Multivariate Calculus - Bruce E. Shapiro

46 LECTURE 6. SURFACES IN 3D
Definition 6.3 A quadric surface is any surface described by a second degree
equation, i.e., by any equation of the form
Ax 2 + By 2 + Cz 2 + Dxy + Exz + F yz + Gx + Hy + Iz + J = 0 (6.1)
where A, B, C, D, E, F, G, H, I, J are any constants and at least one of A, B, C, D, E, F
are non-zero.
Theorem 6.1 Any quadric surface can be transformed, by a combination of rotation
and translation, to one of the two following forms:
Ax 2 + By 2 + Cz 2 + J = 0 or (6.2)
Ax 2 + By 2 + Iz = 0 (6.3)
Quadric surfaces of the form given by equation (6.2) are called central quadrics
because they are symmetric with respect to the coordinate planes and the origin.
Table of Standard Quadric Surfaces.
Ellipsoid
Hyperboloid of one Sheet
Hyperboloid of two Sheets
x 2
a 2 + y2
b 2 + z2
c 2 = 1
x 2
a 2 + y2
b 2 − z2
c 2 = 1
x 2
a 2 − y2
b 2 − z2
c 2 = 1
Elliptic Paraboloid
z = x2
a 2 + y2
b 2
Hyperbolic Paraboloid (saddle)
Elliptic Cone
z = y2
b 2 − x2
a 2
x 2
a 2 + y2
b 2 − z2
c 2 = 0
Revised December 6, 2006. Math 250, Fall 2006