Quadric Surfaces

Calculus 3 · Axiom Academy

Slice a second-degree surface with a plane, read the conic it leaves behind, and stack those cross-sections to rebuild the whole shape. The general equation of a quadric surface in standard position is: a second-degree equation in x , y and z By examining the signs and coefficients of the squared terms, we can identify which type of surface we have. There are six main types of quadric surfaces: All three squared terms positive. Every trace is an ellipse, and the surface is bounded. One negative squared term. A single connected surface with a waist. Two negative squared terms. Two disconnected pieces with an empty band between them. One variable appears linearly; the two squared terms share a sign. A bowl. One variable appears linearly; the two squared terms have opposite signs. A saddle. The degenerate member of the family: its horizontal traces are ellipses that shrink to a single point at the origin. The tool that does all six identifications is the same one. Set a variable to a constant, and what is left is a second-degree equation in the two remaining variables — a conic. xy-trace ( z = 0 ): substitute z = 0 into the equation. xz-trace ( y = 0 ): substitute y = 0 into the equation. yz-trace ( x = 0 ): substitute x = 0 into the equation. Coordinate planes are only the start. Slide the cutting plane to any height z = k and you get a whole family of traces. For the ellipsoid :

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