Gaussian Curvature of Curves on Surfaces

Given a curve (u(t),v( t)) in the domain of a surface, we have a curve

lying on the surface. We also obtain a curve lying on the unit sphere.

Gaussian Curvature related to Curves on Surfaces

In the control panel for the demo, there are two sliders for constants. The c slider controls a constant for the surface while the d slider controls a constant used in UVCurve .

In this demonstration, we select a surface in space and a curve (u(t),v( t)) in the domain of a surface and then show its image on the surface and on the Gauss map. We also graph the Gaussian curvature K(t) of the surface at the point X(t) as a function of the parameter t .

Tangent Space Demo

In this demonstration, we select a point (u0,v0) in the domain of a surface with the middle mouse button and then select a unit vector (cos(q) ,sin(q)) by changing Angle Theta . This defines a straight line

We then see the image of this line on the surface and on the spherical image. We can also see the tangent vectors X'q(t) and N'q(t) in the tangent plane. Length of curve allows you to control how much of the curve you see.

You may make the tangent plane show up in the Screen by clicking on Tangent Plane? . Note that solidifying the surface may help to make its contact with the tangent plane more visible. The X' and N' window clearly displays the vectors Xu , Xv , X' , and N' in the tangent plane at X(u0,v 0) .

The Normal Strip of Curves on Surfaces

The normal strip above the image of a curve Y(t) =(u(t),v( t)) in a mapping X(u,v) is defined in the following way:

This demonstration allows you to define a surface as usual but also allows you to choose a curve in the domain of the surface. Curve Parameters controls the curve display. Progressive? allows one to draw only a portion ofthe curve, this portion being controlled by the tapedeck. Strip? will draw the normal strip of the curve to the surface and its the progressive acts on this also. The tapedeck goes from 0 to 1 scaling between the minimum and the maximum of the curve domain.