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NASA Technical Memorandum 101628 

ROTATION OF IMAGES BY SCAN MIRRORS 

Stephen J. Katzberg 

June 1989 

(?lASA-TH-301628) 

MX330PS #biaSA, Langley Research Center) ROTATIOM Ofp XHAGES BY SCAB 

11 P CSCL 20P 

National Aeromuncs and 

Space AdministfaDon 

langley R- Cmtm 

Hmnpton, Mrginia 23665 

- 

B89-23504 

Uaclas 

63/74 02227 13 
ERRATA 

NASA Technical Memorandum 101628 

Stephen J. Katzberg 

ROTATION OF IMAGES BY SCAN MIRRORS 

June 1989 

Pages 4-5. - Text should read as follows: 

Looking at the case alluded to in the introduction, the following 

conditions defined the problem: With a spacecraft at 824 km altitude, the 

imager was to look at the Earth's limb from 50 km to 550 km above the tangent 

point, both directly ahead (positive "XI1 axis) and 90 degrees to the side of 

the spacecraft direction of travel (i.e. along 'lY'l axis). To accomplish 

this imaging geometry, the imager optical axis was aimed to the rearward 

direction and down. A scan mirror redirected the optical axis towards the 

direction of travel , or to the right or left under comnand. The downward tilt 

with respect to the spacecraft mounting plane was 24 degrees. Rotation of the 

scan mirror was to be about a line in the plane of the front surface of the 

redirecting mirror (approximately along the "Z" axis). 

Page 5. - Last Paragraph should read: 

When the mirror is rotated to move the optical axis to the side of the 

spacecraft, as illustrated in Figure 4b, the condition is different. In this 

case, scanning the mirror normal upward in addition to its side rotation will 

yield an image rotation. The initial movement of the mirror in the plane 

perpendicular to the spacecraft Z axis would simply retain the depression, but 

would also rotate horizontal lines in the imager so that their projection 

would be nearly parallel to the horizon. Now scanning the mirror normal to 

have a Z axis component also causes a Z axis component in the projected 

horizontal lines as well as the vertical ones. The amount of the rotation of 

the horizontal line (which starts with some vertical projection) can be 

developed from the following: Let n be the vector normal to the reflector 

surface and have the form: 

Page 6. - There are two missing equations: 

Cos R,= 2Sin Q Sin a Cos Q 

Page 6. - The equations for tangent of the rotation angle should read: 

a Rx = 2 - Rz Sin Rx tan Rx = -a 

cos Rx' 

- Cos 4 Sin a Sin 4 - Sin 4 Sin 4 Cos a Sin a 

cot 4 

cos a 

- 

tan ~x =,$,cot 4 

= 2 cot 4 f 

Rx =p($ - 6 ) 

Issued January 1990 
ROTATION OF IMAGES BY SCAN MIRRORS 

* 

Stephen J. Katzberg 

NASA Langley Research Center 

Hampton, Virginia 23665 

Introduction 

During the evaluation of proposals submitted in response to the 

Announcement of Opportunity for EOS, a question arose as to image rotation 

effects caused by scan mirrors. 

described the intention of using a pointing mirror to redirect a rearward 

pointing optical axis to the forward direction and to two acute angles on 

either slde of forward. In the technical discussions related to this 

proposal, it was believed that any such redirection would cause a major, and 

uncompensated, rotation of the image. 

It is the purpose of this document to examine the general problem of 

image rotation in electro-optical imagers, to develop the necessary analysis 

to quantify the rotation, and to relate the results to the above mentioned 

example. 

In particular, one instrument proposal 

Theory 

The fundamental principle involved here is what Is comnonly referred to 

as "the angle of incidence equals the angle of reflection.'' The origin of 

this relationship comes from historical practical observation, quantified in 

modern electro-magnetic field theory. 

a reflector is for the electric field to match the boundary condition of "no 

tangential or normal electric fields penetrating the (perfect) reflector'' 

along the interface between the two. Such a requlrement implies that there 

must be another wave, the reflected wave, traveling outward, as illustrated in 

Figure 1. Otherwise, there will be no hope of everywhere meeting the boundary 

condition of no electric field penetrating the (perfect) reflector. Since 

there are no other sources of propagating waves, the reflected wave must have 

the following characteristics: The part of the incident propagation vector 

normal to the surface must change sign to become an outward directed wave. 

The tangential component of the propagation vector will be matched by an equal 

one from the reflected wave to match the boundary conditions at the reflector. 

Moreover, since the material constants for each propagation vector are the 

same, then the magnitude of the two vectors are the same. 

In electro-magnectic field terms, the requirement for a wave impinging on 

In vector terms, this effect can be written: 

+ +++ +++ 

= (n*k)n; 2, = 2 - (n*k)n 

+ +++ 

kP 

2' = k - 2(n*k)n 

1 
+ Where it is the parallel component of the propagation vector, kn is the 

normal component, and ii is the normal to the reflector surface. 

medium, then the magnitude of the two propagation vectors must match. 

P 

Note that since the incident wave and the reflected wave are in the same 

++ ++ ++ ++ 2 = k.k - 4( n.k) (nok) + 4(n*k) 

If now the dot product of the reflected wave propagation vector is taken 

against the normal to the reflector surface, Z, the following results: 

++ ++ *+++ n.k' = n.k - 2(n.k)n.n 

++ = -2n.k 

= -litlCtw 

Where 8' is the angle between the propagation vector and the normal i. 

And, if She dot product of the incident wave is taken against the normal 

vector, n, then it is seen that the cosines of the two angles 8' and 8 are 

opposite in sign. 

++ n.k = IZlcose 

That is, they are 180 degrees from one another and of the same maginitude, 

demonstrating that the "angle of incidence equals the angle of reflectance''. 

The process just described in generating the reflected wave propagation 

vector is linear in the sense that two waves each produce a component from 

the reflector normal proportional to the ot pr duct between the two. Thus, 

given two waves with propogation vectors 1 and 3 the effect of the mirror 

reflection on a third propagation vector b, which is the difference between 

them is evidently: 

dl = d - 2(ii.d)ii 

81 = z1 - 2(ii.z1)ii 

b' = 31-8' = (1-3, - 2(i;.X)i; + 2(ii*8)ii 

= b - 2(ii.b)ii; d = d-b 

Therefore, the effect of reflection from a mirror surface can be 

conveniently described by this vector equation. Some obvious implications 

can be noted. First, if there is no component of the source vector in the 

direction of the normal to the reflector, then there is no effect of the 

mirror on the source vector. Second, the normal vector can contribute 

to affect the source vector in all dimensions of the normal, just so long as 

the source and normal are not orthogonal. Third, even though the normal can 

affect the source vector, the angles among a group of source vectors are 

2 
retained. This can be seen by forming the dot product between any two source 

vectors and their reflected versions: 

Fourth, the magnitude of the reflected vector (a source vector, the 

propagation vector, etc.) is unchanged, which follows from the above equation 

with 6 chosen the same as A. Fifth, the rotation of the reflected source 

vector is twice the angle between the incoming source vector and the 

reflected source vector: 

Where bl 

I is the 

vector. 

is the angle between the reflected vector agd the source vector and 

angle between the reflector normal vector, n, and the source 

Effect in an Imaqins System 

To determine the effect of an external scan mirror in an imaging system, 

three things are important to understand. First, imaging systems have some 

sort of image forming element, lens, pinhole, focussing mirror, etc., and 

that element has the property of more or less maintaining geometric relationships between elements in the object and the image generated. Second, in 

the focal area of the imaglng system is some sort of scanning aperture or 

detector array whose geometry, in conjunction with the optical axis, sets the 

geometric orientation of the imaging system. Third, there is an inversion in 

travelling through the imaging system, left-for-right, top-for-bottom, which 

causes the image vector to be the negative of what impinges on the external 

scan mirror. In all that follows, note the appearance of a sisn reversal in 

the vector equations to account for the fact that vectors are assuned to 

originate in the imase side of the optical device. 

The question, in simplest terms, of whether and how an image is rotated 

is one of picking a vector in the image plane and determining how that vector 

projects against the scene of interest. It is of interest that image rotation is not always a bad thing, and sometimes is essential for proper system 

performance. For example, an imager on a spacecraft may be desired to look 

forward and sideways at the horizon. Pointed in the direction of travel, 

the imager would see vertical lines along an Earth radius vector at the 

tangent point, and horizontal lines perpendicular to the radius vector and 

approximately) perpendicular to the optical axis to the target. Looking to 

the side of the spacecraft, the vertical lines would be again along the 

radius vector at that tangent point, and the "cross-lines" would again be 

3 
perpendicular to both optical axis and radius vector. However, these 

I1cross-linest1 will also be perpendicular to the llcross-lines" in front of the 

spacecraft, having Itrotatedlt from across the spacecraft track to along it. 

In this case, the rotation of the image is desirable and, in fact, the 

imaging scan system must provide the rotation of the imaging coordinates for 

proper operation. 

Undesirable rotation occurs when there is rotation between the optical 

System axes in the image plan and the orientation of the targets of interest, 

referred to the image plane. 

Some Examples 

The simplest case is a mirror set to rotate about one single axis in 

order to scan some field of view. Two cases will be examined: A mirror 

rotating about an axis perpendicular to the optical axis of an imager and a 

mirror rotating about the same axis as the optical axis. In the first case 

illustrated in Figure 2, the mirror normal is perpendicular to one set of 

parallel lines and in line with, but with varying projection against, the 

orthogonal set in the optical image plane. From the above considerations, 

the set of constantly orthogonal lines will project completely una tered 

except for the left-to-right, top-to-bottom inversion: 

dl = -3 + 2(X)3 

= -d; n d 

The other set has a component from the normal which sweeps around in 

orientation from completely in line to completely orthogonal (in the line of 

the optical axis) to this set: 

= (-B + BCosZR)Tx + (BCosnSinR)ty 

= B(Cos2dx + Sin2dy) 

Thus, the llin-linelt set of lines in the image plane form a projected set 

(inverted) which rotate around with the normal vector, but twice as fast. 

axis, as illustrated in Figure 3, the normal has components at various points 

in the rotation which overlap and then become orthogonal to all sets of 

straight lines in the image plane. Thus, a projected image will rotate round 

and round as the normal progresses, but without the "two-times" relationship 

in the former case. With a mirror set at a 45-degree angle to the optical 

axis (still rotating about the optical axis) then the image will project at a 

90-degree angle. With the mirror deflecting the image straight down, say, 

horizontal lines will still be anti-parallel in object and image. But if the 

mirror rotates 90 degrees, the lines will be orthogonal, those in projection 

being tlfront-to-backll to their "side-to-side" source in the image plane 

(vector B and B' in Figure 3). 

Looking at the case alluded to in the introduction, the following 

conditions defined the problem: With a Spacecraft at 824 km altitude, the 

In the case of a mirror rotating about an axis parallel to the optical 

4 
imager was to look at the Earth's limb from 50 km to 550 km above the tangent 

point, both directly ahead and 90 degrees to the side of the spacecraft 

direction of travel. To accomplish this imaging geometry, the imager optical 

axis was aimed to the rearward direction and down. A scan mirror redirected 

the optical axis towards the direction of travel, or to the right or left 

under comnand. The downward tilt with respect to the spacecraft mounting 

plane was of 24 degrees. Rotation of the scan mirror was to be in the plane 

of the front surface of the redirecting mirror. 

Given these parameters, a few relationships can be pointed out. First, 

the Earth's tangent point from 824 km is out at approximately 3342 km and 

represents a depression angle of 27 degrees. If the external plane mirror has 

its normal parallel to the plane of the spacecraft/optical imager, then a rear 

pointing depression of 27 degrees yields a forward reflected depression of the 

required 27 degrees. 

planned to cover the range from 27 degrees depression to about 18. 

The scan from a point 50 km above the Earth's limb to 550 km above the 

limb represents a start at about 0.9 degrees above the tangent point, continuing to about 9.3 above the tangent point. Thus, the depression angle must 

change and a certain amount of image rotation may occur. In terns of what 

has already been presented, the amount of image rotation can be calculated by 

assuming that the scan mirror normal is given a Z axis tilt in an opposite 

direction to the backward tilt of the optical axis and just enough to rotate 

the line to the tangent point upward to start at 50 km and end at 550 km. In 

the forward looking case, a horizontal line in the image plane would have no 

dot product with the mirror normal at any tilt, whil'e a vertical line would, 

yielding a fore and aft tilt in its projection at the tangent, as illustrated 

in Figure 4a. 

When the mirror is rotated to move the optical axis to the side of the 

spacecraft, as illustrated in Figure 4b, the condition is different. In this 

case, scanning the mirror normal upward in addition to its side rotation will 

yield a rotation. The initial movement of the mirror in the plane perpendicular to the spacecraft Z axis simply retained the depression, but also 

rotated horizontal lines in the imager so that their projection was still 

parallel to the horizon. Now rotating the normal to have a Z axis component 

also causes a Z axis component in the projected horizontal lines as well as 

the vertical ones. The amount of the rotation of the horizontal line (which 

Starts with no vertical projection). can be developed from the following: Let 

n be the vector normal to the reflector surface and have the form: 

In the case here, a mirror scan of a few degrees was 

+ 

n - Sin4CosaTx + Sin4Sinat + ~os~t~ Y 

Where the normalization of the direction cosines has been used to express 

the new unit vector components in terms of the previous ones. In the case 

here, the mirror normal started at a 45-degree angle with respect to the 

spacecraft tangential velocity vector, so tQe angles can be evaluated 

easily. Let the source vector be a vector o that is perpendicular to the 

spacecraft velocity and zenith-nadir axis vectors (a vector in Yaw.) 

5 
Then, it remains to determine the X and Z components of the vector o 

Once the components are found, then taking the arctangent will 

when the mirror is pointed out at 45 degrees (remembering the double angle 

rotation to 90 degrees to the spacecraft track in order to look to the side 

horizon). 

yield the ar$ount of rotation. 

product of 0 with a unit vector in the Z direction: 

While the X component can be found in a similar way: 

The Z component can be found by taking the dot 

The tangent of the angle Rx is then found to be: 

ll 2 - Rz si nRx tanRx = m; Rx = 

CosdSi nasi nQ = singsing 

= 2 CoteSina 

For the case here in which a is 45 degrees the approximation to the tangent 

and cotangent can be used, remembering that g is close to 90 degrees: 

2 

2 

tanRx = - Cotg 

= 2 cot 0 

Using the values here for the change in depression angle from 27 degrees 

down to about 18 degrees in scanning from 50 km up to 550 km at the tangent 

height, the amount of rotation is thus seen to be approximately 12.0 degrees. 

Concl usion 

An analysis of the comnonly encountered question of the effects of scan 

mirrors on scene orientation or rotation for image forming systems has been 

presented. Some simple vector relationships have been presented, which when 

combined with the anticipated application conditions, make it possible to 

easily assess image orientation effects resulting from scan mirrors. The 

analysis has been applied to some examples for illustration and to one 

example of current interest. In the latter example, the degree of image 

shift has been presented; and in the process, the danger of falling into 

simple misconceptions has been highlighted. 

6 
Jncident vector 

, 

4 / 

'Reflected vector 

Figure 1. Illustration .-of geometry of reflection 

of a plane wave. I 

Figure 2. Case of a scan mirror rotating about 

an axis perpendicular to the optical axis. 

7 
Ref I 

Figure 3. Case of a scan mirror rotating about an axis 

parallel to the optical axis. 

8 
c 

a. 

b. 

Figure 4.. Case of a scan mirror designed to move 

the target point to the horizon from a 

spacecraft mounted backward pointing imager. 

9 
Report Documentation Page 

1. Report NO. 

NASA TM- 10 1628 

2. Government Accession No. 3. Recipient's Catalog No. 

ROTATION OF IMAGES BY SCAN MIRRORS 

4. Title and Subtitle 

June 1989 

5. Report Date 

7. Author(sl 8. Performing Organization Report No. 

Stephen J. Katzberg 

17. Key Words (Suggested by Author(s1) 

Imaging Systems, Image Rotation 

10. Work Unit No. 

18. Distribution Statement 

Unclassified - Unlimited 

Subject Category 74 

488-80-07-01 

9. Performing Organization Name and Address 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 

U nc 1. a s s i f i ed 

21. No. of'pages 

Unclassified 10 

-- 

NASA Langley Research Center 

22. Price 

A02 

11. Contract or Grant No. 

Hampton, VA 23665 

13. Type of Report and Period Covered 

Technical Memorandum 12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546 I 14. Sponsoring dgency Code 

I 

5. Supplementary Notes 

6. Abstract 

An analysis is presented of the effect of scan mirrors on image rotation in 

imaging systems. Examples are cited to demonstrate application of the analysis.