﻿ Transformation between Catenoid and Helicoid
Geodesics

Geodesics are locally shortest curves on a surface. An equivalent description is (without proof on this page): a curve is a geodesic if the tangential part of the second derivative of the curve vanishes. in tangential direction.Using Christoffel symbols G(ijk) we obtain:

c(t) = x(u(t), v(t)) is geodesic on a hypersurface x <=>

u'' + (u')²G¹¹¹ + 2u'v'G¹²¹ + (v')²G²²¹ = v'' + (u')²G¹¹² + 2u'v'G¹²² + (v')²G²²² = 0

(G(ijk): Christoffel-symbols).

For every pair (initial point, initial direction) there exists a uniquely determined geodesic.

Shadow lines are curves that follow the borderline between light and darkness if the surface would be illuminated by parallel light in a given direction. Therefore the surface normals at the points of a shadow line are orthogonal to the direction of the light. With the Euclidean scalar product <.,.> and the condition <L, N(u(t),v(t))> = const (L: direction of the light; N: surface normals) follows

u'(t) = -<L,dN(u(t),v(t))/dv> and v'(t) = <L,dN(u(t),v(t))/du>.

If you specify an initial point (u,v) and choose L out of the tangent space at x(u,v) (x: hypersurface) you get a shadow line through the point (because <L,N(u,v)> = 0).

A disadvantage of this description is the dependence on dN: Look at a shadow line on the torus in direction 0° (u-direction of the light). The line ends at the upper or lower u-circle though the normals on the complete v-circle lie in the same plane that is orthogonal to the direction of the light. The reason is: dN/du = 0 for the upper and lower u-circle.

Asymptote Lines

Asymptotic lines follow zero-curvature directions on the surface. That means for an asymptotic c(t): At every point the intersection of the plane {c'(t), N(c(t))} with the surface has locally curvature zero. Another description is given by b(c'(t),c'(t)) = 0 (b: second fundamental form).

As you can imagine on many surfaces there exist points with no such direction, e.g. on the sphere or on the torus. So don't wonder when you pick a point and nothing happens.

Also there does not exist an asymptotic for all given initial values c(t),c'(t). The applet searches for the nearest asymptotic direction to the given initial direction. So changing the initial direction doesn't always force the curve to change.

Curvature Lines

At every point of a surface there exist directions in which the curvature maximize or minimize. These are called mean curvature directions. A mean curvature line now follows the nearest mean curvature direction at every point. The problem can be described by: S(c'(t)) || c'(t) with S is the Weingarten-operator.

Like the asymptotics the initial direction is just a guiding rule (see Asymptotics above, third paragraph).