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)