On Growth and Form

by D'Arcy Wentworth Thompson

Fig. 275. An Ammonite, to shew corrugated surface-pattern.

Fig. 276.

Fig. 277.

The first method is theo­ret­i­cal­ly sim­ple, but dif­fi­cult in prac­tice; for it requires great accuracy in de­ter­min­ing the points. Let AD, DB, be two tan­gents drawn to the curve. Then a circle drawn through the points ABD will pass through the pole O; since the angles OAD, OBE (the sup­ple­ment of OBD), are equal. The point O may be de­ter­mined by the in­ter­sec­tion of two such circles; and the angle DBO is then the angle, α, required.

Or we may determine, graphically, at two points, the radii of curvature, ρ₁ρ₂ . Then, if s be the length of the arc between them (which may be determined with fair accuracy by rolling the margin of the shell along a ruler)

cot α = (ρ₁ − ρ₂) ⁄ s.

The following method[516], given by Blake, will save actual determination of the radii of curvature.

Measure along a tangent to the curve, the distance, AC, at which a certain small offset, CD, is made by the curve; and from another point B, measure the distance at which the curve makes an equal offset. Then, calling the offset μ; the arc AB, s; and AC, BE, respectively x₁ , x₂ , we have

ρ₁ = (x₁² + μ²) ⁄ 2μ , ap­prox­i­mate­ly, and

cot α = (x₂² − x₁²) ⁄ 2μs .