Level Surfaces and Quadratic Surfaces
Level Surfaces and Quadratic Surfaces
For a function of three variables, , , and , the level surface of level is defined as the set of points in that are solutions of . A quadratic surface or quadric is a surface that is given by a second-order polynomial equation in the three variables , , and .
f
x
y
z
k
3
f(x,y,z)=k
x
y
z
Let , , and be nonzero constants. We plot level surfaces for quadratic functions in three variables, which give some well-known quadratic surfaces:
a
b
c
⋄ ++=k gives ellipsoids; when , this is a sphere centered at the origin of radius .
2
x
2
a
2
y
2
b
2
z
2
c
a=b=c
k
⋄ +=1 or +=1 give elliptical cylinders with symmetry axes along the axis and axis, corresponding to and .
2
x
2
a
2
y
2
b
2
y
2
b
2
z
2
c
z
x
n=1
n=0
⋄ ++cz gives elliptic paraboloids, opening up or down as or .
2
x
2
a
2
y
2
b
n
(-1)
n=1
n=0
⋄ +- and ++, with , give elliptic cones. For , the level surfaces are hyperboloids of one sheet.
2
x
2
a
2
y
2
b
2
z
2
c
2
x
2
a
2
y
2
b
2
z
2
c
k=0
k≠0
⋄ -- () and () give hyperboloids of two sheets.
2
x
2
a
2
y
2
b
2
z
2
c
n=0
--+
2
x
2
a
2
y
2
b
2
z
2
c
n=1