6.5: Extra Material

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

X(t)=X(u(t),v( t))

lying on the surface. We also obtain a curve

N(t) =N(u(t ),v(t))

lying on the unit sphere.

Gaussian Curvature related to Curves on Surfaces


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. The default curve is a circle of chosen radius centered at a point selected in the parameter domain.


Exercises

    1. In the previous section we saw that the outside part of a torus has positive Gaussian curvature while the inside part of a torus has negative Gaussian curvature. What happens to the graph of K(t) as the circle is moved to different parts of the surface. When will K(t) be 0?

Tangent Space Demo


In this demonstration, we select a point C = (u0,v0) in the domain of a surface using the white dot in the window labeled "Domain," and then draw a line through this point in the direction (cos(a), sin(a)). In the domain, the equation for the line is

(u1(t),v1(t)) = (u 0+tcos(a),v0+tsin(a))

This defines a straight line whose length is set by the interval for t, and whose direction is determined by the value of the angle a.

The image of the line appears both on the surface as X(u1(t),v1(t)) and on the Gauss Map as N(u1(t),v1(t)). In the window labeled "Surface," the plane tangent to the point X(u0,v0) is shown in green, and there are also two tangent vectors. The white vector is X'a(t) and the orange vector is N'a(t). The fourth window displays the tangent plane.

Exercises

    1. How do the vectors change as position and direction of the line change?

    2. For which values of a will X'a(t) and N'a(t) be linearly dependent?

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:

Z(t,w) = X(u(t),v(t)) + wN(u(t),v(t))

The demonstration allows you to define a surface as usual but also allows you to choose a curve in the domain of the surface. The default curve is a straight line and the image of this line is shown on the surface. A progressive normal strip can be added to the picture. The variable t0 uses a tapedeck feature that allows one to draw only a portion ofthe normal strip. The tapedeck goes from 0 to 1 scaling between the minimum and the maximum of the curve domain.

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