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Unformatted text preview: Page 2 13.6 Quadric Surfaces A quadric surface is the graph of a seconddegree equation in three variables x , y , and z . The most general such equation is Ax 2 + By 2 + Cz 2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = . By translating or rotating the graph, the equation can be brought into one of the two standard forms: Ax 2 + By 2 + Cz 2 + J = or Ax 2 + By 2 + Iz = . Quadric surfaces are the three dimensional counterparts to the conic sections in the plane. Example C We examine the traces of x 2 + y 2 9 + z 2 4 = 1 . For the traces parallel to the xyplane, we set z = k for different values of k . These traces satisfy x 2 + y 2 9 = 1 − k 2 4 which describes an ellipse. Similarly, setting y = k gives the ellipses x 2 + z 2 4 = 1 − k 2 9 and likewise for setting x = k . Example D Above is the graph of x 2 4 + y 2 − z 2 4 = 1 a hyperboloid of one sheet. Exercise 1 Use traces to identify the surface given by 4 x 2 − 16 y 2 + z 2 = 16 ....
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 Spring '08
 Hodges
 Conic section, Quadric

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