Sec.11.6 Cylinders and Quadric Surfaces
Sketches: To sketch a surface it is useful to find the traces of cross-sections where the planes passing
through the surface are parallel to the coordinated planes.
A cylinder is a surface that consist of all lines (rulings) parallel to a given line and passing through a
given plane curve (generating curve). If one of the variables x y, or z is missing from the equation of a
surface, then the surface is a cylinder.
EX1 Sketch the 3D graph of : x2 + z = 1
A quadric surface is the graph of a second degree equation in 3 variables say x, y, and z. Quadratic
surfaces are the 3D counter part of conic sections in 2D.
x2 y 2 z 2
Ellipsoid
All three traces are ellipses.
1
+
+ =
a 2 b2 c2
If a = b = c > 0, we have a sphere.
EX 2 Sketch 4x2 + 9y2 + 36z2 = 36
Elliptic paraboloid
For the first formula, horizontal traces are ellipses; vertical traces are parabolas.
The variable raised to the first power indicates the axis. Similar for other two
formulas.
z x2 y 2
x z2 y2
y x2 z 2
,
,
=
+
=
+
=
+
c a 2 b2
a c2 b2
b a2 c2
x^2+y^2=z
50
40
z
30
20
10
1
5
3
1
-1
-3
-5
0
y
-5
x
50
40
z
40-50
30
30-40
20
20-30
10-20
10
1
5
3
1
-1
-3
-5
0
0-10
y
-5
x
EX 3 Sketch 4y = x2 + z2
Hyperbolic paraboloid
z x2 y 2
z y 2 x2
=
−
,
=−
c a 2 b2
c b2 a 2
Horizontal traces are hyperbolas; vertical traces are parabolas.
There are other forms. (See k and i on p. 656.) Beware of technology!
x^2-y^2=z
30
20
10
z
0
-10
20
z
0
-20
-30
-40
2
-5 -4 -3 -2
-1 0 1
2 3 4
5
x
-6
4
2
-20
y
0 y
-4
-2
-2
0
x
2
-4
4
EX 4 Do this example by hand and compare to these.
Cone
z 2 x2 y 2
=
+ ,
c2 a 2 b2
For the first formula, horizontal traces are ellipses; vertical traces in the planes
x = k and y = k are hyperbolas if k ≠ 0, but are pairs of lines if k = 0. Similar
for other two formulas. The axis of symmetry goes with the term on the left of
the equals in the formulas given.
x2 y 2 z 2
=
+ ,
a 2 b2 c2
y 2 x2 z 2
=
+
b2 a 2 c2
EX 5 Sketch: x2 = 4y2 + 9z2
Hyperboloid of one sheet
x2 y 2 z 2
+
− = 1,
a 2 b2 c2
For the first formula, horizontal traces are ellipses; vertical traces are
hyperbolas. The axis of symmetry goes with the variable of negative
coefficient. Similar for other two formulas.
z 2 y 2 x2
+
− = 1,
c2 b2 a 2
x2 z 2 y 2
+ − = 1
a 2 c2 b2
EX 6 Sketch x2 - y2 + z2 - 5 + 2y +6z +5 = 0
Hyperboloid of two sheets
For the first formula, horizontal traces in z = k are ellipses if k > c or k < -c;
vertical traces are Hyperbolas. The two minus signs indicate two sheets.
Similar for other two formulas. The axis of symmetry goes with the positive
coefficient.
x2 y 2 z 2
z 2 y 2 x2
x2 z 2 y 2
− 2 − 2 + 2 = 1, − 2 − 2 + 2 = 1, − 2 − 2 + 2 = 1
a
b
c
c
b
a
a c
b
EX 7 y2 = 4x2 + z2 + 4