QUADRIC SURFACES
Some techniques for sketching a specialized but important
class of surfaces are considered.
The categories of the basic quadric surfaces and the relations
that describe them are listed here.
Ellipsoid :
x
a
2
2
+
y
2
b
2
+
z
c
2
2
=1
Hyperboloid of One Sheet :
x
a
2
z
Hyperboloid of Two Sheets :
c
2
Elliptic Cone : z −
x
a
2
2
−
Elliptic Paraboloid : z −
y
a
2
b
2
2
−
−
2
x
a
z
c
2
2
−
2
=1
2
y
2
b
2
=1
2
b
x
+
2
y
2
=0
2
2
−
Hyperbolic Paraboloid : z +
y
2
b
x
a
2
=0
2
2
−
y
2
b
2
=0
Page 1 of 12
2
2
y
x
.
Example # 1A: Identify the quadric surface: z =
+
4
9
Elliptic Parabaloid
z
y
x
Page 2 of 12
2
y
2
Example # 1B: Identify the quadric surface: z =
−x .
25
Hyperbolic Parabaloid
z
y
x
Page 3 of 12
2
2
2
Example # 1C: Identify the quadric surface: x + y − z = 16.
Hyperboloid of One Sheet
z
y
x
Page 4 of 12
2
2
2
Example # 1D: Identify the quadric surface: x + y − z = 0.
Elliptic(Circular) Cone
z
x
y
Page 5 of 12
2
2
2
Example # 1E: Identify the quadric surface: z − x − y = 1.
Hyperboloid of Two Sheets
z
x
y
Page 6 of 12
2
z
Example # 1F: Identify the quadric surface: x + y +
= 1.
4
2
2
Ellipsoid(Prolate Sphereoid)
z
x
y
Page 7 of 12
Example # 2: Find an equation for and sketch the surface that
2
2
results when the circular parabaloid: z = x + y is reflected about
the plane y = z.
Parabaloid & Reflection Plane
z
x
y
2
z= x +y
2
Interchange "y" & "z".
2
y=x +z
2
Page 8 of 12
Reflected Parabaloid
z
x
y
Example # 3: Find an equation of the trace and identify the
resultant conic section for the surface defined by this
2
2
equation: z = 9 ⋅ x + 4 ⋅ y when cut by the plane: z = 4.
2
2
2
2
z = 9⋅ x + 4⋅ y
4 = 9⋅ x + 4⋅ y
9 2
2
⋅x + y = 1
4
This is an ellipse in the xy-plane.
Page 9 of 12
Example # 4: Identify and sketch the quadric
2
2
2
surface: 9 ⋅ z − 4 ⋅ y − 9 ⋅ x = 36.
2
2
2
9 ⋅ z − 4 ⋅ y − 9 ⋅ x = 36
2
2
2
z
y
x
−
−
=1
4
9
4
This is an Hyperboloid of Two Sheets.
2
2
z =4+x +
4 2
⋅y
9
Find the equation of the trace in the plane z = 4.
2
16 = 4 + x +
2
4 2
⋅y
9
2
x
y
+
=1
12 27
The trace is an ellipse.
Trace in z=4
y
6
4
2
6
4
2 0 2
2
4
6
x
4
6
Page 10 of 12
Note that z ≥ 2.
Here is the "sketch".
Hyperboloid of Two Sheets
z
x
y
Example # 5: Identify the quadric surface:
2
2
2
z = 4 ⋅ x + y + 8 ⋅ x − 2 ⋅ y + 4 ⋅ z and make a sketch that shows its
position and orientation.
2
2
2
z − 4⋅ z = y − 2⋅ y + 4⋅ x + 8⋅ x
Complete the squares.
Page 11 of 12
(z 2 − 4 ⋅ z + 4) − 4 = (y 2 − 2 ⋅ y + 1) − 1 + 4 ⋅ (x 2 + 2 ⋅ x + 1) − 4
2
2
2
( z − 2) − 4 = ( y − 1) − 1 + 4 ⋅ ( x + 1) − 4
2
2
2
4 ⋅ ( x + 1) + ( y − 1) − ( z − 2) = 1
( x + 1)
⎛ 1⎞
⎜ 2⎟
⎝ ⎠
2
2
2
2
+ ( y − 1) − ( z − 2) = 1
This is an Hyperboloid of One Sheet "centered" at the point:
( −1 , 1 , 2).
Page 12 of 12