2/14/2003 QUADRICS Math 21a, O. KnillSection 9.6 16, 20, 22, 26,43 Due Wednesday 2/19.TRACES. To draw surfaces, it helps to look at the traces, the intersections of the surfaces with the coordinateplanes x = 0, y = 0 or z = 0.Quadrics: If f (x, y, z) = ax2+ by2+ cz2+ dxy + exz + fyz + gx + hy + kz + m then the surface f(x, y, z) = 0is called a quadric. Below are some examples.SPHERE.All three traces are circles(x − a)2+ (y − b)2+ (z − c)2= r2PARABOLOID.The z-traces are circles, the x and ytraces are parabolas.(x − a)2+ (y − b)2− c = zPLANE.All three traces are lines.ax + by + cz = dONE SHEETED HYPERBOLOID.The z-traces are circles, the x and ytraces are hyperbolas.(x − a)2+ (y − b)2− (z − c)2= r2CYLINDER.The z-traces are circles, the x and ytraces are lines.(x − a)2+ (y − b)2= r2TWO SHEETED HYPER-BOLOID.The z-traces are circles or empty,the x and y traces are hyperbolas.(x − a)2+ (y − b)2− (z − c)2= −r2ELLIPSOID.All three traces are ellipses.x2/a2+ y2/b2+ z2/c2= 1HYPERBOLID PARABOLOID.The z-trace form two crossed lines,the x- and y- traces are parabolas.x2− y2+ z = 1TWO SHEETED ELLIPTIC HY-PERBOLOID.The z-traces are ellipses or empty,the x and y traces are hyperbolas.x2a2+y2b2−z2c2=
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