Surfaces in 3D 13.6,13.7

Quadric Surfaces and beyond… Plotting the following will result in what is known as a quadric surface… This can be simplified into one of the following two simpler forms: or The shape of the resulting surface depends solely on the choice of the coefficients and can lead to a considerable variety in shapes. Study Table 1 in Chapter 13…

Tips on how to visualize quadric surfaces… Slice the surface with one or more coordinate planes to produce the “traces” for the surface. Example: try to visualize by setting z = constant, then by setting y = constant etc. This generates a series of slices in the xy, xz and yz planes that “trace” the shape of the function. Reduce the equation to “standard form” to allow comparison to one of the quadric surfaces tabulated in Table 1. This is just an extension on the “completing the square” technique that you learned in HS. If you let z = k this generates families of circles If you set either x or y = k, you get a series of hyperbolas

Examples… Pg 874: 11, 21,28, 43 Next stop: cylindrical and polar coordinate systems Link to Maple worksheet on 3d surfaces