r r a rr

θ θ

+− <  − +− > . 

Hence, 

( )

( )

2

12 12 1 2

2

12 12 1 2

1 cos cos 2 2 0, 22 2

1 cos cos 2 2 0.

22 2

R R

R R

rr rr a r r a

a

rr rr a r r a

a

θ θ

θ θ

     − − +− >

   

     + − +− <

   

Putting ( ) 2

12 12 1 2

1 cos cos 2 2

22 2

R R u rr rr a r r a

a

   θ θ

= − − +−      

 and noting that cosh 1, y > we 

surmise that u >1. To prove that this is indeed the case, we have 

( )

( )

( )

( ) ( )

1 2 12 1 2 12

2

2

12 1 2 12

2

12 1 2 12

2 2

12 1 2 12

2 cos 2 4 cos

2 2

cos 2 2 cos 2

2 2

cos 2 2 cos 2

2 2

cos 2 2 cos 2 1 arccosh ln 1 .

2 2

R R

R R

R R

R R

r r rr a r r a rr

rr a r r a rr a

rr a r r a rr a

rr a r r a rr a u y u u u

θ θ

θ θ

θ θ

θ θ

+ >  − + >−

  − +− > −    

 − +− > −

 − + − − >−  >  = = +−
NSWCDD/TR-03/81 

D-4

Set ΙΙ. Given 1, , and , c f r v t   determine the type of trajectory and its solution. 

As in Ι, 

1 2

1

2 −















 = − μ

c v

r

a

and the type of trajectory is determined by the sign of a . Now, from Appendix A, 

1 11

1 1 cos 1 , sin c c r b e E eE

a ah μa

⋅ ⋅ =− = = rv rv

   

. 

Hence, 

( ) () { } { () ()}

( ) () ( ) () ( )

22 1 1 1

1 1

1 1

1 cos 1 cos 2 1 cos cos 2 sin sin 2

1 1 cos 2 sin 2 cos 2 sin 2 . c

c

r a e E a e xE a e E x e E x

r a

a x x a ar x x

a μa μ

=− =− + =− +

     ⋅ = − − + =− − + ⋅    	    
 
 

r v

r v

   

From Equation (1) of the main text, 

{ } ( ) ( )

{ } ( )

( ) { } ( ) ()

3 3

2

1 11

3

2

1 1

3

1 1 2 2

1 1

2 sin cos 2 sin cos cos sin sin

2 sin 2 cos 2 sin sin

2 2 2 1 sin 2 sin 2 sin 2 sin

fe

c

c

a a t xe x xE xe x x E e x E

a xe x E e x E

ar a a

x x x ax a r x x

a a

μ μ

μ

μ μμ μ

= − += − +

=− +

     ⋅ = −− + = −− + ⋅    	    
 
 

r v

r v

   

{ } ( ) () ( ) 11 2

2 2 cos 2 sin 2 2 fe

c

t a aa

a ar x x r

x μ μμ

∂

 = −− + ⋅ =

∂

r v

  . 

Referring to pages 4 and 5 of the main text, 21 21 , c c r r vr r v =+ =+ f g fg      , where 

{ } ( )( ) { } ( )

( )

2

1

3 3

21 2 1

2

12 2

2 1 sin ,

sin sin sin 2 sin 2 ,

2 sin 2 , 1 sin ,

fe

a f x

r

a a

g EE e E E t x x

a a f xg x

rr r

μ μ

μ

= −

= − − − =− −

=− = −  

where x is updated by the Newton-Raphson method as follows: 
NSWCDD/TR-03/81 

D-5

( )

( ) 1 { } ( )

2

1

2

n

f fe n

n n n f fe n

fe n

x x

t tx

x x x t tx

t rx a

x

μ +

=

− =+ =+ −  ∂

    ∂

, 

and a good initial guess is 1 x = 0.

The corresponding equations for a hyperbolic trajectory are: 

( ) () ( )

{ } ( ) ()

{ } ( )

( )

21 1

2

11 2

2

1

2

12 2

cosh 2 sinh 2 ,

2 2 sinh 2 sinh , 2 ,

2 1 sinh , sinh 2 2 ,

2 sinh 2 , 1 sinh ,

c

fh

fh c

fh

a

r a ar y y

a aa t

t ay a r y y r

y

a a f yg t a y y

r

a a f yg y

rr r

μ

μ μμ

μ

μ

− =− − + ⋅

− − ∂ = −− − ⋅ = ∂

− =+ = + −

− =− = +

r v

r v

 

 



where y is updated as follows: 

( )

( ) 1 { } ( )

2

1

2

n

f fh n

n n n f fh n

fh n

y y

t ty y y y t ty t ry a

y

μ +

=

− =+ =+ −  ∂ −     ∂

, 

and a good initial guess is 1 y = 0.
NSWCDD/TR-03/81 

D-6

THIS PAGE INTENTIONALLY LEFT BLANK 
NSWCDD/TR-03/81 

E-1/E-2 

APPENDIX E 

DERIVATION AND COMPUTATION OF THE ELEMENTS 

ARISING IN THE Q-GUIDANCE MATRIX 

NSWCDD/TR-03/81 

E-3

Suppose that the position vector at release is slightly perturbed. Let 0 1 r r and ! ! denote the 

unperturbed and perturbed position vectors at release, respectively, and 2r

! the position vector at 

impact. Then 

( ) ( )

( )

1

0 0 2 0 2 0 0 0 0 0 0 1 10 1 0 1 0

1

222

1 1 1 1 12 1 2 1

11 11 1 1 1

111 1 1 1 1

1 ˆ ˆˆ arccos , csc cot , , , ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ , arccos , csc cot . ˆˆ ˆ ˆ

, , 

R RR r p

r pR R R

r p

r pr

rr r

r

r rrr

rr rr r r r

rrrrr r r

θ

θ

θ

θ

θ θθ

θ θθ

=

= ⋅ = − =× = + +

= ++ = ⋅ = −

∂∂ ∂ ∂   === ⇒   ∂∂ ∂ ∂ r

r r θ r r nr θ r r θ n

r r θ r r

!

{ } ( ) ( )

{ } ( ) ( )

0 1 0 1 0

1 0

1 0

1 1

1 1

1 1 2 2

3 1 1 0 1 10 1 0

11 1

1 10 2 2

3 1 1 0 1 1 0 10

11 1 0

1

1 1

1, 0.

ˆ ˆ 1 ˆ ˆ ˆ 0,

ˆ ˆ ˆ 1 ˆ ˆ ˆ ,

ˆ 1

p

rr p

r r

p r

p

r r

r r

r r rr r

rr r

r r rr r

rr r r

r r

θ

θ

θ θ

θ θ

= =

=

=

  ∂ ∂   = ==     ∂ ∂     

∂ ∂  

= −− + ⇒   = ∂ ∂ 

∂ ∂  

= −− + ⇒   = ∂ ∂ 

∂ = ∂

r r r r r

r r

r r

r r

r θ n

r r θ θ n r

r

! !! ! !

! !

! !

{ } ( ) ( ) 1 0

2 2 1 0

3 1 1 0 1 10 1 0

1 0

ˆ ˆ ˆ ˆ ˆ . p pr

p

r r rr r

r r

θ

=

  ∂

−− + ⇒   =   ∂ r r

r n

n r θ ! !

Differentiating 1 2 cos = rˆ ⋅ rˆ θ R , we have 

{ } ( ) ( )

{ } ( ) 1 0

1

2

1 1

1 2 2

2 1 1 02 1 102 1 02 3

1 11

2 2

3 1 1 0 11 0

1 1

ˆ sin ˆ

ˆ csc ˆ csc ˆ ˆˆ ˆ ˆ ˆ

csc cos sin 0;

R

R

r r

R R

R rr p

r r

R R

r Rr R

r

r r

r r rr r

r rr

r r rr

r r

θ

θ

θ θ

θ θ θ

θ θ θ θ

=

∂ ∂ − =⋅ ∂ ∂

∂ ∂

⇒ =− ⋅ =− − ⋅ − ⋅ + ⋅

∂ ∂

  ∂ =− − − ⇒   = ∂  r r

r

r

r

r rr θ r nr

! !

{ } ( ) ( )

{ } ( ) 1 0

1 2 2

2 1 1 0 2 1 1 0 2 10 2 3

1 11

2 2

3 1 1 0 11 0

1 10

ˆ csc ˆ csc ˆ ˆ ˆ ˆ ˆˆ

csc 1 sin cos ;

R R

R pr

R R

Rr R

r r rr r

r rr

r r rr

r rr

θ θ

θ θ

θ θ

θ

θ θ θ

θ θ θ θ

=

∂ ∂

=− ⋅ =− − ⋅ − ⋅ + ⋅

∂ ∂

  ∂ =− − − ⇒   = − ∂  r r

r

r θ r nr rr

! !

{( ) ( )}

( )

1 0

1 2 2

2 1 1 0 2 1 10 2 1 0 2 3

1 11

1

3 1 01 0

1 1

ˆ csc ˆ csc ˆ ˆ ˆ ˆˆ ˆ

csc

cos sin 0.

R R

R p pr

p p

p R R

rR R

p

r r rr r

r rr

r

r r

r r

θ

θ

θ θ θ

θ θ θ θ

=

∂ ∂

=− ⋅ =− − ⋅ − ⋅ + ⋅

∂ ∂

  ∂

= + ⇒   =   ∂ r r

r

r nr rr θ r

! !
NSWCDD/TR-03/81 

E-4

( )

( )

( )

1 1 2

2 1

1 1 11

1

1 2

1 1

1 1

2 11

11 11

1 1

1

ˆ ˆ ˆ ˆ csc cot csc cot

ˆ csc csc cot cot ˆ ˆ

ˆ ˆ ˆ ˆ csc cot csc sin cos cot ˆ

ˆ ˆ cot

R R

RR R R

r r rr

R

RR R R

r r

R R

R R RR R R

rr rr

R

R

r

r r rr

r r

rr rr

r

θ θ θθ θ θ

θ θθ θ θ

θ θ θ θ θθ θ θ

θ θ

∂ ∂ ∂∂

=− + −

∂ ∂ ∂∂

∂ ∂ = −− ∂ ∂

∂∂ ∂∂ =− − = − − ∂∂ ∂∂

∂ = − ∂

θ r

r r

r

r r

r r θ r θ

r θ

1 0

1 1

1 1

ˆ ˆ cot 0. R

r r r r

θ

=

∂ ∂  − ⇒   = ∂ ∂ r r

r θ

! !

Similarly, 

( )

( )

( )

1 0

1 0

1 1

1 1

1 11

1 00

00 0 0

1 0 00

1 11 0

1 1 0

1 1 11 0

1 1

ˆ ˆ ˆ ˆ cot cot

ˆ 1 cot ˆ ˆ ˆ ˆ cot ,

ˆ ˆ ˆ cot ˆ ˆ ˆ cot cot .

R

R R

R

R

R R

R R

p p pp

ci ci

r

r rr

rr rr

r r rr r

v vr

r r

θ θθ

θ

θ θ θ

θ θ

θ θ θ θ

=

=

∂ ∂∂ =− − ∂ ∂∂

  ∂

⇒   =− − − =− ∂ 

∂ ∂ ∂∂   =− − ⇒   = − ∂ ∂ ∂∂   

∂ ∂∂ = ∂ ∂

r r

r r

θ r

r θ

θ r

r θ θ

θ r θ

r θ n

! !

! !

10 10

1 0 1 0

1

111 1

1

1 11 1 1 0

1

1 11 1 1

,

1 ,

ci R ci ci

r Rr r

ci ci ci R ci ci

R R

ci ci ci R ci

p p Rp p

v vv

rrr r

v vr v v v

r rr r r r

v vr v v

r rr r r

θθθθ

θ

θ

θ

θ θ

θ

θ

= =

= =

∂∂ ∂ ∂  

+ ⇒   = ∂ ∂∂ ∂ ∂  

∂ ∂ ∂ ∂∂ ∂ ∂  

= + ⇒   = − ∂ ∂∂ ∂ ∂ ∂ ∂    

∂ ∂ ∂ ∂∂ ∂

= + ⇒  ∂ ∂∂ ∂ ∂ ∂

rr rr

r r r r

!! !!

! ! ! !

1 0

0.

=



 =  r r

! !

Differentiating 1 1 ˆ ˆ c cr c v v = + θ v r θ ! , we have 

10 10 10

1 1

1 1 00

1 1 11 1 1 1 1

1 1

1 1

11 11 1

1

ˆ ˆ ˆ ˆ ˆ ˆ ;

ˆ ˆ ˆ ˆ

c cr c c cr c

cr c

r r rr r r

c cr c

cr c

c

v v vv

v v

rr rr r r r r

v v

v v

rr rr r

r

θ θ

θ

θ

θ

θθ θθ θ

θ

== =

∂ ∂ ∂∂ ∂ ∂ ∂ ∂     

=+ + + ⇒      = +

∂∂ ∂∂ ∂ ∂ ∂ ∂     

∂ ∂ ∂∂ ∂

=+ + +

∂∂ ∂∂ ∂

  ∂

⇒   ∂ 

rr rr rr

r

v r θ v

r θ r θ

v r θ

r θ

v

!! !! !!

!

! !

!

!

1 0 10 10

00 0 0

0 0

1 1 ˆ ˆ ; cr c

c cr

R R

v v

v v

r r

θ

θ θ θ = = =

        ∂ ∂

=− + − −      ∂ ∂      r    rr rrr θ ! !! !!

( )

1 0

1 1

1 1 0 0 00

11 11 1 1 0

ˆ ˆ 1 ˆ ˆ ˆ cot c cr c c

cr c cr c R

pp pp p p

v v

v v vv

rr rr r r r

θ

θ θ θ

=

∂ ∂ ∂∂ ∂ ∂ 

=+ + + ⇒   = − ∂∂ ∂∂ ∂ ∂   r r

v r θ v

r θ n ! !

! !

. 
NSWCDD/TR-03/81 

E-5

The Q-Guidance matrix is given by 

( )

1 0

1 0 1 0

cot

1 0 0

0 1

0 1

, ,

1

1 1

1 1

1 1 1 1

r r

r r r r

v v v

r

v

! !

! ! ! !

! ! !

!

!

=

= =

















































−

















− 















∂

∂ − 















∂

∂

















+ 















∂

∂ − 















∂

∂

=





































∂

∂

















∂

∂

















∂

∂ =

































∂

∂ =

cr c R

cr

R t

c

t

c

c

R t

cr

t

cr

t p

c

t

c

t r

c

t

c

v v

r

v

v

r r

v

v

v

r r

v

r r r

Q

f f

f f

f f f f

θ

θ

θ

θ

θ θ

θ

θ

. 

It is desired to evaluate the partial derivatives 

f f f t

cr

R t

c

t

c

r

v v

r

v

















∂

∂

















∂

∂

















∂

∂

1 1

, , θ

θ θ , and 

f R t

cr v

















∂

∂

θ . 

Differentiating the formulas 

22 1

11 2

22 2 sin , cos cos , cos cos 22 2

RR R

c cr tr

rr r

v v xv x

rS S r S r θ

μ θ μ    θ μ θ = = −= −   

  

, 

we have 

R c r x

x S

v θ θ μ sin sin

2 = − 3 2 ∂

∂ , (E-1) 





 



 = − + − 















∂

∂

2

2 3 cos cos

2

sin

2 3 3 1 2 1 2

1

2

1

R R

x

c r r r r x

r S

r

r

v θ μ θ θ , (E-2) 

( ) 









 



 



 = + − + 















∂

∂

2

cos 1 cos

2

cos

2

2

3 1 2 1 2

1

2 R R

R x

c r r r r x

r S

v μr θ θ

θ

θ , (E-3) 





 



 = + − ∂

∂

2

cos cos

2

sin sin 2

1 2

2

3 1 2 cr R R x r r r r x

x S

v μ θ θ , (E-4) 





 



 = − + − 















∂

∂

2

sin cos cos

2

sin

2 2 1 2

2

3 1

1

2 R R

R x

cr r x r r r x

r S

v μr θ θ

θ , (E-5)





 



 = − + − ∂

∂

2

cos cos

2

sin sin 2

2 1 2

2

3 1 tr R R x r r r r x

x S

v μ θ θ , (E-6)





 



 = + − 















∂

∂

2

sin cos cos

2

sin

2 1 2

2

3 1 2

2

1 R R

R x

tr r r x r r x

r S

v μr θ θ

θ . (E-7)
NSWCDD/TR-03/81 

E-6

Now 

11 1

, 

f ff f

c cc c cc

t x tt x t RR R

v v vx v v v x

r r xr x

θ θθ θ θθ

θθ θ

          ∂ ∂ ∂∂ ∂ ∂ ∂ ∂           =+ =+

∂ ∂ ∂∂ ∂ ∂ ∂∂          

, 

with similar expressions for 

f

cr

R t

v

θ

  ∂

  ∂ 

 and 

f

tr

R t

v

θ

  ∂

  ∂ 

. Using a well-known theorem in partial 

derivatives, we have 

1

1

1 1 1

1 1

1

f f

f f

f x x

f f t t f r x

r

t t

xr x t r r

rt x r t t

x x

  ∂ ∂

    ∂∂ ∂     ∂ ∂ ∂       = − ⇒ =− =−   ∂∂ ∂ ∂     ∂ ∂         ∂ ∂  

, 

where 

x

t f

∂

∂

 is given by Equation (5) of the main text for elliptical trajectories, and 

( )( ) ( )

{ } ( )( ) ( )

2

12 2 2

2 2 1

12 12

1

3 csc cos 6 cos cos 1 cot

csc 2

2 2 cos 3 3 1 cot 5 sin

2

R

f

R x

rrx x x r x x x

t x

r S r

rr x x rr x

r

θ

μ θ

  + −− −     ∂

  = ∂   + + − −+

 

, 

obtained by differentiating Equation (4) of the main text. Similarly, 

x

t

t

x

f

R x

f

R tf

∂

∂

















∂

∂

= − 















∂

∂ θ

θ , 

where 

R x

f t

















∂

∂

θ is given by Equation (6) of the main text. 

Using the above results, It is straightforward though tedious to verify the following three 

identities: 

1

1

0,

f f

cr c

c

R t t

v v

v r

r

θ

θ θ

  ∂ ∂   ++ =

∂ ∂  

 (E-8)

1 0,

f f

cr c c

cr c

R Rf t t

v vv

vv r

t

θ θ

θ θ θ

  ∂ ∂∂   + +=

∂ ∂∂  

 (E-9)

f

c c f

R t

tr

R t

cr

t

v

v t

v

r

v r

f f

∂

∂

+ = 















∂

∂ − 















∂

∂ θ

θ θ θ 2

3

2

1

1

2 . (E-10)
NSWCDD/TR-03/81 

E-7

From Equation (E-8), we have 

















+ 















∂

∂ = − 















∂

∂

θ

θ

θ c

R t

cr

t

c v

v

r r

v

f f 1 1

1 , (E-8′) 

which proves the symmetry of the Q-Guidance matrix. 

Eliminating 

f R t

cr v

















∂

∂

θ between Equations (E-8) and (E-9), 

f

c cr

R t

c c

t

c cr t

v

v

v

r

v

r

v

v

f f

∂

∂

+

















− 















∂

∂ = 















∂

∂ θ θ θ θ

1 1 θ . (E-11) 

Writing Equation (E-8) for the reverse trajectory, we have 

2

2

0.

f f

tr t

t

R t t

v v

v r

r

θ

θ θ

  ∂ ∂   −− = ∂ ∂  

Now 

1 1

2 2 22 2

, 

f f

t cc

t c

t t

r v rv v

v v

r r rr r

θ θθ

θ θ

    ∂ ∂   = =−     ∂ ∂      

 (E-12) 

2 1

2 2 f f f

tr t c

t

R t tt

v vv

vr r

r r

θ θ

θ θ

   ∂ ∂∂

⇒    =+ =

∂ ∂∂   

. (E-13)

Using Equations (E-8) and (E-13) to eliminate 

f

cr

R t

v

θ

  ∂

  ∂ 

 and 

f

tr

R t

v

θ

  ∂

  ∂ 

 from Equation (E-10), 

















∂

∂

+ 















∂

∂ = − + 















∂

∂

f

c f

t

c c

t

c

t

v

t

r

v

v r

r r

v

f f

θ θ

θ

θ

2

3

2

1 1

1

1

2 2

. (E-14) 

Writing Equation (E-11) for the reverse trajectory, 

f

t tr

R t

t t

t

t tr t

v

v

v

r

v

r

v

v

f f

∂

∂

+

















+ 















∂

∂ = 















∂

∂ − θ θ θ θ

2 2 θ . 

Substituting the system of equations (E-12) into the above equation, 

















∂

∂

+ 















∂

∂ = − 















∂

∂

f

c

R t

c c

tr t

c

t

v

r

v

v

r

r

r r v

v

f f

θ θ

θ

θ

θ 2

2

1

2 2

1 . (E-15) 
NSWCDD/TR-03/81 

E-8

Eliminating 

f t

c

r

v

















∂

∂

2

θ between Equations (E-14) and (E-15), 

1 2

1

2 1

1 3 0.

2 2 f f

cc c c

c f

tr R tr f t t

rv v r v v

v tr

rv v t r

θθ θ θ

θ θ

  ∂ ∂∂     − +− − =   ∂ ∂∂    

Eliminating 

1 f

c

t

v

r

θ   ∂

  ∂  

 between Equation (E-11) and the above equation, 















 −

∂

∂



























 − − −

= 















∂

∂

tr

cr c

f

c f

tr

cr c cr

R t

c

r v

r v

v

t

v

t

v

r

v v r v

v

f

2

1

2

1

1

2

3

2

1

θ

θ

θ

θ

θ . (E-16)

The expression (E-16) is indeterminate when 

1 2 1 2 1 2 12 sin sin 3 cr tr c t rv rv E E E E = ⇒ rv rv ⋅ =⋅ ⇒ = ⇒ + = π ! ! ! !

(see Appendix A), which can occur when E1 is in the third quadrant and E2 is in the fourth 

quadrant. Therefore, it is desirable to derive an expression for the above partial which is analytic 

everywhere. This is done in Appendix F. 

The remaining partials in the Q -Guidance matrix are now readily obtained. Employing the 

equations for cr tr cθ v , namely, and v in terms of v

1

2 1

1 1

21 2

cot csc tan , 2

csc cot tan 1 cot , 2 2

R

cr R R c

c

R R

tr R R c c cr

c

r

v v

r rv

r r

v v vv

r rv r

θ

θ

θ θ

θ

μ θ θ θ

μθ θ θ θ

 

=− +    

   = − − =− −      

we deduce 

1

2

csc 1 tan

2 f f

cr tr R c

tr R

Rc R t t

v vr v

v

v r

θ

θ

θ θ

θ θ

    ∂ ∂         =− + − +       ∂ ∂    

, (E-17) 

















+ 















∂

∂ = − 















∂

∂

θ

θ

θ c

R t

cr

t

c v

v

r r

v

f f 1 1

1 , (E-8′) 
NSWCDD/TR-03/81 

E-9

c R

t

R c c

c

tr

t

R c

c

tr R c R tr

t

cr

v

r r

v

r

v

r

r

v

v

r

v

r

r

v

v

v

r

r

v

r r

v

f

f f

θ θ

θ θ θ

θ

θ θ

θ

θ

θ

θ

cot

1

2

1 tan

2

cot 1 tan

2

1 tan

1

2 1 1 1

1

2 1

1

2

1

1 1

−

































∂

∂

+



























 = − +

















∂

∂





























+ − +





























+ 













 = − + 















∂

∂

1

1 2

1 1 tan cot

2 f

tr R cr

c R

c R t

vr v

v

rv r

θ

θ

θ θ

θ

         ∂

=− − + +          ∂       

, (E-18) 

where Equation (E-8) has been used. Substituting Equation (E-8′) into Equation (E-14), 

















∂

∂ − 















∂

∂ = + 















∂

∂

f

c f

R t

cr c

t

c

t

v

t

v

v

r r

v

f f

θ

θ

θ

θ 2

3

2

1 1

2 2

. (E-19) 

Finally, 

f f t

R c

c

tr c R

t

cr

r

v

r

r

v

v

v

r

r

r

v

















∂

∂



























 = + − + 















∂

∂

2 2

1

2

2

1

2 2

csc 1 tan θ

θ

θ

θ θ . (E-20) 
NSWCDD/TR-03/81 

E-10

THIS PAGE INTENTIONALLY LEFT BLANK 
NSWCDD/TR-03/81 

F-1/F-2 

APPENDIX F 

DERIVATION OF AN ANALYTIC EXPRESSION FOR 

f R t

c v

















∂

∂

θ

θ

NSWCDD/TR-03/81 

F-3

As stated in Appendix E, the expression (E-16) is indeterminate when 1 2 cr tr rv rv = . An 

analytic expression for 

f R t

c v

















∂

∂

θ

θ will now be derived in two ways. 

First method: From Appendix A, 

 ra e E ra e E 1 12 2 =− =− ( )( ) 1 cos , 1 cos . (F-1) 

From Equation (C-1), 

 

1 2

sin sin

2

R b

x

r r

θ = . (F-2) 

From page 2 of the main text, 

 { } ( ) 1

1 2

cos cos cos

2

R a

x e xE

r r

θ

= −+ . (F-3) 

{ } ( ) 1

1 2

2 sin 2sin cos sin cos cos

2 2

R R

R

ab

x xe xE

r r

θ θ θ = = −+

 { } ( )( ) 21 2 1

1 2

sin sin sin ab E Ee E E

rr = −− − . (F-4) 

{ } ( )( ) ( )

2

2 2

1 2

2

2 2

1 2

1 2

2 cos 1 2sin 1 sin

2

1 cos 1 cos 2 1 sin

R

R

b

x

r r

a eE eE e x

r r

θ θ =− =−

= − − −−

 { } ( )( ) ( )

2

2

21 1 2 1 2

1 2

cos cos cos 1 sin sin a E Ee E Ee E E

rr

= −− + + − . (F-5) 

{ } ( ) ()

{ } ( ) ()

2 2

1 2 12 1 2 12 1 2 12

2

12 1

12 1

2 cos 2 cos 2 cos

2 2

2 cos cos 2cos 2 cos .

2 cos cos 2cos 2 cos

R R

R r r rr r r rr r r rr

a e E E x e xE

e E E x e xE

θ θ θ    + − = ++ +−      

=− + + − +

− + −+ +

 { ( )} 22 2 2

1 = −+ 4 sin 1 cos a x e xE . (F-6) 

From Appendix A, 

 1 1

111

c

h b p v r

r r ra

θ

μ μ = == = θ

! , (F-7) 
NSWCDD/TR-03/81 

F-4

 11 1

1 1

sin sin cr

ah a v r eE eE

br r

μ == = ! , (F-8) 

 2 2

2

sin tr

a vr eE

r

μ = = ! . (F-9) 

Substituting Equations (F-3) and (F-7) into Equation (11) of the main text and using Equation (F-4), 

{ } ( )

2

2

1 12 1

1

2 2 cos sin sin cos cos

2

R

c

b F r rr v x x x e x E

r

θ

θ μ = = −+

 { } ( )( )

2

21 2 1

1

sin sin sin b E Ee E E

r

μ= −− − (F-10) 

 F

br

a

R

2

sin

μ

⇒ θ = . (F-4′) 

From Equations (F-2), (F-6), (F-7), and (F-10), 

( ) { } ( )

{ } ( )( )

2 4

24 2 2 22 4 2 2 2 2

1 1 2 12 2 2 1

1

2 4

2 2 2

2 21 2 1

1

4 2 cos 4 sin sin cos cos

2

sin sin sin

R

c R

b r v r r rr r x x e x E

r

b EE e E E

r

θ

θ μ θ μ

μ

+− − = − +

= −− −

{ } ( )( )

2

21 2 1

1

sin sin sin b F EE e E E

r

μ= −+ − . (F-11) 

Also, 

 { } ( ) 2 1 2 1

3

E E e sin E sin E at f = − − − μ, (F-12) 

( ) 2

2

2 sin

arctan 2 1

2

2

2

1 1 2

E E

x

r r r v r

F

R c

− = =

+ − θ

θ μ

 (Equation (12) of the main text). 

Substituting Equations (F-2), (F-4′), (F-7), (F-11), (F-12), and (12) of the main text into 

Equation (14) of the main text, namely, 

( )

( )

24 2 2 22 4

1 1 2 12 2

2342 3 2

332 3 1 2

1 2

2

2 2

11 2 2

1 2 cos 4 sin .

2

16 sin sin 4 sin 2 arctan ,

2 sin

2

f R

c R

c

R

c R f c R

c R

c

t

r v r r rr r

v F

rrv t rrv F

v F F F rr rv r

θ

θ

θ

θ

θ

θ

θ θ μ

θ μ θ μ θ

θ

μ

∂   = +− −   ∂  

 

    + −

  + −  
NSWCDD/TR-03/81 

F-5

we obtain 

 { ( )( )}

{ } ( )( )

2

21 2 1

2

21 2 1

sin sin sin .

3 sin sin 16sin

f

c

t a b EE e E E

v θ F E Ee E E x

∂   −+ − =   ∂ −− − −  

. (F-13) 

Substituting Equations (F-7), (F-8), (F-9), and (F-12) into Equation (E-16), namely, 















 −

∂

∂



























 − − −

= 















∂

∂

tr

cr c

f

c f

tr

cr c cr

R t

c

r v

r v

v

t

v

t

v

r

v v r v

v

f

2

1

2

1

1

2

3

2

1

θ

θ

θ

θ

θ , 

we obtain 

( ) ( )

( )

1 2

2 2

1 22 1 2

1 21 1 2

1 22 1 2 1

sin sin

1 2 sin sin

2 sin sin

3 sin sin sin sin f

c

c

R c t

f

be E E

v r Er E v rv E E r

ae E E E E e E E t

θ

θ

θ

μ

θ

 

    ∂ =     −     ∂   ∂ −   −  

+ −− −   ∂        

. 

From Equations (F-1), 

{( )( ) }

( )( ) { } ( )

2 2 22 2

1 22 1 1 2 2 1

2 2

2 1 1 2 21

sin sin 1 cos sin 1 cos sin

sin sin 1 1 sin sin 2 sin

r Er Ea e E E e E E

a E E e E E e EE

− = − −−

= − ++ − −    

( ) { ( )}

( )( ) { } { } () ()

22 2

1 22 1 1 22 1 2 1

2 2

2 1 1 2 21 1 2 21

2 sin sin 3 sin sin sin sin

sin sin 2 2 sin sin 3 sin sin 4sin .

r E r E ae E E E E e E E

a E E e E E eEE E E EE

⇒ − + −− −

= − +− + − − −    

( ) ( )( )

() ()

1 2

2 2

2 1 12 1 21 2

1

21 1 2 21

sin sin

1

2 sin sin sin sin 2 2 sin sin

3 sin sin 4sin

f

f

c

c c

Rc f t

t

be E E

v

v v

rv E E t E E e EE

a r

e EE E E EE

θ

θ θ

θ

μ

θ

 ∂    ∂   ∂ ∂  

∴ = ⋅      ∂ −∂ − +−         −     +− − −     

   
NSWCDD/TR-03/81 

F-6

Substituting Equations (F-10) and (F-13) into the above equation, 

( )

( )( )

( )( )

{ } ( )( )

( )( )

( )

21 2 1

1 2 2

21 2 1

2 2

2 21 2 1

1 21

2

2 1 12

21 1 2

sin sin sin .

sin sin

3 sin sin 16sin

sin sin sin .

2 sin sin

sin sin 2 2 sin sin

3 sin sin 4s

f

c c

R f t c

EE e E E

eE E

E Ee E E x

v ab v EE e E E

rv E E F t

E E e EE

e EE E E

θ θ

θ

μ

θ

  −+ −     

      −− − − 

  ∂ ∂   = − −− −

  ∂ ∂ −

− +−    

+− − ( )

( )( )( ) ( )

( )

2 1

22 2 2 2

2 1 1 2 21

2 2

1 21 1 2

in

sin sin 3 sin sin 3 1 sin . 3 sin sin

c

c f

E E

ab v e E E ee E E e EE

rv F t eE E E E

θ

θ

μ

                         −         

∂   − +− − + −   =   ∂   + −  

Using Equation (F-7) in the above equation, 

( )( )

( ) ( )( ) 















− + − + −

− + −

∂

∂ = 















∂

∂

2 1 1 2

2

2 1

2

1 2

2 2

2 1

3

3 1 sin 3 sin sin

sin sin 3 sin sin

e E E e E E E E

e E E e e E E

t

v

v

F

v a

f

c c

R t

c

f

θ

θ

θ

θ . (F-14)

Note that a and b can be computed from Equations (3), (9), (11) of the main text, and (F-7). We 

have 

222 2 2 2

2 12 12

1

1 sin sin

csc , 2

c R cR

c

r r v a r rv a S x b rv

F F

θ θ

θ

μ θ θ

μ = = == . 

Second method: Equation (F-14) may also be derived from the identity 

= −1 















∂

∂

















∂

∂

















∂

∂

R c f c

f

v f

R

R t

c

v

t

t

v

θ θ

θ

θ

θ

θ . 

R c

f

v

t

θ θ

















∂

∂

 is given by Equation (F-13). Differentiating Equation (13) of the main text, namely, 

( ) ( )

( )

2 22 2

2 1 2 1 1 2 12

1 2 222 2

1 2

2 2

11 2 2

2 sin 2 cos

2 sin

2 sin

arctan

2 sin

2

R

c R

c R

f c R

R

c

r r r rv r r rr

rrv

t rrv F F

F rr rv r

θ

θ

θ

θ

θ μ θ

θ

μ θ

θ

μ

  + − +−  

=  

  +

+ −  

, 

we obtain 
NSWCDD/TR-03/81 

F-7

( )

( )

( )

2

22 2 2

12 1 2

1

2 2

1 2 2 2

2 11 2 2

1

2 2

1 2

2

11 2 2

1

cot

2 2 sin

2

sin tan 2 sin

22 2

sin 1 4cot arctan

2 sin

c

f

f R

R R v

R

c

c

cR R R

c

c R

c R

R

R c

t F

t

F

r

r r rv

rv

rr v F

rr rv r

F rv F

r rv F F

F F

rr rv r

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ θ

μ θ μ

θ μθ θ μ θ

μ θ θ

θ μ

   ∂ ∂    = −+

∂ ∂   

+− +

  ∂

+ +−     ∂

  ∂

+ −   ∂   + − 2

.

2

θ R

 

 

 

 (F-15) 

Differentiating the expression for F on page 11 of the main text, namely, 

22 2 2

2 1 1 2 12 1 1 2 12 2 2 sin 2 cos 2 cos 2 sin

2 2 22

R R RR F r rv r r rr rv r r rr r c c θ θ

θ θ θθ

μ μ

      = − +− ++ −             , 

we have 

( ) 34 2 2 2

2 1 11 2 2 2 sin 2 sin

2

R

Rc c

R

F

r rv r r r v r θ θ

θ θμ μ θ∂   = −+ + −   ∂   . 

Substituting Equations (F-1), (F-2), (F-4′), and (F-7) into the above equation, 

 { } ( )( ) 2

21 1 2

1

2

cos cos cos

R

F ab F EE e E E e

r

μ

θ

∂

= −− + +

∂ . (F-16) 

From Equations (F-1), (F-2), and (F-7), 

( ) { } ( )( )

2

22 2 2 2

12 1 2 1 1 2 2

1

2 2 sin 1 cos cos cos 2 .

2

R

c

c

r r r rv a E E e E E e

rv

θ

θ

μ θ μ μ +− + = − − − + + (F-17) 

From Equations (F-2) and (F-4′), 

 ar F

b x

R

R

R

R

R

1

3 2

2

2 sin

sin

2

2sin

2

cos

2

sin

2

tan μ

θ

θ

θ

θ

θ = = = . (F-18) 

From Equations (F-7), (F-16), (F-18), and (12) of the main text, 

( ) () ( ) ( ) { }

2 2

2 2 2

2 11 2 2 2 1 1 2

1 1

tan 2 sin sin 2 cos cos cos .

22 2

R R

c

c R

F ab rr rv r x E E e E E e

rv F r F θ

θ

μθ θ μ μ θ

  ∂   + − = −− + +   ∂
NSWCDD/TR-03/81 

F-8

Substituting Equation (F-10) into the above equation, 

( )

( )

( )( ) { } ( )( )

2 2

2 11 2 2

1

2 1 2

21 1 2

21 2 1

tan 2 sin

22 2

sin

cos cos cos .

sin sin sin

R R

c

c R

F

rr rv r

rv F

a EE

E Ee E Ee

EE e E E

θ

θ

μθ θ μ θ

μ

  ∂   + −   ∂

− = −− + + −− −

 (F-19) 

From Equations (F-4′), (F-5), and (F-16), 

{ } ( )( ) ( )

2

2

21 1 2 1 2

1 2

1 2 4cos sin cos cos cos 1 2sin sin R R

R

F a E Ee E Ee E E

F rr

θ θ

θ

∂

− = −− + + −

∂ . 

Dividing the above equation by sinθ R and using Equation (F-4), 

{ } ( )( ) ( )

2

2

21 1 2 1 2

1 2

1 2 4 cot cos cos cos 1 2sin sin

sin R

R R

F a E Ee E Ee E E

F rr

θ

θ θ

∂

− = −− + + −

∂

 ( )( ) ( )

( )( )

2

21 1 2 1 2

21 2 1

2 cos cos cos 1 2sin sin

sin sin sin

a E Ee E Ee E E

b EE e E E

−− + + − = −− − . (F-20) 

From Equations (F-4′), (F-7), (F-20), and (12) of the main text, 

( )

( ) ( )( ) ( )

( )( )

2 2

1 2

2 2

11 2 2

2

21 1 2 1 2

2 1

21 2 1

sin 1 4cot arctan

2 sin

2

cos cos cos 1 2sin sin . sin sin sin

c R

R

R R

c

r rv F F

F F

rr rv r

E Ee E Ee E E

aE E

EE e E E

θ

θ

μ θ θ

θ θ

μ

μ

  ∂   − ∂   + −

−− + + − = − −− −

 (F-21) 

From Equations (F-4′) and (F-10), 

{ } ( )( )

22 2 2

12 1

2 2 21 2 1

sin sin sin sin cR c c rr v a rv a v E Ee E E

F b F

θ θθ θ

μ μ = = −− − . (F-22) 

From Equations (F-4′), (F-5), and (F-16), 

{ } ( )( ) ( )

2

2

21 1 2 1 2

1 2

1 cot cos cos cos 1 sin sin

sin R

R R

F a E Ee E Ee EE

F rr

θ

θ θ

∂

− = − −+ + − +

∂

{ } ( )( ) ( ) 2

21 1 2 1 2

1

cos cos cos 1 sin sin ab E Ee E Ee E E

r F

μ = − −+ + − + . (F-23) 
NSWCDD/TR-03/81 

F-9

From Equations (F-7), (F-12), and (F-23), 

{ } ( )

{ } ( )( ) ( )

2

21 2 1

1

2

21 1 2 1 2

1 cot sin sin .

cos cos cos 1 sin sin

f R

R

F ab a t E Ee E E

F rF

E Ee E Ee E E

μ θ

θ

  ∂   − = −− − ∂  

− −+ + − +

{ } ( ) { } ( )( ) ( )

3

2

21 2 1 21 1 2 1 2 sin sin cos cos cos 1 sin sin . c a v EEe E E E E e E E e E E

F

θ = −− − − − + + − +

 (F-24) 

Substituting Equations (F-17), (F-19), (F-21), (F-22), and (F-24) into Equation (F-15), 

{ ( )}{ ( )( ) ( )}

{ } ( )( ) { } ( )( )

( ) ( )( ) { }

( ) ( )( )

2

21 2 1 21 1 2 1 2

2 3 21 2 1 21 1 2

2

21 21 1 2

2

21 21 1 2 1

sin sin cos cos cos 1 sin sin

sin sin sin 1 cos cos cos 2

sin cos cos cos

cos cos cos 1 2sin sin

c

f c

R v

E Ee E E E E e E E e E E

EE e E E EE e E E e t a v

F EE EE e E E e

EE EE e E E e E

θ

θ

θ

−− − − − + + − +

∂ + −− − − −− + +  

  =

  ∂ + − −− + +

+− −− + + − { } ( )

() ( ) { }

( ) ( )( ) { } ( )

2

2 3 21 1 2

2 2

2 1 21 1 2 21 1 2

sin 3 1 2 cos cos . sin sin 1 2cos 1 sin sin 3 sin sin

c

E

EE e e E E a v

F e E E E E e E E E Ee E E

θ

               

 − +− + 

=   − − − −+ − − −   

Now 

() () ( )

( )( ) ( ) ( )

{ } ( ) ( )

( )( )

1 2 1 1 21

2

21 1 2 2 1 2 1

21 2 1

22 2 3

2 1 12

cos cos 2cos cos 2cot sin cos cot sin sin

sin cos cos 2sin cos cot sin sin 2cos sin sin

1 cos sin sin

sin sin 3 sin sin 3

c

f c

R v

E E x xE x x xE x E E

E E E E xxx E E x E E

EE E E

t a v e E E ee E E e

F θ

θ

θ

+ = += += −

⇒ − + = −= −

=+ − −

  ∂ − − +− + +

⇒   = ∂ 

( ) ( )

( )

2 1

2

21 1 2

1 sin

3 sin sin

f c

c c f

R fR t v

E E

eE E E E

v v t

t θ

θ θ

θ θ

  −    

  − −  

  ∂ ∂ ∂

⇒   = − ∂ ∂∂  

which yields Equation (F-14). 

Equation (F-14) is analytic everywhere provided that 0. f

c

t

v θ

∂

≠

∂

 It will now be shown that 

< 0

∂

∂

cθ

f

v

t

 universally. From page 18 of the main text, 

( ) ( )

1 2 12

2 2

1 2 12 12 1 2 0

2

2 2 cos

2 lim cos cos 0, 5 sin 2 2

R

f R R

x c R

r r rr

t

r r rr rr r r

v r θ

θ

θ θ

→ μ θ

    + − ∂     =− + − + + <   ∂  
NSWCDD/TR-03/81 

F-10

which proves the proposition for a parabolic trajectory. For an elliptical trajectory, since 

x

v

x

t

v

t

c

f

c

f

∂

∂

∂

∂

= ∂

∂

θ θ

, where < 0

∂

∂

x

vcθ (from Equation (E-1)), it is desired to prove that 0. f t

x

∂

>

∂ Now, 

from Equation (5) of the main text, 

() ( ) { }

{ } ( )

( ) { } ( )( )

2 2

1 2

22 2

1 2

2 2

12 1 2

3csc 1 cot 1

1 csc 2 cos 6csc 5 3 cot 2 csc 1

2 2

cos 3 csc 4 csc 3 12 csc 1 cos

2

f R

R

rr x x x

t

x rr x x x x

x S

rr r r x x x x x

θ

μ

θ

 

  + −−

∂

= + −− −

∂

+ + −− −  

. 

To prove that the above expression is positive, the following theorem will be employed:∗

 For all 

real values of , u the expression Au + Bu + C 2 has the same sign as , A except when the roots of 

the equation 0 2 Au + Bu + C = are real and unequal, and u has a value lying between them. 

Putting 

( ) ( )( )

{ } ( )

1 2 2 22

1 2

2 2

sec , 3csc 1 cot 1, 3 csc 4csc 3 12csc 1 cos , 2

2 6csc 5 3 cot 2csc 1 ,

R r r u A x xx Bxx x x x

rr

C x xx x

+ θ = = − − = −− −

= −− −

we have 

x( ) Au Bu C

S

r r

x

t f R = + +

∂

∂ 1 2 2 2 csc

2

cos

2

θ

μ . 

Clearly, u > 1. Since A > 0 (from Equation (19) of the main text), the above theorem implies 

that > 0

∂

∂

x

t f when either 2 B AC − ≤ 4 0, or 0 4 2 B − AC > and , u w> where 

( ) B B AC

A

w 4

2

1 2 = − + − . Performing the computations, 

( )

( )( )

2 2 22 2 4 9 csc 6 cot cos 16 3 csc cos 16

3 csc cos 4 3 csc cos 4 .

B AC x x x x x x x x

xx x xx x

− = + + −= + −

= ++ +−

Now 

( )( ) 3 csc cos 4 3csc 1 cot sin sin 0, 0 d x x x x x x xA x x

dx

+ − = − − = > << π . 

 

∗

 Hall, H. S. and Knight, S. R., Higher Algebra, Fourth Edition, Macmillan, London, 1936. 
NSWCDD/TR-03/81 

F-11

Thus, 3x csc x + cos x − 4 increases monotonically in the interval [0 . Since ,π )

( ) 0

lim 3 csc cos 4 0,

x

xx x →+ −= it follows that 3x csc x + cos x − 4 ≥ 0, where 0 ≤ x < π . Hence, 

2 B AC − > 4 0. By expanding cos x, csc x, and cot x in powers of , x it can be shown that 

2

5 1 lim

0

− = →

w x. 

Also, 

() ( ) ( )

{ } ( )

2 2 22 2

3

12 sin cos sin 3 4 3sin sin 3 cos sin 16sin

lim lim 2

2 3 sin cos sin x x

x xxx x xx xx x

w

xx x x → → π π

− −− + + − = =− ⋅ − −

Plotting w versus x for 0 < x < π reveals that w decreases monotonically in the interval [0 . ,π )

Hence, 

5 1 2 1 0.

2

f t

w u

x

− ∂

−≤ ≤ << ⇒ >

∂

For a hyperbolic trajectory, 

y

v

y

t

v

t

c

f

c

f

∂

∂

∂

∂

= ∂

∂

θ θ

. Differentiating 2

sin 2

1

2 R c r S

r

v μ θ

θ ′ = , where 

2

2 cosh cos 1 2 1 2

R S r r r r y

θ ′ = + − , we have 3 2 sinh sin 0.

2

c

R

v

r y y S

θ μ θ ∂

= >

∂ ′ Thus, it is desired 

to prove that 0. f t

y

∂

<

∂ From page 16 of the main text, 

() ( ) { }

{ } ( )

( ) { } ( ) ()

( )

2 2

1 2

2 22

1 2

2 2

12 1 2

1 2 2 2

3csch coth 1 1

1 csch 2 cos 3 coth 2csch 1 6csch 5

2 2

cos 12csch 1 cosh 3 csch 4csch 3

2

cos csch ,

2 2

f R

R

R

r r yy y

t

y rr y y y y

y S

rr r r y y y y y

rr y Au Bu C S

θ

μ

θ

θ

μ

 

  + −−

∂

= + +− −

∂ ′

++ + − +  

= ++ ′ ′′ ′

where 

( ) ( ) ()

{ } ( )

2 22

2 2

3csch coth 1 1, 12csch 1 cosh 3 csch 4csch 3 ,

2 3 coth 2csch 1 6csch 5 .

A yy y B y y y y y

C yy y y

′ ′ = −− = + − +

′ = +− −
NSWCDD/TR-03/81 

F-12

Since A′ < 0 (from Equation (22) of the main text), the theorem on page F-10 implies that 

< 0

∂

∂

y

t f when either 2 B AC ′ ′′ − ≤ 4 0, or 0 4 2 B′ − A′C′ > and , u w> ′ where 

( ) 1 2 4

2

w B B AC

A ′ ′ ′ ′′ = −− − ′ . Performing the computations, 

( )

( )( )

2 2 22 2 4 9 csch 6 coth cosh 16 3 csch cosh 16

3 csch cosh 4 3 csch cosh 4 .

B AC y y y y y y y y

yy y yy y

′ ′′ − = + + −= + −

= ++ +−

Now 

( ) () 3 csch cosh 4 sinh 3csch coth 1 sinh 0, 0. d y y y y yy y A y y dy

+ − = − − =− > > ′

Thus, 3 csch cosh 4 yy y + − increases monotonically in the interval [0 . Since ,∞)

( ) 0

lim 3 csch cosh 4 0,

y

yy y →+ −= it follows that 3 csch cosh 4 0, where 0. yy y y + −≥ ≥ Hence, 

2 B AC ′ ′′ − > 4 0. By expanding cosh , csch , and coth yy y in powers of y, it can be shown that 

2

5 1 lim

0

− ′ = →

w

y

. 

However, 

() ( ) ( )

{ } ( )

2 2 22 2

3

12 sinh cosh sinh 3 4 3sinh sinh 3 cosh sinh 16sinh

lim lim , y y 2 sinh 3 cosh sinh

y y yy y y y y y y

w →∞ →∞ yy y y

+ −+ + + − ′ = = ∞ − −

and a snag is encountered. This snag can be circumvented by observing that 

( ) 1 2 12 12 2 cosh cos cos 2cosh 0 2cosh , 2 2

R R S r r rr y rr u y u y θ θ ′ =+− = − > ⇒ >

which suggests finding limy cosh

w

→∞ y

′ . Using the above expression for , w′ it can be shown that 

lim 1.

y cosh

w

→∞ y

′ = Since 

0

limcosh 1,

y

y →= it follows that 

0

5 1 limy cosh 2w

→ y

′ − = . Plotting w′sech y

versus y for 0 < y < ∞ reveals that w′sech y increases monotonically in the interval [0 . ,∞)

Hence, 

5 1 sech 1 cosh 0.

2 2

f u t

wy w y u

y

− ∂

≤ < ′ ′ ⇒ < << ⇒ <

∂
NSWCDD/TR-03/81 

F-13

Thus, it has been proven that < 0

∂

∂

cθ

f

v

t

 universally. A heuristic argument that this 

inequality is indeed valid was presented to the author by William Davis in a private 

communication. If either cr v or cθ v is increased while holding the other constant, θ R will 

increase. Therefore, to keep θ R constant, either cr v has to be increased while decreasing cθ v or 

vice versa. However, simultaneously increasing cr v and decreasing cθ v tends to increase f t , 

while simultaneously decreasing cr v and increasing cθ v tends to decrease f t . In either case, 

0. f

c

t

v θ

∂

<

∂

To recapitulate, given cθ , , and v 1 2 r r " " , the desired algorithm is 

( ) 1 1

1 2

21 2

22 2 2

2 1 1 2 12 1 1 2 12 2

2 2

1 2

arccos , cot csc tan , 1 cot , ˆ ˆ 2 2

2 sin 2 cos 2 cos 2 sin , 2 2 22

R R

R cr R R c tr c cr

c

R R RR

c c

c

r r

v v v vv

r rv r

F r rv r r rr rv r r rr r

rrv

a

θ θ

θ

θ θ

θ

μθ θ θ θθ

θ θ θθ

μ μ

μ

  

= ⋅ = − + =− −      

      = − +− ++ −            

=

r r

( )

{ } ( )( ) { } ( )( )

2 2 22 2

1 2 2

2

1 2

1 2 21

2 2

11 2 2

2

2 2 1

21 2 1 21 2 1

3

sin sin , , 1 ,

sin , sin , 2arctan ,

2 sin

2

sin sin sin 3 sin sin 16sin , 2

f

R cR

cr tr

R

c

f

c

c

R t

r rv b b e

F a F

rv rv F eE eE EE

a a rr rv r

t ab E E EE e E E EE e E E

v F

v a

F

θ

θ

θ

θ

θ θ

μ μ θ

μ

θ

= =−

= = −=

+ −

∂  −  = −+ − −− − −   ∂  

  ∂

  = ∂ 

( )( )

( ) ( )( )

2 2

2 1 12

2 2

21 21 1 2

sin sin 3 sin sin . 3 1 sin 3 sin sin

c

c

f

e E E ee E E v

v

t e E E eE E E E

θ

θ

  − +− ∂     ∂   − + −+ −  
NSWCDD/TR-03/81 

F-14

THIS PAGE INTENTIONALLY LEFT BLANK 
NSWCDD/TR-03/81 

G-1/G-2 

APPENDIX G 

A GENERAL EXPRESSION FOR THE NULL-MISS DIRECTION 

NSWCDD/TR-03/81 

G-3

The null-miss direction is defined by 

f

c

f

c

DK

dt

d

dt

d

v

v

P !

!

ˆ = − , 

where 

2

2

1 1

2

2 2

,

ˆ ˆ ,

,

f

cc c

ff f t

c cr c

ff f

f

d d

dt t dt

v v

tt t

d

dt

θ

∂ ∂ 

= +   ∂ ∂  

∂∂ ∂

= +

∂∂ ∂

= −∆ = ×

vv v r

r

v

r θ

r

r Ω r

! ! ! !

!

!

! ! ! !

2 2

2 22

2

1 0

cos sin 0

1 0 sin cos 0

0 01

0 0

sin

f f

ff f

cr cr

t t R

R R

cc c

R R

tt t R

c

R

v v

r r

v v

r r

v

r

θ θ

θ

θ

θ θ

θ θ

θ

θ

    ∂ ∂     ∂ ∂  

       ∂∂ ∂        = − ∂∂ ∂       

 

v

r

!

! . (G-1) 

f R t

c v

















∂

∂

θ

θ is given by either Equation (E-16) or (F-14), 

f R t

cr v

















∂

∂

θ is given by Equation (E-17), 

f t

c

r

v

















∂

∂

2

θ is given by Equation (E-19), and 

f t

cr

r

v

















∂

∂

2

 is given by Equation (E-20). We will now 

show that c

f

c

dt

d

v

v ! !

= −∆ , where vc

! ∆ is given by either Equation (7) or (8) of the main text. 

Suppose that the target vector 2r

! is perturbed by a small amount 2 δ r

! (Figure G-1). Let 2r′

!

be the new target vector, , and R RR δθ θ θ δ =− =− ′ ′ βββ . Then 2 222 2 ˆ ˆ R p δ δθ δ rrr =−≈ + ′ r r θ n ! ! ! , 

where sgn sgn ( ) δ δ rp = − ( ) β , the minus sign being attributed to the fact that a positive change in 

β points in the opposite direction as nˆ . From Figure G-1, we have 
NSWCDD/TR-03/81 

G-4

Figure G-1. Representation of a small perturbation 

in the target vector 

( )

2

sin sin sin p

R

r

r

δ = − θ δ ′ β . 

If these finite changes are made infinitesimal, then 

( )

2 22

2 2

sin sin , sin sin , 

sin , or sin .

p pp

RR RR

p

pR R

r dr dr

dd d

r rr

dr

dr r d r

d

δ δβ ββ θθ θθ

θβ θ β

= = = = =+ = ′

⇒ =− =−

From Equation (G-1), we have 

n

v ˆ

2 sin R

c

p

c

r

v

r θ

θ = ∂

∂

!

 ˆ c c p

c

p

dr

v

r d θ β β

∂ ∂

⇒ = =− ∂ ∂

v v

n

! !

. (G-2) 

We will now evaluate R

f

d

dt

θ and 

f

d

dt

β for which we need the following identities:∗

 θ R φ φT φ φT λ D cos = sin sin + cos cos cos , (G-3) 

 θ R β φT λ D sin sin = cos sin , (G-4) 

 cos cos cos cos sin sin cos , φTD R R λ = − φ θ φ θ β (G-5) 

 cos cos sin sin cos cos cos , φβ φ Tb R R = − θ φ θ β (G-6) 

 sin sin cos cos sin cos , φ φ T RR = + θ φ θ β (G-7) 

 

∗

 Sofair, Isaac, K40 Training Guide 4085.1, A Detailed Derivation of Formulae Arising in Spherical Trigonometry, 

Naval Surface Weapons Center, Dahlgren, VA, May 1986 (currently Naval Surface Warfare Center, Dahlgren 

Division). 

δβ

θ R

′

nˆ

2r

rδ p

δθ R

θ R

β β ′

2

2

r

r

!

δ

λ D

φ T

π − 2 φ

π − 2
NSWCDD/TR-03/81 

G-5

where , and λ λ λλ λ DT T = − being the longitudes at release and impact, respectively; T φ andφ

are the corresponding spherical latitudes (i.e., latitudes corresponding to a spherical earth), and 

β b is the back bearing. Differentiating Equation (G-3) with respect to f t and using 

Equation (G-4) and the fact that = Ω

f

D

dt

dλ , we have 

φ θ β λφ φ λ θ sinθ cos cos sin cos sin R sin

f

D

T D

f

R

R dt

d

dt

d − = − = −Ω

 φ β θ

⇒ = Ωcos sin

f

R

dt

d . (G-8) 

Differentiating Equation (G-4) with respect to f t and using Equations (G-5), (G-6), and (G-8), 

we have 

R T D

f

R dt

d

φ θ β φ λ β sinθ cos β cos cos sin cos cos 2 + Ω = Ω

( )

( )

( )

2

2

2

sin cos cos cos cos cos sin

cos cos cos cos cos cos cos

cos cos cos sin sin cos

cos cos cos

R TD R

f

TD R R

R R

T b

d

dt

β θ β φ λ φθ β

φ λ φ θ φ θ β

φ θ β φθ β

φ ββ

⇒ =Ω −

=Ω − +

=Ω −

= −Ω

 R T b dt f

d θ φ β β ⇒ = −Ωcsc cos cos . (G-9) 

Now if Pˆ is a unit vector in the direction of the North Pole, then Ω = ΩPˆ ! . 

( )

( )

2 2 21 2 1 2 ˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ csc cot csc cot

sin csc sin cot

sin csc sin cos cos sin cos cot (from Equation (G-7))

sin sin cos cos cos cos cos (from Equa

RR R R

R TR

R R RR

R R Tb

θθ θ θ

φθ φ θ

φθ φ θ φθ β θ

φθ φ θ β φ β

× ⋅ = ⋅ × =− ⋅ = ⋅ − = ⋅ − ⋅

= −

=− +

=− =

P r n Pr n P θ P r r Pr Pr

tion (G-6))

2 22 2 ˆ ˆ ˆˆ sin (from Equation (G-9)) R

f

d

r r

dt

β ⇒ Ω × ⋅ =Ω × ⋅ =− rn Prn θ ! ! . (G-10) 

Since β is the angle between 1 12 ˆ

Pr r r × × ˆ ˆˆ and , it follows that 

( ) ( )

( )( ) ( )( )

1 1 2 11 2

121 112 1 21 1 12

1 2

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ sin cos sin sin

ˆˆ ˆˆ ˆˆˆ ˆˆˆ ˆ ˆˆ ˆ ˆˆ

ˆˆ ˆ ˆˆ ˆ ˆ sin cos sin .

R

R

β φθ β

θ φβ

× × × =× × =

= ×⋅ − ×⋅ = ⋅× − ⋅×

=⋅× =⋅ ⇒ ⋅ =

P r r r P rr r

P r r r P r r r Pr r r Pr r r

Pr r Pn Pn
NSWCDD/TR-03/81 

G-6

Referring to page 7 of the main text and Equation (G-8), we have 

 ˆ ˆ ˆ cos sin R

R

f

d

dt

θ ∆ = − ⋅ = −Ω ⋅ = −Ω = − θ φβ Ω n Pn ! , (G-11) 

 































∂

∂ − ∂

∂ = −

∂

∂

∆ 















∂

∂

+

∆ = −

f

R

R x

f

f f

R

R x

f

dt

t d

t

x

x

t

t

x

θ

θ

θ

θ

1

1

. (G-12) 

From page 8 of the main text and Equations (G-10) and (G-11), 

( ) 2

2 2 12 1 2 ˆ sin cos ˆ ˆˆ RR RR

f

d

r r

dt

= ×= ∆ − ∆ + ×⋅ θθ θθ r Ω r r θ Ω r nn

! ! ! ! !

 2 12 12 ˆ sin cos sin ˆ ˆ R R

RRR

f ff

ddd

rr r

dt dt dt

θ θ β =− + − θθθ r θ n . (G-13) 

From Equations (G-1), (G-2), and (G-13), it is easy to see that 

2

1 1

2

ˆ ˆ ˆ

ff f f

c cr R c R c R c

c

f Rf Rf f Rf f tt t t

d vd vd d d d

v

dt dt dt dt dt dt

θ

θ

θ θ β θ β

θ θ θβ

       ∂ ∂ ∂ ∂∂        = + −= +

∂∂ ∂ ∂∂       

vr v v

r θ n

r

! !! !

! . 

2

2 f f

c c c c c Rc

ff ff R f f t t

d d dd

dt t dt t dt dt

θ β

θ β

∂∂ ∂∂ ∂   

∴ =+ =+ +    ∂∂ ∂∂ ∂   

vv v rv v v

r

!! ! ! ! ! !

!

 1 1 ˆ ˆ ˆ cr c

c

ff f

dv dv d

v

dt dt dt

θ

θ

β =+ − r θ n . (G-14) 

Now 

1

f f

f

cr cr cr R cr cr cr R

ff Rf f R R f t xt

cr R cr R cr R f

f R f Rf f Rf tx x

dv v v d v x v v x d

dt t dt x t x dt

v x x d v d vx d t

x t dt dt x t dt

θ θ

θ θθ

θθ θ

θθ θ

  ∂ ∂ ∂∂ ∂ ∂ ∂      =+ = + +      ∂ ∂ ∂∂ ∂ ∂ ∂       

∂ ∂ ∂ ∂ ∂∂         ∂

=+ + = −      ∂ ∂ ∂ ∂ ∂∂ ∂        

 (from Equations (G-11) and (G-12)).

cr R

R f x

cr cr

R

R x

v d

dt

v v

x

x

θ

θ

θ

θ

   ∂  +     ∂   

∂ ∂  =− ∆ − ∆   ∂ ∂ 

Substituting for cr v

x

∂

∂

 and cr

R x

v

θ

  ∂

  ∂ 

 from Equations (E-4) and (E-5) and using Equation (8) of the 

main text, we obtain 
NSWCDD/TR-03/81 

G-7

2

3 1 2 12

2 2

3 1 2 12

1

2 sin sin cos cos

2 2

 sin sin cos cos

22 2

cr R R

f

R R

R

dv

x r r rr x x

dt S

r

r x r rr x

r S

μθ θ

μθ θ θ

  =− + − ∆    

  + +− ∆    

 1

ˆ = −∆ ⋅ c v r

! . (G-15) 

Similarly, from Equations (E-1) and (E-3), we have 

( ) 2 2

3 3 2 1 2 12

1

sin sin cos cos 1 cos

2 22 2

cc c

R

f R x

R R

R R

dv v v

x

dt x

r

r x x r r rr x

S rS

θθ θ θ

θ

μ μθ θ θ θ

∂ ∂  =− ∆ − ∆   ∂ ∂  

    = ∆− + − + ∆        

1

ˆ = −∆ ⋅ c v θ ! . (G-16) 

From Equation (F-7) and the expression for g in Appendix C, namely, 

R

p

r r

g θ

μ

sin 1 2 = , 

we have 

 

θ

θ

c

R

v

r

g

sin 2 = . (G-17) 

Substituting Equation (G-17) into Equation (G-10), we obtain 

2 ˆ ˆ c c

f

d

v

g dt θ

× ⋅ β = − = −∆ ⋅ Ω r n

v n

! ! !

 ˆ c c

f

d

v

dt θ

β ⇒ =∆ ⋅ v n

! . (G-18) 

Substituting Equations (G-15), (G-16), and (G-18) into Equation (G-14), we finally obtain 

( )11 1 1 ( ) ( ) ˆ ˆ ˆˆ ˆˆ c

cc c c

f

d

dt

= − ∆ ⋅ − ∆ ⋅ − ∆ ⋅ = −∆ v

v rr v θ θ v nn v

! !! ! ! , 

which proves the identity. 

Note that the expression for PDK

ˆ on page G-3 is general in that it is applicable to all three 

types of trajectories. 
NSWCDD/TR-03/81 

G-8

THIS PAGE INTENTIONALLY LEFT BLANK 
NSWCDD/TR-03/81 

H-1/H-2 

APPENDIX H 

DETERMINATION OF FOOTPRINT ACHIEVABILITY IN CONNECTION WITH 

TIME OF FLIGHT AND CANTING ANGLE 

NSWCDD/TR-03/81 

H-3

Assume a scenario consisting of two targets whose spacing is δ s. Let 1r

! be the position 

vector at first release, 2 r

! the first target vector, 2 r′! the second target vector, and θ R β the range 

and bearing corresponding to the first target, and and θ R

′ ′ β the range and bearing corresponding 

to the second target (Figure H-1). 

Figure H-1. Canting angle as a measure of the 

deviation of 2r′

! from the downrange arc, R θ

The canting angle, αc , measured clockwise from the downrange direction at first impact, is 

the angle between the trajectory plane formed by 1 2 r r and ! ! and the plane formed by 2 2 r r and ′ ! ! . 

Its domain is 0 2 ≤ < α π c . It is easy to show that 

( )( ) ( )

( )( )

( )

( )

12 22

12 22

sgn ˆˆ ˆˆ sin sin

arctan arctan ˆˆ ˆˆ cos sin cos sin cos

R

c

RR R R

δβ θ δβ α

θθ δβ θ θ

××× ′ ′ = = ×⋅× − ′ ′′rr rr

rr rr . 

For a given canting angle, αc , it is possible to predict the relative capability of a missile for 

different times of flight by determining the relative sizes of the velocity gains necessary to 

produce a given increment in impact position. Minimum velocity gain reflects optimum missile 

capability. Specifically, the magnitudes of the minimum velocity changes necessary for the 

given times of flight must be compared. The minimum velocity change is achieved by 

considering the effect of changing f t . 

Suppose that a definite trajectory defined by 1 2 , , and f r r t ! ! has been selected, as well as a 

fixed small change in impact position, 2 δ r

! . For any small change in f f δ t t , we have, using the 

results of Appendix G, 

αc

δβ

θ R

′

nˆ

2r

rδ p

θ R

β β ′

δθ R

2

s

r

δ
NSWCDD/TR-03/81 

H-4

2

2

cc c c c ˆ c f DK f R

f fR

d d

t t

dt dt

δ δ δ δ δθ δβ θ β

∂ ∂∂

= + =− + +

∂ ∂∂

vv v v v v rP

r

!! ! ! ! ! ! ! , 

where it is implied that and c c

θ R β

∂ ∂

∂ ∂

v v

! !

 are evaluated at constant f t . The value of f δ t that will 

minimize c δ v

! is the one for which c δ v! is perpendicular to , c

f

d

dt

v

!

 i.e., 

1

c cc c 0, or 0 ˆ ˆˆ c c DK f DK R DK

f fR

d d

t

dt dt

δ δ δ δθ δβ θ β

−

  ∂ ∂ ⋅= ⋅= ⇒ = ⋅ +⋅   ∂ ∂  

v vv v v vP P P

! !! ! ! ! . 

Substituting this value of f δ t into the expression for c δ v

! , we obtain 

( )min

c cc c ˆˆ ˆˆ c DK DK R DK DK

R R

δ δθ δβ θ θβ β

    ∂ ∂∂ ∂     

= − ⋅ +− ⋅            ∂ ∂∂ ∂      

v vv v v PP PP

! !! ! ! . 

If these finite changes are made infinitesimal, then 

( )

( ) { } ( )

min

min

ˆ ˆ ,

ˆˆ ˆˆ ˆ ˆ , using Equation (G-2).

c c c

DK DK

RR R

c c c

DK DK c DK DK

d

d

d

v

d θ

θθ θ

ββ β

∂ ∂   =− ⋅   ∂ ∂  

∂ ∂   = − ⋅ = ⋅−   ∂ ∂  

v v v P P

v v v P P PPnn

! ! !

! ! !

From Figure H-1, we have 

( ) ( )

2 22 2

sin sin sin sin sin , cos sin sin cos p p

R c Rc

r r s s

r rr r

δ δ δ δ =− =− = θ δ ′ β α δθ α . 

If these finite changes are made infinitesimal, the above equations reduce to 

2 2

2 2

cos sin sin sin , cos , sin

Rc c

p R cR c

R

dr r d ds r d ds

s r sr

θ α β α θβ α θ α θ∂ ∂ =− =− = ⇒ = = ∂ ∂ . 

Hence, 

( ) ( ) ( ) ( ) ( )

{ } ( )

min min min min min

2

2

1 sin

cos

sin

1 sin ˆˆ ˆˆ cos . ˆ ˆ sin

cc c c c R c

c

R RR

c cc

c DK DK c DK DK

R RR

dd d d d

ds d s d s r d d

v

r

θ

θβ α α

θ β θ θβ

α

α

θ θθ

∂ ∂  

= += +   ∂ ∂  

      ∂ ∂  

= − ⋅ + ⋅−         ∂ ∂      

vv v v v

v v P P PPnn

!! ! ! !

! !
NSWCDD/TR-03/81 

H-5

With the knowledge that a lower value of ( ) c min d

ds

v

!

 implies a greater missile capability, 

the values of ( ) c min d

ds

v

!

 corresponding to a specific canting angle for different times of flight can 

be compared. Thus, ( ) c min d

ds

v

!

 can be used to determine the dependence of missile capability on 

the time of flight. 
NSWCDD/TR-03/81 

H-6

THIS PAGE INTENTIONALLY LEFT BLANK 
NSWCDD/TR-03/81 

I-1/I-2 

APPENDIX I

COMPUTATION OF THE RATE OF CHANGE OF THE NULL-MISS VECTOR 

NSWCDD/TR-03/81 

I-3 

Since ˆ

PDK is the direction in which the guidance system is attempting to steer the missile, 

DK

f

d

dt

P



 predicts how the missile is turning, i.e., how the direction of thrust is changing. In this 

appendix, a closed-form expression for DK

f

d

dt

P



 in the local coordinate frame will be derived. The 

results of Appendices E, G, J, and K will be used. In Appendix G, it was shown that 

1 1 ˆ ˆ ˆ c cr cr R c c R

DK c

f f Rf f Rf f

d v vd v vd d

v

dt t dt t dt dt

θ θ

θ

θ θ β

θ θ

  ∂∂ ∂∂

=− =− + − + +    ∂∂ ∂∂   

v P r θ n

  , (I-1) 

where and cr c

R R

v v θ

θ θ

∂ ∂

∂ ∂ imply and 

f f

cr c

R R t t

v v θ

θ θ

∂ ∂   ∂ ∂  

, respectively. The derivative of PDK

 with 

respect to f t can be expressed as 

1 1

1 1

DK DK DK DK DK DK R DK b

f f f f f Rf bf

d d dr d d

dt t dt t r dt dt dt

θ β

θ β

∂∂ ∂∂ ∂ ∂

=+ =+ + +

∂∂ ∂∂ ∂ ∂

P P Pr P P P P

r

    



1

DK DK DK R DK b

cr

f Rf bf

d d

v

t r dt dt

θ β

θ β

∂ ∂∂ ∂

=+ + +

∂ ∂∂ ∂

P PP P

  

, (I-2) 

where the partials are with respect to a fixed target 2 r

 and a changing initial position 1r . 

We now proceed to derive expressions for 

1

, , , and DK DK DK DK

f Rb t r θ β

∂∂∂ ∂

∂ ∂∂ ∂

PPP P

  

. The first two 

partials are straightforward. A change in f t without changing 1r

 does not change the local 

coordinate frame. Since 1 2 r r and   are both constant in this case, and θ R β – and therefore 

 and R

f f

d d

dt dt

θ β – remain constant  0. R

f f ff

d d

t dt t dt

∂ ∂ θ β = = ∂ ∂ Hence, 

22 22

2 2 1 1 ˆ ˆ ˆ DK cr cr R c c R c

f f Rf f f Rf f f f

v v d v v d vd

t t t dt t t dt t dt

θ θ θθ θ β

θ θ

∂ ∂∂ ∂∂ ∂   

=− + − + +    ∂ ∂ ∂∂ ∂ ∂∂ ∂   

P

r θ n



. (I-3) 

Similarly, a change in 1

r does not change the local frame. Since and θ R β – and hence their 

derivatives – are independent of 1

r , 

1 1

0. R

f f

d d

r dt r dt

∂ ∂ θ β = = ∂ ∂ Hence, 

22 22

1 1

1 11 11 1

ˆ ˆ ˆ DK cr cr R c c R c

f Rf f Rf f

v v d v v d vd

r r t r dt r t r dt r dt

θ θ θθ θ β

θ θ

∂ ∂∂ ∂∂ ∂   

=− + − + +    ∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂   

P

r θ n



. (I-4) 
NSWCDD/TR-03/81 

I-4 

In computing DK

θ R

∂

∂

P



, observe that a change in R R δθ θ is due to a change in 1r

 so that the 

local coordinate frame changes. Let 1 1 ˆ rˆ ˆ ′ ′ , , θ n be the new local coordinate frame. Then, 

11 1 () () ˆ ˆ ˆ cos sin δθ δθ R R r r ′ ≈ − θ , the minus sign being attributed to the fact that a positive change in 

θ R points in the opposite direction as 1 θˆ . If R δθ is made infinitesimal, then 

1 1 1 1 1 11 ˆˆ ˆ ˆ ˆ ˆˆ ˆ , R R r r ′ ′′ =− =×= + θ θ d d θ θ n r θ r . Let DK P′

 be the counterpart of PDK in the new frame. 

Neglecting terms in ( )2

R dθ , we have 

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

1 1

1 1 11

ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ

DK

DK DK R

R

DK r DK DK p

DK R DK R DK R r p

RR R

DK r DK

DK R R DK R R r

R R

DK p

DK R p

R

d

P P P

P dP d P d

P P

P d dP d d

P

P d

θ

θ

θ

θ

θ

θ

θθ θ

θθ θ

θθ θθ

θ θ

θ

θ

∂ ′ = +

∂

 ∂  ∂  ∂   =+ ++ ++      ′ ′ ∂∂ ∂       

 ∂  ∂

= + −+ + +    ∂ ∂   

 ∂   + +   ∂    

P P P

r θ n

r θ θ r

n

  

( ) ( ) ( ) ( ) ( )

1 1 ˆ ˆ ˆ DK r DK DK p

DK DK DK R r

R RR

P P P

P Pd θ

θ θ

θ θθ

 	  ∂  ∂ ∂

=+ + + − + 
     ∂ ∂∂  
   P r θ n 

( ) ( ) ( ) ( ) ( )

1 1 ˆ ˆ ˆ DK DK r DK DK p

DK DK r

RR R R

P P P

P P θ

θ θθ θ θ

∂  ∂  ∂ ∂

 =++−+    ∂∂ ∂ ∂   

P

r θ n



 

2 2

2 1

2 2

2 1

ˆ

ˆ

+ . ˆ

cr c cr c R

Rf f R R f

c cr c cr R

Rf f R R f

c

c

Rf Rf

v v v vd

t t dt

v v v vd

t t dt

vd d

v

dt dt

θ θ

θ θ

θ

θ

θ

θ θθ

θ

θ θθ

β β

θ θ

   ∂ ∂ ∂∂  

=− + + +    	   ∂∂ ∂ ∂ ∂  
 
 

   ∂ ∂∂∂  

− −+ −    	   ∂∂ ∂ ∂ ∂  
 
 

  ∂ ∂  	 +

∂ ∂ 
 

r

θ

n

 (I-5) 

Note that Equation (I-5) should contain additional terms involving R

R f

d

dt

θ

θ

∂

∂ ; however, it is 

shown in Appendix K that 0. R

R f

d

dt

θ

θ

∂ = ∂
NSWCDD/TR-03/81 

I-5 

Similarly, in computing DK

βb

∂

∂

P



, a change in βb is due to a change in 1r

 . Let 1 1 ˆ rˆ ˆ ′′ ′ , , θ n be 

the new local coordinate frame. Consider a perturbation 1 δ r

 resulting in a small change b δβ

without changing θ R . By an analysis similar to that on pages G-3 and G-4 for a perturbation in 

the target vector, 1 11 1 ˆ p δ δ rrr n =−≈ ′ r  . Since a positive change in βb points in the same direction 

as nˆ , ( ) ( ) ( ) 1

1

1

sgn sgn sin sin sin p

p b Rb

r

r

r

δ δ δ = β  = θ δβ . Also, 2 2 () () ˆ ˆ cos sin ˆ θ θ ′ ≈ − δβb b n δβ . 

If these finite changes are made infinitesimal, then 

1 1 11

22 1 1 1 1

sin sin , ˆˆˆ

ˆˆ ˆ ˆ ˆ ˆ ˆˆ cos sin cos sin

p Rb Rb

b R R R Rb

dr r d d

d d

θ β θ β

β θθ θθ β

=  ′ = +

′ ′′ = −  −= −−

rrn

θ θ n θ r θ r n

( ) ( ) 2

1 1 11 1

2

1 11

ˆˆ ˆ cos cos sin cos 1 sin ˆˆ ˆ ˆ

ˆ ˆˆ cos cos cos , ˆ ˆ

R R R b R Rb

R Rb Rb

d d

d d

θθ θ β θ θ β

θ θβ θβ

 ′ ′ = +− − = − −

= −  ′ = −

θ θ rr n θ n

θ n θ θ n

11 1 1 1 1 2 ( ) ( )( ) ˆ ˆˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆ sin cos cos sin Rb Rb R R b b n r ′′′ =× = + × − =+ − =+ θ r n θ d d dd β θ n n θ β θ θ θ r n β β θ . 

Again letting DK P′

 be the counterpart of PDK in the new frame and neglecting terms in ( )2

b dβ , 

we have 

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )() ( ) ( )

( ) ( )

1 1

1 1

ˆ ˆ ˆ

ˆ ˆˆ ˆ sin cos

DK

DK DK b

b

DK r DK DK p

DK b DK b DK b r p

bb b

DK r DK

DK b R b DK b R b r

b b

DK p

DK b p

b

d

P P P

P dP d P d

P P

P d dP d d

P

P d

θ

θ

θ

θ

β β

ββ β ββ β

β θβ β θβ β β

β β

∂ ′ = +

∂

 ∂  ∂  ∂   =+ ++ ++      ′′ ′ ∂∂ ∂       

 ∂  ∂

=+ + ++ −    ∂ ∂   

 ∂  + +  ∂

P P P

r θ n

r n θ n

  

{ } ( )

( ) ( ) ( ) ( )

( ) () ()

1 1

1 1

ˆ ˆ ˆ cos sin

ˆ sin cos ˆ

sin cos ˆ

R Rb

DK r DK

DK R DK R p p

b b

DK b

DK p

DK R DK R r

b

d

P P

P P

d

P

P P

θ

θ

θ θ β

θ θ

β β

β

θ θ

β



 + −

   

 	  ∂  ∂ 
     − ++

∂ ∂   

= +  ∂  ++ −   ∂  
    

n θ r

r θ

P

n


NSWCDD/TR-03/81 

I-6 

( ) ( ) ( ) ( )

( ) () ()

1 1 ˆ sin cos ˆ

sin cos .ˆ

DK DK r DK

DK R DK R p p

bb b

DK p

DK R DK R r

b

P P

P P

P

P P

θ

θ

θ θ

ββ β

θ θ

β

∂  ∂  ∂

 =− ++    ∂∂ ∂   

 ∂  ++ −   ∂    

P

r θ

n



Since and cr c v v θ – and hence their derivatives – do not change with either or β βb when θ R is 

fixed, it follows that 

0. c cr cr c c

b bf bR bf b R

vvvvv

t t

θ θθ

β β βθ β βθ

∂ ∂∂ ∂∂ ∂∂ ∂∂ ===== ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂

Hence, 

1 1 ˆ sin cos ˆ

+ cos sin cos sin . ˆ

DK cr R c R

cR c R

b Rbf f R bf f

c cr c cr R

c RR RR

bf f f R R f

vd d vd d

v v

dt dt dt dt

dv v v v d

v

dt t t dt

θ

θ θ

θ θ

θ

θβ θ β θ θ

β θβ θβ

β θ θθ θθ

β θθ

∂ ∂∂ ∂∂   

=− + − −    ∂ ∂∂ ∂∂   

	 	   ∂∂ ∂ ∂ ∂  
  + −+ −   	 	 ∂∂ ∂ ∂ ∂  
  

P

r θ

n



 (I-6) 

Substituting Equations (I-3), (I-4), (I-5), and (I-6) into Equation (I-2), we finally obtain 

22 2 2 2

2 2

1 1

1

2 2

2

2

ˆ

sin

cr cr cr cr c cr c R R

cr cr

f f Rf R f R R f f DK

f cr b R

c R

Rbf f f

c c

cr

f

v v v v v vv d d

v v

t r t t r t dt dt d

dt v d d d

v

dt dt dt

v v

v

t

θ θ

θ

θ θ

θ θ

θ θ θθ

θ β β θ

θ β

 ∂ ∂ ∂ ∂ ∂ ∂∂        	 + + + ++ + 
   
 ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂       = −   ∂ ∂

+ +  
 ∂ ∂    

∂ ∂

+

∂ −

P

r



22 2

2

1 1

1

2

ˆ

cos

+

c c cr c cr R R

cr

f Rf R f R R f f

c b R

c R

Rbf f f

R b c

c

f Rf f bf

v v v vv d d

v

r t t r t dt dt

v d d d

v

dt dt dt

d d d d v

v

dt dt dt dt

θθ θ

θ

θ

θ

θ θ

θ θ θθ

θ β β θ

θ β

θ β β β

θ β

     ∂ ∂ ∂∂∂    	 + + −+ − 
   
 ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂      

  ∂ ∂

+ −  
 ∂ ∂    

  ∂ ∂ ∂  
 + +

∂ ∂  

θ

1

ˆ,

cos sin cos sin

c c R

cr

f Rf f

c cr c cr b R

RR RR

f f R R ff

v v d d

v

t dt r dt

vv vv d d

t t dt dt

θθ θ

θ θ

θ β

θ

θ β θθ θθ

θ θ

   ∂ ∂  	  
 + +

∂∂ ∂  

    ∂∂ ∂∂  

+ −+ − 
   
   ∂∂ ∂∂      

n

where all the required derivatives are derived in Appendices E, G, J, and K. 
NSWCDD/TR-03/81 

J-1/J-2

APPENDIX J 

DERIVATION OF SECOND PARTIALS OF c cr v v θ AND 

NSWCDD/TR-03/81 

J-3

Put ( ) ( ) 1 1 ,,,, cr c f R R v vt r r θ = ζ θθ . Using the equations 

1 1

21 2

cot csc tan , 1 cot 2 2

R R

cr R R c tr c cr

c

r r

v v v vv

r rv r

θ θ

θ

μθ θ θ θ   

= − + =− −      

, 

we have 

1

2

1 1

2 2

2 2

2

1 tan , csc (from Equation (E-17)), 2

1 cot 1 tan , 2 2

1 , 

tr R

tr R

cc R

tr R cr tr cr R

c c c cc

tr tr

c c c c cR

v r

v

vv r

v r vv v r

v r v v vv r

v v

vv vv v

θ θ

θ θ θ θθ

θ θ θθ θ

ζ θζ θ

θ

θζ ζ θ

ζ ζ

θ

∂ ∂  

= − + =−   ∂ ∂  

∂ ∂ +∂  

=− − = − = ++   ∂ ∂∂  

∂∂ ∂   = − =−   ∂ ∂ ∂∂  

csc , tr

R

c

v

v θ

θ ∂

∂

( ) ( )

{ } ( )( )

( )

2

1 2

2

2

2

2

1 cot 1 csc csc

2 2

1

cot csc tan csc csc

2 22

cot csc csc csc

cos csc .

R

tr R c R

R R

R R

tr R R cr tr R

tr R R cr tr R R

cr tr R R

r

v v

r

v vv

v vv

v v

θ

ζζ θ θ θ

θ θ

θ θ θθ θ

θ θ θθ

θ θ

∂ ∂      

= ++−     ∂ ∂      

  = − ++    

= − ++

= +

1 1

1

1

2 2

, ,

1 , cot (Equations (E-17) and (E-18)),

1 3

2 2

cr c tr tr c

f cf f cf

cr c cr cr

c R

R R cR cR

tr c cr c

c f

R Rf

v v v vv

t vt t vt

v vv v

v

v r rv

v vr v v

r vt

rr t

θ θ

θ θ

θ

θ

θ θ

θ θ

θ

ζ

ζζ ζ θ

θθ θ θ

θ θ

∂ ∂∂ ∂ ∂∂ = = ∂ ∂∂ ∂ ∂∂

∂ ∂ ∂∂ ∂ ∂∂  

= + =− +   ∂ ∂ ∂∂ ∂ ∂∂  

∂∂ ∂ ∂  

= = +−  ∂∂ ∂ ∂ 

 

 (by Equations (E-13) and (E-19)). 

Another expression for tr

R

v

θ

∂

∂ is obtained by direct differentiation of the expression for tr v above. 

We have 

1 1 2

2 2

1 1 2

2 2

1 1 csc 1 cot

22 2

1 1 csc 1 cot

222

csc .

tr cr R c R

c

RR R

R Rc

c

R cR

tr c

cr R

c R

v v r rv

v

r r

r rv

v

r rv

v v

v

v

θ

θ

θ

θ

θ

θ

θ

θ θ

θθ θ

ζ θ θζ

θ θ

θ

θ

∂∂ ∂  

=− − − + −   ∂∂ ∂  

∂ ∂∂       =− − − + − −     ∂ ∂∂      

∂ ∂

=− +

∂ ∂
NSWCDD/TR-03/81 

J-4

2

2

f

c

t

v θ

∂

∂ is obtained by differentiating Equation (14) of the main text. To facilitate the computation, 

it is convenient to define some recurring terms. Put 

( ) 2 33 3

1 1 2 1 12 2 1 2 , 2 cos , 2 sin , 4 sin 2 2

R R A r r r B r rr C r D r r R

θ θ

=+ = = = μ μ θ . 

Then { ( ) }{( ) } 2 2 F =−− + − C A Bv A Bv C c c θ θ and Equations (13) and (14) of the main text can be 

expressed as 

( )

( )

2 22 2

2 2

2 24 2 2

2

2 2

arctan , 2

2

arctan .

c c c

f

c

f f c c

c

c cc

Dv v F AC A B v

t

F B C Av C F

t t A Bv C Dv F CD

v

v F v Av C F F F

θ θ θ

θ

θ θ

θ

θ θθ

    − − = +   −    

∂ − −  

=+−   ∂ −  

Differentiating the second expression yields 

( ) ( )( )

( )( )

( )

3 2 2 8 2 2 2 2 24

2 2 3 2 22 4 2 2

2 3

2 24

2 24 2 2

2

1 2

2

1 3 27

8 2 38 .

10 3 2 5 arctan

c c

f c c

c

c

c c

c

c c f

f ff f c

A B v CA B A Bv

t Dv B C AC A B v C A B

v F

BCv F F A B v ACv C

F Av C

v v t

t tt t v

θ θ

θ θ

θ

θ

θ θ

θ

θ θ

θ

−

    −−− −       ∂   + −−+ = −    

∂  

+ − + +−     −  

∂ ∂∂ ∂   ∂ = =   ∂ ∂∂ ∂ ∂ 

1 3 2

2 . f f c c

cc f f c

t t v v

vv t t v

θ θ

θθ θ

−

∂ ∂∂   ∂ ∂   = =−     ∂∂ ∂ ∂ ∂      

Differentiating Equation (E-16), namely, 

2

1

1

2

1 3

2 2

1

c

cr c cr f

tr f c

R cr

c

tr

r v

vv r v t

v v t

rv

v

r v

θ

θ

θ

θ

θ

    ∂ −− −     ∂ ∂     = ∂  

  −

 

, 

with respect to and f R t θ , respectively, we obtain 

2 2

2

1 12

2 2

2

2 1

2 2

1

1 1

1

+

1 1

2

c c cc f

Rf f c c f R

cr cr tr c cr cr tr c

cr

cr f tr f R c cr f tr f f

cr c f

cr tr f c

v v vv t

t tv vt

v rv v v r v rv v v

v

v t rv t v v t v t t

rv v t

v r v t v

θ θ θθ

θ θ

θ θ

θ

θ

θ

θ θ

θ

∂ ∂ ∂∂   ∂ = −   ∂∂ ∂ ∂ ∂ ∂  

     ∂ ∂∂ ∂ ∂ ∂      − + −−

∂ ∂∂ ∂ ∂ ∂     

 ∂ ∂ − −  ∂ ∂ 

,






NSWCDD/TR-03/81 

J-5

2 2

2

1 12

2 2

2

2 1

2

1

1 21

1

+

1 1

2

c c cc f

R Rf c c R R

cr cr tr cr c cr cr tr c

cr R tr R c R c cr R tr R f

cr c f

cr tr Rf c

v v vv t

tv v

v rv v v v r v rv v v

v rv v v v v t

rv v t

v r v t v

θ θ θθ

θ θ

θ θ

θ θ

θ

θ

θ θ θθ

θθ θ θθ

θ

∂ ∂ ∂∂   ∂ = −   ∂ ∂∂∂ ∂ ∂  

     ∂ ∂ ∂ ∂ ∂∂    −+ + −

∂ ∂ ∂ ∂ ∂∂      ∂ ∂ −  −  ∂∂∂ 

.











2

22 2

22 2

2 2 22

2

22 2 2 2

22 2 2

,

,

2 .

cr c c

f c f cf

cr c c c

Rf c Rf c R c R f

cr c c c

R R c R cR c R R

vv v

t v t vt

v v vv

tv t v v t

v v vv

v vv

θ θ

θ θ

θ θθ

θ θθ

θ θθ

θ θθ

ζ ζ

ζ ζζ

θ θ θθ

ζζ ζ ζ

θ θ θ θ θθ

∂ ∂ ∂ ∂∂  

= +   ∂ ∂ ∂ ∂∂    

∂ ∂ ∂ ∂ ∂∂ ∂  

= ++   ∂∂ ∂ ∂∂ ∂∂ ∂ ∂ ∂  

∂ ∂ ∂∂ ∂ ∂ ∂ ∂  

=+ + +   ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂  

Differentiating Equation (E-8′), namely, 

1 1

1 c cr

c

R

v v

v

r r

θ

θ θ

∂ ∂ 

=− +   ∂ ∂ 

, 

we obtain 

2 2 22

2

11 1 1

1 1 , c cr c c cr c

f Rf f R R R

v v v v vv

rt r t t r r

θ θθ θ

θ θ θθ

∂ ∂ ∂ ∂ ∂∂    

=− + =− +     ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂    

. 

Differentiating Equation (E-18), namely, 

1 1

1

cot cr cr

c R

c R

v v

v

r rv

θ

θ

ζ θ

θ

∂ ∂∂  

=− +   ∂ ∂∂  

, 

we obtain 

2 22

2

1 1

2 22 2

2

2 2

1 1

1

cot ,

1

cot csc .

cr cr cr c

R

f c Rf c R f

cr c cr cr c

Rc R

R cR c R R c R R

v vv v

rt r v t v t

v vv v v

v

r rv v v

θ

θ θ

θ θ

θ

θθ θ

ζ ζ θ

θ θ

ζζ ζ θ θ

θ θ θθ θ θ

∂ ∂ ∂ ∂∂ ∂      

=− + +     ∂∂ ∂ ∂ ∂ ∂ ∂ ∂      

∂ ∂ ∂ ∂ ∂ ∂∂ ∂      

=− + + + −     ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂      
NSWCDD/TR-03/81 

J-6

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NSWCDD/TR-03/81 

K-1/K-2

APPENDIX K 

DERIVATION OF , , , R R

R f b f Rf bf

d d d d

dt dt dt dt

θ θ β β

θβθ β

∂∂∂ ∂

∂∂∂ ∂

AND 

NSWCDD/TR-03/81 

K-3

The following identities will be needed∗ (Appendix G): 

 cos sin sin cos cos cos θ R T TD = + φφ φ φ λ , (K-1) 

 cos sin cos sin sin cos cos β θ R T TD = − φφ φ φ λ , (K-2) 

 sin sin cos sin θ Rb D β φ = − λ , (K-3) 

 cos cos cos sin sin cos λD b bR =− − β β ββ θ , (K-4) 

 sin sin cos sin θ R TD β φ = λ , (K-5) 

 cos sin cos sin φβ φ β = − T b , (K-6) 

 cos cos sin sin cos cos cos . φβ φ Tb R R = − θ φ θ β (K-7) 

Note that Equations (K-1), (K-5), and (K-7) are identical to Equations (G-3), (G-4), and (G-6), 

respectively. It is clear that , , and φ φT D λ are independent of each other so that 

0. T TD

D

φ φ λ

φλ φ

∂∂∂ === ∂∂ ∂ We have from Equations (G-8) and (G-9), respectively, 

1 1 cos sin , csc cos cos . R R

RT b

D f Df

d d

dt dt

θ θ β β φ β θ φ β λ λ∂ ∂ = = = =− ∂Ω ∂Ω

Differentiating Equation (K-1) with respect to φ and using Equation (K-2), we have 

sin cos sin sin cos cos cos sin cos . R R

R T TD R

θ θ θ φφ φ φ λ β θ β

φ φ

∂ ∂ −= − = ⇒ = − ∂ ∂

Differentiating Equation (K-3) with respect to λD and using Equation (K-4), we have 

( )

cos sin sin cos cos cos

sin cos cos cos sin sin cos cos cos cos

csc cos cos csc cos cos .

R b

Rb R b D

D D

b

R b D bR b

D

b b

R R

D f

d

dt

θ β θ β θ β φλ λ λ

β θ β φ λ β β θ φββ λ

β β θ φβ θ φβ λ

∂ ∂

+ =−

∂ ∂

∂

⇒ =− + =

∂

∂

⇒ = ⇒ = Ω

∂

Differentiating Equation (K-3) with respect to φ and using Equations (K-5), (K-6), and (K-7), 

we have 

cos sin sin cos sin sin R b

Rb R b D

θ β θ β θ β φ λ

φ φ

∂ ∂

+ =

∂ ∂

 

∗

 Sofair, Isaac, K40 Training Guide 4085.1, A Detailed Derivation of Formulae Arising in Spherical Trigonometry, 

Naval Surface Weapons Center, Dahlgren, VA, May 1986 (currently Naval Surface Warfare Center, Dahlgren 

Division). 
NSWCDD/TR-03/81 

K-4

( )

T

sin sin cos sin sin cos cos sin = sin sin cos cos cos

cos

sin cos csc sin .

b

Rb D R b R R

b

b R

β β θ β φ λ θ ββ φ θ φ θ β

φ φ

β ββ θβ

φ

∂

⇒ =+ −

∂

∂ = ⇒ = ∂

2

2

2

cos cos csc cos cos cos cos .

sin sin sin cos cos csc cos sin cos

R

RT b

D D

R

RT b

D D

θ β φβ θ φφ β β λ λ

θβ β β φβ φ β θ φ ββ λφ λ φ

∂ ∂ = =− ∂ ∂

∂∂ ∂

= =− + =−

∂∂ ∂ ∂

( )

( )

cos cos sin sin csc cos cos

csc sin sin sin cos cos

cot cos cos sin (from Equation (K-7))

RT b

R R Tb

R

β φβ β φ θ φ β

φ

θ β φθ φ β

θ φββ

∂

⇒ = − ∂

= −

=

cot sin . R

β θ β

φ

∂

⇒ = ∂

( )

( )

2

2

2

cos csc cos cos csc cos cot sin

csc cos cos cos sin sin cos cos .

R b

T R b T R bR b

D D D D

R T b bR

β θ β φ θ β φ θ βθ β λ λ λλ

θ φφ ββ β β θ

∂ ∂ ∂∂  

=− = +   ∂ ∂ ∂∂  

= +

( )

( )

2

2

2

cos csc cos

csc cos cos cos sin sin

csc cos sin sin cos cos cos .

T Rb

D

R b

R T R b Rb

R T b bR

β φ θβ λφ φ

θ β θφ θ β θβ

φ φ

θ φ ββ β β θ

∂ ∂ = − ∂∂ ∂

  ∂ ∂

= +     ∂ ∂

= −

It is desired to compute , , , and R R

RD bD RD bD

θ θ β β

θ λ β λ θλ β λ

∂∂ ∂∂ ∂ ∂ ∂ ∂

∂∂ ∂∂ ∂∂ ∂∂ , where the partials 

with respect to θ R imply a constant value of βb , and the partials with respect to βb imply a 

constant value of θ R . Since θ R and β – and hence their derivatives – depend on , , and φ φT D λ , 

these partials cannot be found by direct differentiation. In general, if a function is defined by 

ρ ( ) y xx y xx xx 1 12 2 12 12 ( )( ) , , , , , constant, = then 

12 12

11 2 1 1 12 2 2 2

0, 0. yy yy

yx y x x yx yx x

∂∂ ∂∂ ∂ ∂∂ ∂∂ ∂ ρ ρ ρρ ρ ρ + += + +=

∂∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂
NSWCDD/TR-03/81 

K-5

Solving these two equations simultaneously yields 

22 11

21 12 12 21

12 12 12 12 1 2

12 21 12 21

, 

yy yy

x x xx xx xx

y y yy yy yy yy

x x x x xx x x

ρρ ρρ

ρ ρ

∂∂ ∂∂ ∂∂ ∂∂ − − ∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂∂ = = ∂ ∂ ∂∂ ∂∂ ∂∂ ∂∂ − − ∂∂ ∂ ∂ ∂∂ ∂ ∂

. 

Putting 12 1 2 , , , , and R

DR b

D

xx y y

θ φλ θ β ρ λ∂ == = = = ∂ , we obtain 

2 2

2

2 2

2

0,

cos cos .

Rb R b

R D DD

R D Rb Rb

D D

R R RR

R D DD

T b

Rb Rb b D

D D

θβ θ β

θ λ φ λφλ

θ λ θβ θβ

φλ λ φ

θ θ θθ

θ λφλ λ φ φ β β λ θβ θβ

φλ λ φ

∂∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∂∂∂ = = ∂ ∂ ∂∂ ∂∂ − ∂∂ ∂ ∂

∂ ∂ ∂∂ − ∂ ∂ ∂∂∂ ∂ ∂ = = ∂ ∂ ∂∂ ∂∂ − ∂∂ ∂ ∂

Similarly by putting 

D

β ρ λ∂ = ∂ , we obtain 

2 2

2

2 2

2

csc cot cos cos ,

csc cos sin .

b b

D DD

RR T b

R D R R b b

D D

R R

D DD

RTb

b D R R b b

D D

β β β β

β λ φ λφλ θ θ φ β θ λ θ θ β β

φλ λ φ

β β θ θ

β λφλ λ φ θφβ β λ θ θ β β

φλ λ φ

∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∂∂∂ = =− ∂ ∂ ∂ ∂ ∂ ∂ − ∂∂ ∂ ∂

∂ ∂ ∂ ∂ − ∂ ∂ ∂∂∂ ∂ ∂ = =− ∂ ∂ ∂ ∂ ∂ ∂ − ∂∂ ∂ ∂

Hence, 

0, cos cos ,

csc cot cos cos , csc cos sin .

R R

T b

Rf bf

RR T b R T b

Rf bf

d d

dt dt

d d

dt dt

θ θ φ β θ β

β β θ θ φ β θ φ β θ β

∂ ∂ = =Ω

∂ ∂

∂ ∂ = −Ω = −Ω

∂ ∂
NSWCDD/TR-03/81 

K-6

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NSWCDD/TR-03/81 

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