WOLFRAM|DEMONSTRATIONS PROJECT

Level Surfaces and Quadratic Surfaces

​
level
k
1.25
function
f(x,y,z)
2
x
2
a
+
2
y
2
b
+
2
z
2
c
n
2
x
2
a
+
2
y
2
b
+
(1-n)
2
z
2
c
2
x
2
a
+
2
y
2
b
+c
n
(-1)
z
n
(-1)
2
x
2
a
+
2
y
2
b
+
n+1
(-1)
2
z
2
c
n
(-1)
2
x
2
a
-
2
y
2
b
+
n+1
(-1)
2
z
2
c
constants
a
2
b
2
c
2
n
For a function
f
of three variables,
x
,
y
, and
z
, the level surface of level
k
is defined as the set of points in
3

that are solutions of
f(x,y,z)=k
. A quadratic surface or quadric is a surface that is given by a second-order polynomial equation in the three variables
x
,
y
, and
z
.
Let
a
,
b
, and
c
be nonzero constants. We plot level surfaces for quadratic functions in three variables, which give some well-known quadratic surfaces:
⋄
2
x
2
a
+
2
y
2
b
+
2
z
2
c
=k
gives ellipsoids; when
a=b=c
, this is a sphere centered at the origin of radius
k
.
⋄
2
x
2
a
+
2
y
2
b
=1
or
2
y
2
b
+
2
z
2
c
=1
give elliptical cylinders with symmetry axes along the
z
axis and
x
axis, corresponding to
n=1
and
n=0
.
⋄
2
x
2
a
+
2
y
2
b
+
n
(-1)
cz
gives elliptic paraboloids, opening up or down as
n=1
or
n=0
.
⋄
2
x
2
a
+
2
y
2
b
-
2
z
2
c
and
2
x
2
a
+
2
y
2
b
+
2
z
2
c
​
, with
k=0
, give elliptic cones. For
k≠0
, the level surfaces are hyperboloids of one sheet.
⋄
2
x
2
a
-
2
y
2
b
-
2
z
2
c
(
n=0
) and
-
2
x
2
a
-
2
y
2
b
+
2
z
2
c
(
n=1
) give hyperboloids of two sheets.