Drawing a Logarithmic Spiral

a

1.1

1.2

1.3

1.4

1.6

n

12

16

24

draw!

show labels for A, B, C

This Demonstration shows an approximation of the logarithmic spiral

ρ=

θ

a

. Draw

n

equally spaced rays from the origin

O

. The point

A

is on the polar axis at a distance 1 from the origin. The point

B

is on the ray

2π/n

at the distance

b=

2π/n

a

. So

A

and

B

are on the spiral. Draw a point

C

so that the triangles

OAB

and

OBC

are similar. From

OA:OB=OB:OC

, we get

OC=

2

b

. On the ray

4π/n

, draw a point at distance

OC

; this point will be on the spiral. Continue in this way. Note that 1,

b

,

2

b

, … form a geometric sequence.