quadric surfaces - math 212quadric surfaces math 212 brian d. fitzpatrick duke university january...

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)