Quadric SurfacesMath 212
Brian D. Fitzpatrick
Duke University
January 23, 2020
MATH
Overview
Level SetsDefinitionExamplesGraphs
Quadric SurfacesRotational SymmetrySpheresParaboloidsHyperboloids (One Sheet)Hyperboloids (Two Sheets)Double ConesCylinders
ExamplesRotationsShifts
Level SetsDefinition
DefinitionThe level set of f : Rn → R at c ∈ R is
Lc(f ) = {x ∈ Rn | f (x) = c}
Note that Lc(f ) ⊂ Domain(f ).
Level SetsExamples
Example
x2 + y2 = 1← f (x , y) = x2 + y2 c = 1 Lc(f ) ⊂ R2
Example
r2 − z2 = −1← f (r , z) = r2 − z2 c = − 1 Lc(f ) ⊂ R2
Example
sin(z − xy) = 0← f (x , y , z) = sin(z − xy) c = 0 Lc(f ) ⊂ R3
Example
x1x2x3x4 = 7← f (x1, x2, x3, x4) = x1x2x3x4 c = 7 Lc(f ) ⊂ R4
Level SetsExamples
Example
f (x , y) = x2 + y2
x
y
f=0
f=1
f=3
f=5
f=7
Level SetsExamples
Example
f (x , y) = x2 − y
x
y
Level SetsExamples
Example
f (x , y) = x2 − y2
x
yf=0
f=−1
f=1
f=0
f=−1
f=1
Level SetsExamples
Example
f (x , y) = x2
x
yf=1 f=1f=3 f=3f=9 f=9
Level SetsGraphs
DefinitionThe graph of f : Rn → R is the level set
z − f (x1, x2, . . . , xn) = 0
Note that Graph(f ) ⊂ Rn+1.
ObservationThe level sets Lc(f ) are the “cross sections” of Graph(f ).
Level SetsGraphs
Example
f (x , y) = x2 + y2
x y
z
Quadric SurfacesRotational Symmetry
QuestionHow can we visualize level sets in R3?
DefinitionSuppose r2 = x2 + y2 eliminates all x ’s and y ’s from a level set
f (x , y , z) = c
Then Lc(f ) has rotational symmetry about the z-axis.
Quadric SurfacesSpheres
Example
Consider the level set r2 + z2 = 1.
r
z
rotate−−−→
z
The level set x2 + y2 + z2 = 1 is a sphere of radius one.
Quadric SurfacesParaboloids
Example
Consider the level set r2 − z = 0.
r
z
rotate−−−→
z
The level set x2 + y2 − z = 0 is a paraboloid.
Quadric SurfacesHyperboloids (One Sheet)
Example
Consider the level set r2 − z2 = 1.
r
z
rotate−−−→
z
The level set x2 + y2 − z2 = 1 is a hyperboloid of one sheet.
Quadric SurfacesHyperboloids (Two Sheets)
Example
Consider the level set r2 − z2 = −1.
r
z
rotate−−−→
z
The level set x2 + y2 − z2 = −1 is a hyperboloid of two sheets.
Quadric SurfacesDouble Cones
Example
Consider the level set r2 − z2 = 0.
r
z
rotate−−−→
z
The level set x2 + y2 − z2 = 0 is a double cone.
Quadric SurfacesCylinders
Example
Consider the level set r2 = 1.
r
z
rotate−−−→
z
The level set x2 + y2 = 1 is a cylinder.
ExamplesRotations
Example
Consider the level set z2 + y2 − x2 = 0
x2 + y2 − z2 = 0 with x ↔ z
.
zx2+y2−z2=0
x↔z−−−→
xz2+y2−x2=0
z2 + y2− x2 = 0 describes a double cone opening about the x-axis.
ExamplesRotations
ObservationThe variable swaps x ↔ z and y ↔ z change orientation.
Example
Consider the level set x2 + z2 − y = 0
zx2+y2−z=0
y↔z−−−→
yx2+z2−y=0
x2 + z2 − y = 0 describes a paraboloid opening about the y -axis.
ExamplesShifts
Example
Consider the level set (x − 1)2 + (y + 2)2 + (z − 3)2 = 4.
1
O
x2+y2+z2=1 x↔x−1y↔y+2
z↔z−3−−−−−−→
2
(1,−2, 3)
(x−1)2+(y+2)2+(z−3)2=22
ExamplesShifts
ObservationThe replacements
x ↔ x − x0 y ↔ y − y0 z ↔ z − z0
shift the origin O(0, 0, 0) to the point P(x0, y0, z0).
ExamplesShifts
Example
Consider the level set −(x − 1)2 + (y + 2)2 + (z − 1)2 = 1.
z
O
x2+y2−z2=1
x↔z
x↔x−1y↔y+2
z↔z−1−−−−−−→
x
(1,−2,1)
−(x−1)2+(y+2)2+(z−1)2=1
ExamplesShifts
ObservationRecall the equations
x2 − cx =(x − c
2
)2−(c
2
)2x2 + cx =
(x +
c
2
)2−(c
2
)2
Using this algebraic trick is called completing the square.
ExamplesShifts
Example
Consider the level set
x2 + 2 x
(x + 1)2 − 1
−y2 + 14 y
−(y − 7)2 + 49
+z2 + 4 z
(z + 2)2 − 4
= 43
Completing the square gives
(x + 1)2 − 1− (y − 7)2 + 49 + (z + 2)2 − 4 = 43
which reduces to
(x + 1)2 − (y − 7)2 + (z + 2)2 = −1
y
(−1, 7,−2)