Calc 2 Lecture Notes
Section 10.6
Page 1 of 10
Section 10.6: Surfaces in Space
Big idea: Quadratic equations in three variables produce some interesting-looking surfaces that
can be understood by thinking of what the graph of the intersection of the surface with various
planes looks like.
Big skill: You should be able to sketch graphs of the basic three-dimensional cylindrical and
quadric surfaces.
Cylindrical Surfaces
An equation that is supposed to represent a 3D graph, but that only has two of the three variables
in it (like x2 + y2 = 1) creates a graph called a cylindrical surface. To visualize what is going on,
it is common to draw sketches of the intersection of the graph of the equation with various planes
perpendicular to the unspecified variable’s axis (like z = 0, z = 1, z = 2, z = -1, etc.). Such
sketches of the intersection of the 3D graph with a plane are called traces. It turns out that all
the traces on planes parallel to the two variables for cylindrical surfaces are identical, which
means that to sketch the 3D surface, just draw the trace, then extend it along both directions of
the third, unspecified variable.
Practice:
Sketch the 3D surface described by the equation x2 + y2 = 1.
y
y
x
Trace on the plane
z=0
y
y
x
x
Trace on the plane
z=1
Trace on the plane
z=2
z
x
y
x
Trace on the plane
z = -1
Calc 2 Lecture Notes
Section 10.6
Page 2 of 10
Sketch the 3D surface described by the equation z = y2.
Trace on the plane
Trace on the plane
Trace on the plane
Trace on the plane
z
x
y
Sketch the 3D surface described by the equation x = sin(z).
Trace on the plane
Trace on the plane
Trace on the plane
z
x
y
Trace on the plane
Calc 2 Lecture Notes
Section 10.6
Page 3 of 10
Quadric Surfaces
A quadric surface is any 3D surface described by a quadratic equation in three variables:
ax 2  by 2  cz 2  dxy  eyz  fxz  gx  hy  jz  k  0 .
Amazingly, all possible quadric surfaces fall into one of seventeen general categories. (just like
the graphs of all possible quadratic equations in two variables ax 2  by 2  cxy  dx  ey  f  0
are one of the conic sections). Of the seventeen categories, only twelve can be graphed using
real-valued coordinate axes. See http://mathworld.wolfram.com/QuadraticSurface.html for more
details.
In this section, we will restrict our discussion to cases where d = e = f = 0. The effect of these
coefficients is to produce rotations of the basic quadric surfaces.
To graph quadric surfaces by hand, you should look at traces parallel to the xy, yz, and xz planes
to get a feel for what is going on, then try to assemble all the “slices” into a 3D sketch. To graph
quadric surfaces in Winplot, you should solve the equation for z, then generate an explicit graph
in the 3D window.
The Twelve Graphable Quadric Surfaces:
1. Ellipsoids (includes Spheres)
2
2
2
a. Spheres
 x  x0    y  y0    z  z0   r 2
Notice that this corresponds to the case of a = b = c in the general quadric
equation.
2
2
2
Practice: Sketch the surface described by the equation  x  1   y  2    z  2   9 .
Trace on the plane
Trace on the plane
Trace on the plane
z
x
y
Trace on the plane
Calc 2 Lecture Notes
Section 10.6
 x  x0 
2
 y  y0 
2
Trace on the plane
Trace on the plane
2
Trace on the plane
z
x

 z  z0 
1
bx 2
by 2
bz 2
 Notice this corresponds to the case where a, b, and c all have the same sign.
( ax 2  by 2  cz 2  dxy  eyz  fxz  gx  hy  jz  k  0 )
 2bx, 2by, and 2bz are the lengths of the axes of the ellipsoid.
 Notice that traces in the planes x = constant, y = constant, or z = constant are
ellipses.
 Notice that if two of the intercepts are equal, then the ellipsoid is a “circular
ellipsoid” that is the surface of revolution of a 2D ellipse about one of its axes.
y2 z2
Practice: Sketch the surface described by the equation x 2    1 .
9
4
b. Ellipsoids

Page 4 of 10
y
Trace on the plane
Calc 2 Lecture Notes
Section 10.6
2. Elliptic Paraboloids z  z0 
 x  x0 
2

 y  y0 
Page 5 of 10
2
bx 2
by 2
 Notice this corresponds to cases where a and b have the same sign and c = 0.
( ax 2  by 2  cz 2  dxy  eyz  fxz  gx  hy  jz  k  0 )
 Notice that the “bowl” this surface generates has its minimum at (x0, y0, z0).
 Notice that traces in planes z = constant are ellipses, and traces in the planes
x = constant and y = constant are parabolas.
Notice that if bx and by are equal, then the paraboloid is a “circular paraboloid” that is
the surface of revolution of a parabola about its axis of symmetry.
x2 y 2
Practice: Sketch the surface described by the equation z   .
4
9
Trace on the plane
Trace on the plane
Trace on the plane
z
x
y
Trace on the plane
Calc 2 Lecture Notes
3. Elliptic Cones
Section 10.6
 z  z0  
2
 x  x0 
2

Page 6 of 10
 y  y0 
2
bx 2
by 2
 Notice this corresponds to cases where a and b have the same sign, but c has the
opposite sign ( ax 2  by 2  cz 2  dxy  eyz  fxz  gx  hy  jz  k  0 ).
 Notice that the “touch point” of the two cones is at (x0, y0, z0).
 Notice that traces in the planes z = constant are ellipses, and traces in the planes
x = x0 and y = y0 are lines.
 Notice that if bx and by are equal, then the cone is a “circular cone” that is the surface
of revolution of intersecting lines about an axis bisecting their intersection point.
y2
Practice: Sketch the surface described by the equation z 2  x 2 
.
4
Trace on the plane
Trace on the plane
Trace on the plane
z
x
y
Trace on the plane
Calc 2 Lecture Notes
Section 10.6
Page 7 of 10
 x  x0 
2
 y  y0 
2
Trace on the plane
Trace on the plane
Trace on the plane
z
x
y

 z  z0 
2
1
bx 2
by 2
bz 2
 Notice this also corresponds to cases where a and b have the same sign, and c has the
opposite sign, but the constant left over after completing the squares is negative
( ax 2  by 2  cz 2  dxy  eyz  fxz  gx  hy  jz  k  0 ).
 Notice that traces in the planes z = constant are ellipses, and traces in the planes
x = constant and y = constant are hyperbolas.
 Notice that if bx and by are equal, then the hyperboloid is a “circular hyperboloid” that
is the surface of revolution of a hyperbola about an axis perpendicular to the foci axis.
x2
z2
Practice: Sketch the surface described by the equation
 y2   1 .
4
2
4. Elliptic Hyperboloids of One Sheet

Trace on the plane
Calc 2 Lecture Notes
Section 10.6
Page 8 of 10
 x  x0 
2
 y  y0 
2
Trace on the plane
Trace on the plane
Trace on the plane
z
x
y

 z  z0 
2
 1
bx 2
by 2
bz 2
 Notice this also corresponds to cases where a and b have the same sign, and c has the
opposite sign, but the constant left over after completing the squares is positive
( ax 2  by 2  cz 2  dxy  eyz  fxz  gx  hy  jz  k  0 ).
 Notice that traces in the planes z = constant are ellipses, and traces in the planes
x = constant and y = constant are hyperbolas (that open the opposite way of a hyperboloid
of one sheet).
 Notice that if bx and by are equal, then the hyperboloid is a “circular hyperboloid of
two sheets” that is the surface of revolution of a hyperbola about an axis parallel to the
foci axis.
x2
z2
 y 2   1 .
Practice: Sketch the surface described by the equation
4
2
5. Elliptic Hyperboloids of Two Sheets

Trace on the plane
Calc 2 Lecture Notes
Section 10.6
z  z0 
6. Hyperbolic Paraboloids
Page 9 of 10
 x  x0 
2

 y  y0 
2
bx 2
by 2
 Notice this corresponds to cases where a and b have opposite signs, and c = 0
( ax 2  by 2  cz 2  dxy  eyz  fxz  gx  hy  jz  k  0 ).
 This quadric surface is “saddle shaped”.
 Notice that traces in the planes z = constant are hyperbolas that open in one direction
for z > z0 and that open in the opposite direction for z < z0. The traces in the planes
x = constant and y = constant are parabolas that open in opposite directions and that are
perpendicular to each other.
Practice: Sketch the surface described by the equation z  2 y 2  x 2 .
Trace on the plane
Trace on the plane
z
x
y
Trace on the plane
Trace on the plane
Calc 2 Lecture Notes
Section 10.6
Page 10 of 10
x2 y 2

1
a 2 b2
Note: we already looked at an example of this in the cylindrical surface section.
7. Elliptic Cylinder
8. Parabolic Cylinder x 2  2rz  0
Note: we already looked at an example of this in the cylindrical surface section.
x2 y 2

 1
a 2 b2
Note: this is a cylindrical surface.
9. Hyperbolic Cylinder
10. Coincident Planes
x2  0
11. Parallel Planes
x2  a2
12. Intersecting Planes
x2 y 2

0
a 2 b2