Math 2204 Multivariable Calculus – Chapter 12: Vectors and the Geometry of
Sec. 12.6: Cylinders and Quadratic Surfaces
I.
Review of Conic Sections
A. Parabolas
y = ±x 2
or
x = ±y 2
B. Ellipses
x 2 y2
+
=1
a2 b2
C. Hyperbolas
If
a=b, then we have a circle.
x 2 y2
y2 x 2
−
=
1
−
=1
or
a2 b2
a2 b2
II.
Cylinders
A. A cylinder is the surface composed of all the lines (rulings) that lie parallel to a given line
in space and pass through a given plane curve (a generating curve).
B. An equation in any two Cartesian coordinates defines a cylinder parallel to the axis of the
third (unrestricted) coordinate.
C. Examples
1. Graph
y=x2.
Runs along:
Coordinate Plane
Trace
xy (z = 0)
xz (y = 0)
yz (x = 0)
z=k
y=k
x=k
2. Graph
9y2+z2=16.
Coordinate Plane
Runs along:
Trace
xy (z = 0)
y 2 = 169 ⇒ y = ± 43 ⇒ lines
xz (y = 0)
z 2 = 16 ⇒ z = ±4 ⇒ lines
yz (x = 0)
9y 2 + z 2 = 16 ⇒ ellipse
x=k
III.
Quadratic Surfaces
(See p. 641 for pictures)
A. A quadratic surface is the graph in space of a second-degree equation in
x, y, and z.
B. Quadratic surfaces are the 3D counter part of conic sections in 2D. See page 655.
C. Types
1.
Ellipsoid
x2 y2 z2
+
+
=1
a 2 b2 c 2
a. all variables are squared, all coefficients are positive
b. All three traces are ellipses.
c. If any two of the semiaxes
revolution.
d. If
a, b, and c are equal, the surface is an ellipsoid of
a=b=c, we have a sphere.
e. Example
Sketch
9x 2 + 4y 2 + 36z 2 = 36
Coordinate Plane
xy (z = 0)
xz (y = 0)
yz (x = 0)
Trace
1
4
x 2 + 19 y 2 = 1 ⇒ ellipse
1
4
x 2 + z 2 = 1 ⇒ ellipse
1
9
y 2 + z 2 = 1 ⇒ ellipse
Runs along:
2. Elliptic Paraboloid
z x2 y2
=
+ ,
c a 2 b2
x z2 y2
= + ,
a c 2 b2
y x2 z2
=
+
b a2 c2
a. One variable is linear, coefficients for the two squared terms are the same sign
b. The variable raised to the first power indicates the axis.
c. For the first formula,
1) horizontal traces are ellipses; vertical traces are parabolas.
2) If a=b, then we have a circular paraboloid.
d. Similar for other two formulas.
e. Example
Sketch
y = x 2 + 4z 2
Coordinate Plane
Runs along:
Trace
xy (z = 0)
y = x 2 ⇒ parabola
xz (y = 0)
x 2 ≠ −4z 2 ⇒ No graph
yz (x = 0)
y = 4z 2 ⇒ parabola
y=k>0
y=k<0
3. Elliptic Cone
x2 y2 z2
+
− = 0,
a 2 b2 c 2
x2 y2 z2
−
− = 0,
a 2 b2 c 2
x2 y2 z2
−
+ =0
a 2 b2 c 2
a. all variables are squared, 1 or 2 negative coefficients, RHS=0 in standard form
b. 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.
c. Example
Sketch
x 2 = 4y 2 + 9z 2
Coordinate Plane
Runs along:
Trace
xy (z = 0)
x = ±2y ⇒ lines
xz (y = 0)
x = ±3z ⇒ lines
yz (x = 0)
0 = 4y 2 + 9z 2 ⇒ pt(y, z) = (0,0)
x=k
k 2 = 4y 2 + 9z 2 ⇒ ellipses
4. Hyperboloid of One Sheet
x2 y2 z2
+
− = 1,
a 2 b2 c 2
z2 y2 x2
+
−
= 1,
c 2 b2 a 2
x2 z2 y2
+ −
=1
a 2 c 2 b2
a. all variables are squared, 1 negative coefficient, RHS=1 in standard form
b. 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.
c. Example
Sketch
9x 2 − 4y 2 + 36z 2 = 36
Coordinate Plane
Trace
xy (z = 0)
9x 2 − 4y 2 = 36 ⇒ hyperbola
xz (y = 0)
9x 2 + 36z 2 = 36 ⇒ ellipse
yz (x = 0)
−4y 2 + 36z 2 = 36 ⇒ hyperbola
y=k
9x 2 + 36z 2 = 36 + 4k 2 ⇒ ellipses
Runs along:
5. Hyperboloid of Two Sheets
x2 y2 z2
z2 y2 x2
x2 z2 y2
− 2 − 2 + 2 = 1, − 2 − 2 + 2 = 1, − 2 − 2 + 2 = 1
a
b
c
c
b
a
a
c
b
a. all variables are squared, 2 negative coefficients, RHS=1 in standard form
b. 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.
c. Example
Sketch
−9x 2 − 4y 2 + 36z 2 = 36
Coordinate Plane
Runs along:
Trace
xy (z = 0)
−9x 2 − 4y 2 = 36 ⇒ no graph
xz (y = 0)
−9x 2 + 36z 2 = 36 ⇒ hyperbola
yz (x = 0)
−4y 2 + 36z 2 = 36 ⇒ hyperbola
z=k
k 2 < 1 (−1 < k < 1)
k 2 > 1 (k < −1 or k > 1)
k 2 = 1 (k = ±1)
−9x 2 − 4y 2 = 36 − 36k 2 ⇒ 9x 2 + 4y 2 = 36k 2 − 36
9x 2 + 4y 2 = 36k 2 − 36 < 0 ⇒ No graph
9x 2 + 4y 2 = 36k 2 − 36 > 0 ⇒ ellipses
9x 2 + 4y 2 = 36k 2 − 36 = 0 ⇒ pt(x, y) = (0,0)
6. Hyperbolic Paraboloid
z x2 y2
=
− ,
c a 2 b2
z y2 x2
=
−
c b2 a 2
a. One variable is linear, coefficients for two squared terms are the different signs
b. Horizontal traces are hyperbolas; vertical traces are parabolas.
There are other forms. (See HW p. 642. k & l)
c. Example
Sketch
z = −x 2 + y 2
Coordinate Plane
Runs along:
Trace
xy (z = 0)
y = ±x ⇒ lines
xz (y = 0)
z = −x 2 ⇒ parabola, opens downward
yz (x = 0)
z = y 2 ⇒ parabola, opens upward
z=k>0
−x 2 + y 2 = k > 0 ⇒ hyperbola, opens along y
z=k<0
−x 2 + y 2 = k < 0 ⇒ x 2 − y 2 = k > 0
⇒ hyperbola, opens along x
D. Extra Example:
1. Sketch
z = 4x 2 + y 2 − 2y
Coordinate Plane
Runs along:
Trace
xy (z = 0)
xz (y = 0)
yz (x = 0)
z=k>
z=k<
2. Identify each of the following surfaces:
a.
4x 2 + 4y 2 + z 2 − 1 = 0
b.
−4x 2 + y 2 − 2z 2 = 0
c.
x 2 + 6x − y 2 − z 2 = 0