On Growth and Form

by D'Arcy Wentworth Thompson

Or, changing the origin to the vertex of the figure

x² ⁄ a² − 2x ⁄ a + y² ⁄ b² = 0,

giving

(x − a)² ⁄ a² + y² ⁄ b² = 1.

Then, transferring to polar coordinates, where r · cos θ = x, r · sin θ = y, we have

(r · cos² θ) ⁄ a² − (2 cos θ) ⁄ a + (r · sin θ) ⁄ b² = 0,

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which is equivalent to

r = 2 a b² cos θ ⁄ (b² cos² θ + a² sin² θ),

or, eliminating the sine-function,