Paraboloid

This is particularly easy. Consider

[Graphics:../Images/index_gr_21.gif]

and notice that the cross-sections perpendicular to the z-axis are ellipses. The cross-sections perpendicular to the other axes are paraboas. Specifically, if [Graphics:../Images/index_gr_22.gif] (and [Graphics:../Images/index_gr_23.gif]), then we have the ellipse (in two dimensions)

[Graphics:../Images/index_gr_24.gif],

which motivates the parameterization

[Graphics:../Images/index_gr_25.gif],
[Graphics:../Images/index_gr_26.gif]. 

[Graphics:../Images/index_gr_27.gif]
[Graphics:../Images/index_gr_28.gif]

[Graphics:../Images/index_gr_29.gif]

[Graphics:../Images/index_gr_30.gif]

Converted by Mathematica      October 10, 2001