# 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