logarithmic spiral

logarithmic spiral

The equation of the

logarithmic spiral

in

polar coordinates

r

,

φ

is

r

=

C

⁢

e

k

⁢

φ

(1)

where

C

and

k

are constants (

C

>

0

).  Thus the

position vector

of the point of this curve as the

coordinate vector

is written as

r

→

=

(

C

⁢

e

k

⁢

φ

⁢

cos

⁡

φ

,

C

⁢

e

k

⁢

φ

⁢

sin

⁡

φ

)

which is a parametric form of the curve.

Perhaps the most known of the logarithmic spiral is that any line emanating from the origin the curve under a constant angle

ψ

.  This is seen e.g. by using the vector

r

→

and its

derivative

d

⁢

r

→

d

⁢

φ

=

r

→

′

,  the latter of which gives the

direction

of the

tangent line

(see

vector-valued function

):

r

→

′

=

(

C

⁢

e

k

⁢

φ

⁢

k

⁢

cos

⁡

φ

-

C

⁢

e

k

⁢

φ

⁢

sin

⁡

φ

,

C

⁢

e

k

⁢

φ

⁢

k

⁢

sin

⁡

φ

+

C

⁢

e

k

⁢

φ

⁢

cos

⁡

φ

)

.

One obtains

r

→

⋅

r

→

′

=

k

⁢

r

2

,

|

r

→

|

=

r

,

|

r

→

′

|

=

r

⁢

1

+

k

2

,

whence

cos

⁡

ψ

=

r

→

⋅

r

→

′

|

r

→

|

⁢

|

r

→

′

|

=

k

1

+

k

2

=

constant.

It follows that

k

=

cot

⁡

ψ

.  The angle

ψ

is called the

polar tangential angle

.

The logarithmic spiral (1) goes infinitely many times round the origin without to reach it; in the case

k

>

0

one
may state that

lim

φ

→

-

∞

⁡

C

⁢

e

k

⁢

φ

=

0

but

C

⁢

e

k

⁢

φ

≠

0

⁢

∀

φ

∈

ℝ

(the

exponential function never vanishes

).

The

arc length

s

of the logarithmic spiral is

expressible in closed form

; if we take it for the

interval

[

φ

1

,

φ

2

]

,  we can calculate in the case

k

>

0

that

s

=

∫

φ

1

φ

2

r

2

+

(

d

⁢

r

d

⁢

φ

)

2

⁢

𝑑

φ

=

∫

φ

1

φ

2

C

2

⁢

e

2

⁢

k

⁢

φ

+

C

2

⁢

e

2

⁢

k

⁢

φ

⁢

k

2

⁢

𝑑

φ

=

1

+

k

2

k

⁢

C

⁢

(

e

k

⁢

φ

2

-

e

k

⁢

φ

1

)

,

thus

s

=

1

+

k

2

k

⁢

(

r

2

-

r

1

)

=

r

2

-

r

1

cos

⁡

ψ

.

Letting

φ

1

→

-

∞

one sees that the arc length from the origin to a point of the spiral is finite.

Other properties

•

Any curve with constant polar tangential angle is a logarithmic spiral.

•

All logarithmic spirals with equal polar tangential angle are

similar

.

•

A logarithmic spiral rotated about the origin is a spiral

homothetic

to the original one.

•

The

inversion

z

↦

1

z

causes for the logarithmic spiral a

reflexion

against the

imaginary axis

and a

rotation

around the origin, but the image is

congruent

to the original one.

•

The

evolute

of the logarithmic spiral is a congruent logarithmic spiral.

•

The

catacaustic

of the logarithmic spiral is a logarithmic spiral.

•

The families

r

=

C

1

⁢

e

φ

and

r

=

C

2

⁢

e

-

φ

are

orthogonal curves

to each other.

Title

logarithmic spiral

Canonical name

LogarithmicSpiral

Date of creation

2013-03-22 19:02:26

Last modified on

2013-03-22 19:02:26

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Topic

Classification

msc 14H45

Synonym

Bernoulli spiral

Related topic

AngleBetweenTwoCurves

Related topic

EvoluteOfCycloid

Related topic

PolarTangentialAngle2

Related topic

AngleBetweenTwoLines

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