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Just as we can use functions of one parameter to generate a curve and functions of two parameters to develop a surface, we can use functions of three parameters to generate a solid.
When we kept one of the parameters constant for parametric surfaces, the result was a parametric curve. This time, were we to keep one parameter constant and allow the other two to vary, the result would be a surface. Keeping two parameters constant would generate a curve. Such a curve would trace out a surface, which in turn would trace out a parametric volume. This process can be extended for any number of variables (volumes trace out objects in 4-space, which trace out objects in 5-space, etc.).
This demo allows you to parametrize a volume by defining functions x,y, and z of parameters u, v, and w. By changing a, b, and c, you can see the process by which parametrized surfaces trace out parametrized volumes.
Exercises1. Parametrize the solid region contained in the irregular tetrahedron whose vertices are (0,0,0), (0,0,1), (1,0,1), and (1, 1, 1).
2. Parametrize a solid sphere of radius 1 centered at the origin.
3. Parametrize a solid torus whose points are distance less than or equal to 1 from a circle of radius 2 in the x-y-plane, centered at the origin.