Derivation of the Conical Spiral

Conical spiral, alternative name: tapered hlix.

The same final results can also be found in the Wolfamealpha.com

The purpose is to derive the curvature and torison for

First derive the three vectors

syms b z a t_end;

assume([b, z, a], 'positive')

x = b*z*cos(a*z);

y = b*z*sin(a*z);

p = [x; y; z]

p =

Tangent vector is to take the derivative with respect to the t

T = simplify(diff(p, z)/norm(diff(p, z)))

T =

simplify(norm(T))

ans =

Arc length function

S = simplify(int(norm(diff(p, z)), z))

S =

The curvature can be calculated using the following formular.

Please refer to (https://www.whitman.edu/mathematics/calculus_online/section13.03.html)

kappa = simplify( norm(cross(diff(p, z), diff(p, z, 2)))/ norm(diff(p, z))^3, 'Step', 20)

kappa =

The normal vector is to take derivative of T, and normalize it.

N = simplify(diff(T, z)/norm(diff(p, z))/kappa)

N =

The binormal vector is the cross product of T and N

B = simplify(cross(T, N), 'Steps',80)

B =

Torsion is

tau = simplify(-diff(B, z)./N/norm(diff(p, z)));

tau = simplify(tau(1))

tau =