Differential Geometry and Its Application - Exercise 2.1.19

Problem:

Find the standard patch for the standard cone $ \sqrt{x^2 + y^2} $ and standard cylinder $ x^2 + y^2 = 1 $ which are ruling patches in the send of Example 2.1.17 and Example 2.1.18 above. This explain why the names cones and cylinder are used for the more general patches given above.

Example 2.1.17 said a cone is $ x(u, v) = p + v\delta(u) $, where $ p $ is a fixed point.
Example 2.1.18 said a cylinder is $ x(u, v) = \beta(u) + vq $, where $ v $ is a fixed direction.

Solution:

For the standard cone, the fixed point is obviously the apex, the direction is the angle going up the cone.

$ x(u, v) = (v\cos u, v\sin u, 1)v = (0, 0, 0) + v(\cos u, \sin u, 1) $.

The the standard cylinder, the fixed direction is the vertical direction.

$ x(u, v) = (\cos u, \sin u, v) = (\cos u, \sin u, 0) + v(0, 0, 1) $.