MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 18: Rotation and Quadric Surfaces
If we rotate the xy-coordinate axes counter-clockwise through an angle  about the origin then we obtain
a uv-coordinate system which is related to the xy-coordinate system by
u = x cos  - y sin 
v = x sin  + y cos 
One reason such rotations are useful is that they allow us to identify the graph of any quadratic equation
Ax2
+ Bxy + Cy2
+ Dx + Ey + F = 0
as a conic whose equation in standard form is
A u2
+ C v2
+ D u + E v + F = 0
(that is, whose whiuch has no uv-term). (The required rotation angle is  = 1
2 Cot-1 A-C
B .) Thus we may
classify the graphs of all second degree equations in x and y as conic sections.
Can the same thing be done for equations of the form
Ax2
+ By2
+ Cz2
+ Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
and quadric surfaces in R3
?