Lecture Notes for 120 - UCLA Department of Mathematics

4.1. SURFACES 76
(6) Many classical surfaces are of the form
F (x, y, z) =ax 2 + by 2 + cz 2 + dx + ey + fz + g =0
These are called quadratic surfaces if one of a, b, orc 6= 0.
(a) Give conditions on the coefficients such that it generates a surface
(g =0takes special care).
(b) Under what conditions does it become a surface of revolution around
the z-axis?
(c) Show that when the equation does not depend on one of the coordinates,
then we obtain a generalized cylinder.
(d) When, say c =0,butabf 6= 0we obtain a paraboloid. It is elliptic
when a, b have the same sign and otherwise hyperbolic. Drawpictures
of these two situations.
(e) When abc 6= 0show that it can be rewritten in the form
F (x, y, z) =a (x x 0 ) 2 + b (y y 0 ) 2 + c (z z 0 ) 2 + h =0
(f) When all three a, b, c have the same sign show that it is either empty
or an ellipsoid.
(g) When not all of a, b, c have the same sign and h 6= 0we obtain a
hyperboloid. Showthatitmightbeconnectedorhavetwocomponents
(called sheets) depending of the signs of all four constants.
(h) When not all of a, b, c have the same sign and h =0we obtain a cone.
(i) Given constants a x ,a y ,a z determine when
q (u, v) = (a x sin u sin v, a y sin u cos v, a z cos u)
q (u, v) = (a x sinh u sin v, a y sinh u cos v, a z cosh u)
q (u, v) = (a x sinh u sinh v, a y sinh u cosh v, a z sinh u)
q (u, v) = a x u cos v, a y u sin v, a z u 2
q (u, v) = a x u cosh v, a y u sinh v, a z u 2
yield parameterizations and identify them with the appropriate quadratics.
(7) Let q (z,µ) = p 1 z 2 cos µ, p 1 z 2 sin µ, z with 1