PLOT 3D

Plot 3D

Exercise 14.7.7 (Page 994):

This figure is showing the intersection of the paraboloid and the ellipsoid . Each of these surfaces can be fairly easily parameterized. First : , , . For the top half of the ellipsoid :, , and for the bottom half: , , . A first attempt of this graph is shown below.

This figure gives a good depiction of each of the individual surfaces, but it does not show the intersection well. This is the problem of surface elements from two different functions that overlap. The following addresses this problem by using a total of 9 different parameterizations. First, the bottom of the ellipsoid is parameterized as before.

When , the two surfaces intersect

at and when , the two surfaces intersect at . So for starters, graph the top part of the ellipsoid for ( and graph the paraboloid for (). Reduce the s-direction resolution factor to 5 for the top part of the ellipsoid and to 2 for the paraboloid. The resulting surface is depicted in the top picture to the right. Note the gap between the top part of the ellipsoid and the bottom of the paraboloid. It is in this gap that 5 more parameterizations will be used to handle the intersection of the two surfaces.

Next parameterize a piece of the paraboloid as ,

, . Note that when , the expression goes from 1.47 () down to 1.36 () as s goes from 0 to 1. Also the expression

remains a constant 1.47 when . Reduce the s-direction resolution factor to 1 and the t-direction factor to 5. Also parameterize a piece of the ellipsoid as ,

, with the resolution factors reduced as before. These two surface segments are pictured in the middle graph to the right. The bottom figure to the right show these two segments combined with the other surface pieces already graphed in the top picture to the right.

For , parameterize the piece of the parabola as

and the ellipsoid as , . Neither of these surfaces are visible for

and for , only the ellipsoid is visible. Parameterize it as , . Finally to get the top of the ellipsoid use its original

parameterization with and , set the s-direction resolution to 5 and the t-direction resolution to 20. Putting all of these parameterizations together gives the graph below.

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