12.6 A survey of quadric surfaces Quadric surfaces are the surface analogs of conic sections. Recall that a conic section is a curve in R^2 defined by a quadratic equation in two variables A quadric surface is defined by a quadratic equation in three variables: Ax2 + By 2 + Cz 2 + Dxy + Eyz + F zx + ax + by + cz + d = 0 When the coordinate axes are chosen to coincide with the axes of the quadric, the equation of the quadric has a simple form. The quadric is then said to be in standard position The surface analogs of ellipses are the egg-shaped ellipsoids. in standard form, and ellipsoid has the equation: z ³ x ´2 a + ³ y ´2 b + Ellipsoid ³ z ´2 c =1 9 Surfaces are often represented graphically by a mesh of curves called traces, obtained by intersecting the surface with planes parallel to one of the coordinate planes, yielding certain cross sections of the surface The surface analogs of the hyperbolas are the hyperboloids, which come in two types, depending on whether the surface has one or two components that we refer to as sheets z ³ x ´2 One sheets: a ³ x ´2 Two sheets: a + + ³ y ´2 b ³ y ´2 b = = ³ z ´2 c ³ z ´2 c y +1 ¡1 hyperboloids Notice that a hyperboloid of two sheets does not contain any points whose z-coordinate satisfies -c < z < c because the right-hand side is negative for such z, but the left-hand side of the equation is greater than or equal to zero 2 ³ x ´2 a + ³ y ´2 b = ³ z ´2 c Elliptic cone y An elliptic cone is a transition case between a hyperboloid of one sheet and a hyperboloid of two sheets. The hyperboloid of one sheet is pinched at its narrowest part to form an elliptic cone, and then the two parts of the elliptic cone separate to form the two components in a hyperboloid of two sheets The third main family of quadric surfaces are the paraboloids. There are two types - elliptic and hyperbolic. in standard position, their equations are ³ x ´2 ³ y ´2 Elliptic: z= + Paraboloids a b ³ x ´2 ³ y ´ hyperbolic: z= ¡ z a b Further examples of quadric surfaces are the quadratic cylinders. We use the term "cylinder" in the following sense: given A curve c in the xy-plane, the cylinder with base C is the surface consisting of all vertical lines passing through C The Ellipsoids, hyperboloids, paraboloids, and quadratic cylinders are called non-degenerate quadric surfaces. There are also a certain number of "degenerate" quadric surfaces