Quadric Surfaces
§12.6
09 September 2013
Review: Completing The Square.
Question: Fill in the blank to make a perfect square:
x 2 + Ax +
Review: Completing The Square.
Question: Fill in the blank to make a perfect square:
x 2 + Ax +
Answer:
x 2 + Ax + (A/2)2 = x 2 + Ax + A2 /4 = (x + A/2)2 .
Review: Completing The Square.
Question: Fill in the blank to make a perfect square:
x 2 + Ax +
Answer:
x 2 + Ax + (A/2)2 = x 2 + Ax + A2 /4 = (x + A/2)2 .
Question: Fill in the blank to make a perfect square:
9x 2 + 30x +
Review: Completing The Square.
Question: Fill in the blank to make a perfect square:
x 2 + Ax +
Answer:
x 2 + Ax + (A/2)2 = x 2 + Ax + A2 /4 = (x + A/2)2 .
Question: Fill in the blank to make a perfect square:
9x 2 + 30x +
Answer:
9x 2 + 30x +
= (3x + B)2
so 2 · 3B = 30 =⇒ B = 5 =⇒ fill in 52 :
9x 2 + 30x + 25 = (3x + 5)2 .
Clicker Question: Completing The Square.
Fill in the blanks to make perfect squares:
(x 2 − 10x +
)
+
(y 2 + 3y +
)
A. 25, 9/2
B. −25, 9/2
C. 25, 9/4
D. −25, 9/4
E. I don’t understand the question
receiver channel: 41
session ID: bsumath275
Example: Quadric Plane Curve.
Question: What does this equation define:
x 2 + 10x + y 2 − 4y = 7
Example: Quadric Plane Curve.
Question: What does this equation define:
x 2 + 10x + y 2 − 4y = 7
Answer: First, complete squares:
(x 2 + 10x + 25) + (y 2 − 4y + 4) = 36
Example: Quadric Plane Curve.
Question: What does this equation define:
x 2 + 10x + y 2 − 4y = 7
Answer: First, complete squares:
(x 2 + 10x + 25) + (y 2 − 4y + 4) = 36
or
(x + 5)2 + (y − 2)2 = 36
It is the equation of
Example: Quadric Plane Curve.
Question: What does this equation define:
x 2 + 10x + y 2 − 4y = 7
Answer: First, complete squares:
(x 2 + 10x + 25) + (y 2 − 4y + 4) = 36
or
(x + 5)2 + (y − 2)2 = 36
It is the equation of a circle, centered at
Example: Quadric Plane Curve.
Question: What does this equation define:
x 2 + 10x + y 2 − 4y = 7
Answer: First, complete squares:
(x 2 + 10x + 25) + (y 2 − 4y + 4) = 36
or
(x + 5)2 + (y − 2)2 = 36
It is the equation of a circle, centered at (−5, 2) with radius 6.
Clicker Question: Circle Equation.
Find the center and radius of the circle defined by
x 2 − 6x + y 2 + 2y = 6
A. Center: (−6, 2)
Radius: 6
B. Center: (6, −2)
Radius:
C. Center: (−3, 1)
Radius: 16
D. Center: (3, −1)
Radius: 4
√
6
E. I don’t understand the question.
receiver channel: 41
session ID: bsumath275
Quadric Plane Curves.
Equations in R2 :
Curve
equation
form
Line
linear
Circle
(x − h)2 (y − k)2
+
=1
r2
r2
X 2 + Y 2 = const.
Ellipse
(x − h)2 (y − k)2
+
=1
a2
b2
X 2 + Y 2 = const.
Hyperbola
(x − h)2 (y − k)2
−
=1
a2
b2
X 2 − Y 2 = const.
Parabola
x2 = y
X2 = Y
Example: Quadric Surface
x 2 − 5x + y 2 + 8y + z 2 + 4z = 4
Complete squares:
Example: Quadric Surface
x 2 − 5x + y 2 + 8y + z 2 + 4z = 4
Complete squares:
(x 2 − 5x + 25/4) + (y 2 + 8y + 16)+(z 2 + 4z + 4)
= 4 + 25/4 + 16 + 4
= 121/4
Example: Quadric Surface
x 2 − 5x + y 2 + 8y + z 2 + 4z = 4
Complete squares:
(x 2 − 5x + 25/4) + (y 2 + 8y + 16)+(z 2 + 4z + 4)
= 4 + 25/4 + 16 + 4
= 121/4
or
(x − 5/2)2 + (y + 4)2 + (z + 2)2 = (11/2)2
Example: Quadric Surface
x 2 − 5x + y 2 + 8y + z 2 + 4z = 4
Complete squares:
(x 2 − 5x + 25/4) + (y 2 + 8y + 16)+(z 2 + 4z + 4)
= 4 + 25/4 + 16 + 4
= 121/4
or
(x − 5/2)2 + (y + 4)2 + (z + 2)2 = (11/2)2
Sphere centered at (5/2, −4, −2), radius 11/2.
Example, Changed.
Sphere:
(x − 5/2)2 + (y + 4)2 + (z + 2)2 = (11/2)2
Changed:
2
(x − 5/2) +
y +4
3
2
+ (z + 2)2 = (11/2)2
Example, Changed.
Sphere:
(x − 5/2)2 + (y + 4)2 + (z + 2)2 = (11/2)2
Changed:
2
(x − 5/2) +
y +4
3
2
+ (z + 2)2 = (11/2)2
Stretched by a factor of 3 in the y direction.
Example, Changed.
Sphere:
(x − 5/2)2 + (y + 4)2 + (z + 2)2 = (11/2)2
Changed:
2
(x − 5/2) +
y +4
3
2
+ (z + 2)2 = (11/2)2
Stretched by a factor of 3 in the y direction.
Ellipsoid (stretched sphere)
Sphere and Ellipsoid Equation Form.
x2 + y2 + z2 = r2
x 2 y 2 z 2
+
+
=1
r
r
r
Sphere
Ellipsoid
x 2
a
+
y 2
b
+
z 2
c
=1
Spheres and ellipsoids have equations of the form
X 2 + Y 2 + Z 2 = const.
Other forms.
Form
Quadratic
Signs
Surface
X2 + Y 2 + Z2 = 1
+++
sphere/ellipsoid
X2 + Y 2 = Z2 + 1
X2 + Y 2 = Z2 − 1
X2 + Y 2 = Z2
++−
+−−
++−
hyperboloid of one sheet
hyperboloid of two sheets
cone
X2 + Y2 = Z
X2 − Y2 = Z
++0
+−0
elliptic parabaloid
hyperbolic parabaloid
Others:
cylinders & degenerate
Other forms.
Form
Quadratic
Signs
Surface
X2 + Y 2 + Z2 = 1
+++
sphere/ellipsoid
X2 + Y 2 = Z2 + 1
X2 + Y 2 = Z2 − 1
X2 + Y 2 = Z2
++−
+−−
++−
hyperboloid of one sheet
hyperboloid of two sheets
cone
X2 + Y2 = Z
X2 − Y2 = Z
++0
+−0
elliptic parabaloid
hyperbolic parabaloid
Others:
cylinders & degenerate
Other forms.
Form
Quadratic
Signs
Surface
X2 + Y 2 + Z2 = 1
+++
sphere/ellipsoid
X2 + Y 2 = Z2 + 1
X2 + Y 2 = Z2 − 1
X2 + Y 2 = Z2
++−
+−−
++−
hyperboloid of one sheet
hyperboloid of two sheets
cone
X2 + Y2 = Z
X2 − Y2 = Z
++0
+−0
elliptic parabaloid
hyperbolic parabaloid
Others:
cylinders & degenerate
*May switch X /Y /Z (e.g., cone Y 2 + Z 2 = X )
**Challenge: “mixed” terms xy , xz, yz (not used in class).
Traces.
I
Traces: Intersections with vertical planes x = d or y = e,
or horizontal planes z = f
I
For example: ellipsoid
I
intersection with x = d:
y 2
b
+
x 2
a
z 2
c
+
y 2
b
+
c
2
d
=1−
,
a
ellipse.
I
z 2
traces on y = e, on z = f : also ellipses
=1
Clicker Question: Hyperboloid Traces.
In a hyperboloid of one sheet, the
vertical and horizontal traces are:
A. Vertical: lines
Horizontal: ellipses
B. Vertical: ellipses
Horizontal: hyperbolas
C. Vertical: hyperbolas
Horizontal: ellipses
D. I don’t understand the
question.
receiver channel: 41
session ID: bsumath275