algebraic geometry - university of washington€¦ · algebraic geometry a personal view cse 590b...

Algebraic Geometry A Personal View

CSE 590B

James F. Blinn

[email protected]

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University of Michigan

1967-1974

Gordon Romney (U Utah) 1969

+ Appendix

University of Utah 1974-1977

JPL/Caltech 1977-1995

JPL/Caltech 1977-1995

Voyager Cosmos

The Mechanical Universe Mathematics!

Render 3D Objects

Planar Polygons = First Order Surfaces

Render 3D Objects

Second Order Surfaces

Render 3D Objects

Third (and higher) Order Surfaces

UM, UU, JPL, Microsoft and Now

1962-present

Studying Algebraic Geometry

Algebraic Equations

Geometric Shapes

Making Algebraic Geometry

More Understandable

Jim Blinn’s Corner Articles 1987 - 2007

Many of them on Algebraic Geometry

Why Am I Here

• Share my enthusiasms

• Help me organize my ideas

I work better if I have an audience (M.B.)

Updates to old articles

Unpublished articles

Keep me from repeating myself

Publish on web site

• One Session every 2 weeks

• Later meetings may get more sketchy

• Discuss open questions

Why Are You Here

•Varied Audience

• Go slowly at first

• Prerequisites:

• vectors and matrices

• homogeneous coords

•See old stuff in new ways

•See new stuff

What I will talk about

•Real Algebraic Projective Geometry

• Real is more complex than Complex

• Projective is simpler than Euclidean

•Dimension 1,2,3

•Lowish Order Polynomials

•Notation, notation, notation

•Lots of Pictures

Pictures? Why no

can fool you

show only special cases

hard to generalize to high dimensions

hard to make

forces you to think (visualize internally)

Why yes

intuition

see patterns

I am visual thinker (see patterns)

pretty

Hartshorne vs. Abraham&Shaw

2 2 1X Y

Relation Between

Algebra and Geometry

2 24 1X Y

Relation Between


Relation Between


2 24 1X X Y

General Quadratic Curve

2 22

2 2

0

A X B XY CY

D X EY

F

Quadratic Curve

2 22

2 2 0

A X B XY CY

D X EY F

2 2 2... 2ACF BED D C E A B F D

Discriminant , , , , , 0A B C D E F D

Cubic Curve 3 2 2 3

2 2

3 3

3 6 3

3 3 0

AX BX Y CXY DY

EX FXY GY

HX JY K

1

Discriminant of Cubic

, , , , , , , , , 0A B C D E F G H J K D

G. Salmon (1879):

1

4 4 4 4 3 3

4 2 2 2 2 3 3 3 3

2 3 3 2 3 3 2 2

3 3 3 3

12

36 64

192 192

64 ...

A D K A D K GJ

A D K G J A D K F

A D K F BE AD K FB E

D K B E

D

D has over 10,000 terms


3 264S T D

S: degree 4 in A…K

has 25 terms

T: degree 6 in A…K

has 103 terms

1

Want Better Notation

Notation = Creative Abbreviation

a c b e

f h d k

ab cd e

fb hd k

M v w

N M v NM v

Review of Typical Notation

And some snags

2D Euclidean Geometry

X

Y

X Y X YP

What Went Wrong?

Top View Front View (Post Perspective)

2D Projective Geometry

3D Algebraic Objects

X

Y

x y

w w

x y wP

x y w

Equation of a Line

L

x y wP

0ax by cw

0

a

x y w b

c

Equation of a Line

a

b

c

L

x y wP

0

a

x y w b

c

0 P L

Row/column standardization?

0ax by cw

Two Points Make A Line

S

L

P

L P S

0

0

P P P

S S S

ax y w

bx y w

c

P P P

S S S

a x y w

b

c x y w

, ,P S P S P S P S P S P Sa y w w y b w x x w c x y y x

Two Lines Make A Point

M

P

L

0 0

L M

L M

L M

a a

x y w b b

c c

P L M

L M

L M

L M

a a

x y w b b

c c

Transforming Points

ˆPT P

11 12 13

21 22 23

31 32 33

ˆ ˆ ˆ

T T T

x y w T T T x y w

T T T

Transforming Lines

0 P L

1 0 PT T L

1 T L L

1 0 P TT L

0 P L

PT P

Matrix Adjugate (fka Adjoint)

1

2

3

R

R

R

T *

2 3 3 1 1 2R R R R R R

T

*

det 0 0

0 det 0

0 0 det

T

TT T

T

1 0 0

det 0 1 0

0 0 1

T

?

Transforming Points and Lines

* T L L

* * *11 12 13

* * *21 22 23

* * *31 32 33

ˆ

ˆ

ˆ

aT T T a

T T T b b

T T T c c

PT P 11 12 13

21 22 23

31 32 33

ˆ ˆ ˆ

T T T

x y w T T T x y w

T T T

Point on Quadratic Curve 2

2

2

2 2

2

0

Ax Bxy Cxw

Dy Eyw

Fw

0

A B C x

x y w B D E y

C E F w

0T P Q P

P

Q

Transforming a Quadratic

0T PQP

* * 0T

T P TT Q TT P

* *T T QT Q

* * 0TT PT T QT PT

0T PQP

PT P

Given Point, Find Tangent

P

Q

T P QP

0 T PQP

P L

Given Point, Find Tangent

P

Q

L = Q PT T P QP

0 T PQP

P L

Is a Line Tangent to Q

L

Q

*0 T L Q L

Given Tangent, Find Point

L

Q

P L

*T L Q L

*0 TL Q L

Given Tangent, Find Point

L

Q

LT Q

* =P

*

*

0 T

T

L Q L

L Q L

P L

Three Kinds of Matrix

point point T

point lineT

Q

*line pointT Q

Point on Cubic Curve

3 2 2 3

2 2

2 2

3

3 3

3 6 3

3 3

0

Ax Bx y Cxy Dy

Ex w Fxyw Gy w

Hxw Jyw

Kw

P=[x,y,w]

C

Forms of Cubic Curve Equation 3 2 2 3

2 2

2 2

3

3 3

3 6 3

3 3

0

Ax Bx y Cxy Dy

Ex w Fxyw Gy w

Hxw Jyw

Kw

0

A B E B C F E F H x x

x y w B C F C D G F G J y y

E F H F G J H J K w w

0T T PCP P

Forms of Cubic Curve Equation 3 2 2 3

2 2

2 2

3

3 3

3 6 3

3 3

0

Ax Bx y Cxy Dy

Ex w Fxyw Gy w

Hxw Jyw

Kw

0




, ,

, ,

0i j k i j k

i j k

PP P C

Two Problems With Notation

A B E B C F E F H

B C F C D G F G J

E F H F G J H J K

C

point lineT

Q

Row vs. Column Confusion

Handing More Than Two Indices

The Solution

• Steal Notational Tricks from Physics

– General Relativity

– Quantum Mechanics

• Tuned to Algberaic Geometry

Old Index Types

1 2 3P P PP

1

2

3

L

L

L

L

Column

Row

New Index Types

1 2 3P P P P

1 2 3L L LL

CoVariant

ContraVariant

The Multiplication Machine

1 2 3

1 2 3

i

i

i

P L P L P L

P L

P L

Einstein

Index

Notation

1

1 2 3 2

3

L

P P P L

L

P L


ijQPure covariant

*ij

Q

Pure contravariant

i

jTMixed


i

ij jP Q L

*ij

j

iL Q P

j i i

jP T P

General Tensor Contraction

k lu j l

ij km u imF H R S W

Taking More Ideas from Physics

Writing Tensor Contraction in

Diagram Form

P L

P L


i

ij jP Q L

*ij

j

iL Q P

ˆj i i

jP T P Pij

=~ i

P T

ji=

jP Q L

ji=

jL Q* P

General Tensor Contraction

k lu j l

ij km u imF H R S W

=k

i

lF

R j

H

u

S

m

W

il

m

Don’t need index labels

=F

R

H

S

W

Just be careful about matching dangling arcs

Sum of Terms

= +P R T S

P RT S

i j i i

jP R T S

Consistent type evident

Scalar Product

P R S

R S

P R S

Only Connectivity Matters

=P T L

L P

T

Rearranging internal arcs/nodes doesn’t change value

Now Back To Geometry

Point on a Line

0ax by cw 0

a

x y w b

c

0 P L

0i

iP L P L = 0

L

P

Point on a Quadratic Curve

2

2

2

2 2

2

0

Ax Bxy Cxw

Dy Eyw

Fw

0

A B C x

x y w B D E y

C E F w

0T P Q P

0i j

ijP Q P P Q P = 0

P

Q

Point on a Cubic Curve 3 2 2 3

2 2

2 2

3

3 3

3 6 3

2 3

0

Ax Bx y Cxy Dy

Ex w Fxyw Gyw

Hxw Jyw

Kw

0




0i j k

ijkP P P C P = 0C

P

P

P

C

Transforming a Point

PT P

i j j

iP T P

P T = P~

Transforming a Line

* T L L

*i

i jjT L L

=LT*

L~

Transforming A Quadratic Curve

* *T

T Q T Q

* *i j

ij klk lT Q T Q

= QQT*

T* ~

Transforming A Transformation

* T MT M

*i lj l

i kjkT M T M

= MMT* T ~

Transforming a Cubic Curve

* * *i j k

ijk lmnl m nT T T C C

=CT*

T*

T*

C~

General Transformation Rule

DT*

T*

T

TÞ D ~

D

Dot and Cross Product

L

P

S

?

becomes

P L = s

Levi-Civita Epsilon

123 231 312

321 132 213

1

1

0 otherwiseijk

0 0 0 0 0 1 0 1 0

0 0 1 0 0 0 1 0 0

0 1 0 1 0 0 0 0 0

Cross Product

0 0 0 0 0 1 0 1 0

0 0 1 0 0 0 1 0 0

0 1 0 1 0 0 0 0 0

S

P P P S

S

P S P S P S P S P S P S

x

x y w y

w

y w w y w x x w y x x y

i j

ijk kP S L

Levi-Civita Epsilon Diagram

ijk

jk

i

Cross Product

1

1 2 3 1 2 3

2

3

L

P P P S S S L

L

P S L

i j

ijk kP S L

P

S

= L

i

j k k

Anti-Symmetry and Epsilon

P S S P

P S S P

Mirror Reflections flip sign

AxA=0

A A A A

A A

0

Two Types of Epsilon

ijk

jk

i

ijkjk

i

CONTRAvariant

COvariant

The Other Cross Product

1 1

1 2 3

2 2

3 3

L M

L M P P P

L M

L M P

ijk k

i jL M P

L

M

= P

i

j k k

Triple Product

L M N M N L N L M

L

M

N

P R S R S P S P R

P

R

S

P

R

S

L

N

M

PRS

LMN

Generating Algebraic

Relations Between Diagrams

A

B C

D

A

B C

A

B C

D D

? ?

Linear Combinations of Points

B

BR

R

B

R

BR gG

BR

G

g

Linear Combinations of Points

g

g

B

W R

G

det

,

det

W

R

G

B

R

G

det

,

det

B

W

G

B

R

G

det

,

det

g

B

R

W

B

R

G

Cramer’s Rule

Basic Linear Relationship

det

,

det

W

R

G

B

R

G

det

,

det

B

W

G

B

R

G

det

,

det

g

B

R

W

B

R

G

det

B

R

G

det

W

R

G

det

B

W

G

det

B

R

W

g

Note Symmetry

Grassman-Plucker relation

Arc Swapping Identity

Arc Swapping Identity - Variations

Swap heads

Swap tails

Swap heads (dual)

An Application of Arc Swapping

M

M

M


M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M


M

M

M

M

M

M3

* detMM M I

M

M

M

M

M

M

M

M

M

M

M

M

Compare with:

Relation of Diagram to Adjugate

M

Mn

i

k

m

l

jij kl

mn ikn ljmD M M

M* 2

M

M

33 22 33 22321 231

22 33 22 33

231 321 22 33 32 23

11 23 32 23 32

231 231

32 2332 23

321 321

2

M M M M

M M M MD M M M M

M M M M

M MM M

Example element:

Constant factor

Actual adjugate

Adjugate and Determinant

M

M

M MM =det M

M* 2

Another Arc Swap Application


2


2 2

Epsilon-Delta Rule

jklm

jl

km

jm

kl

m

j

j j

l

k k k

l lm m

j

ii

j

Algebraic Interpretation

A B C A C B A B C a f a f a f

AA

B BC C

A

B C

Projection from L thru C onto G

L

S

C

G


L

S

C

GR

G

L C

=S

G R S

L C R


L

S

C

S L C


L

S

C

G

G

L C

G

L C

S

0 S G

S C G L L G C

C G

L G

S L C

0 L G C G

Shadow Projection Matrix

L

S

C

G

GL

C

= S

GL

T

TC S

An Important Identity

T

TT

What is transformed Epsilon?

Use Modification of EpsDel

Apply to Transformed Epsilon

T

TT

T

TT

T

TT

Mirror Reflection = Change Sign

T

TT

T

TT

T

TT

T

TT

T

TT

T

TT

Move over = sign

T

TT

T

TT2

T

TT

T

TT

T

TT

Recall previous diagram

T

TT

T

TT2

2

T

TT det T

T

TT 2

2det T

}

An Important Identity

T

TT

=

det T

Diagram of

Transformed Quadratic Determinant

Q

Q

QT

T

T

T

T

T Q

Q

Q

Q

Q

Q

(det T)2

T

T

T

T

T

T Q

Q

Q

Weight

MAJOR PUNCHLINE

Of all the Gazillion possible

polynomials in the

coefficients

Tensor Diagrams express

only those that represent

Invariant Properties Q

Q

Q

Trace of Matrix

trace i

i

i

TT =T

=Q

?trace ii

i

QQ


1

C C C 0

C

C


1

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C C

CC C

C


C

C

C

C

124S

3 264S T D

S: degree 4 in A…K

has 25 terms

T: degree 6 in A…K

has 103 terms

C C

CC C

C

16T

1

“Phase space” of cubics

C C

CC C

C

C

C

C

C

C C

CC C

C

C

C

C

C

C

C

C

C

D =

C C

CC C

C

C

C

C

C

, C C

CC C

C

C

C

C

C

C

C

C

C

{ ,}

3 264S T D

History of Diagrammatic

Invariant Notation

1878 Sylvester & Clifford

1885 Kempe

.

.

1989 Olver & Shakiban

1990 Stedman

1992-2007 Blinn

2011 Richter-Gebert

Effect of Changes

Q

Q

Q

Geometric Transformation

Homogeneous Scale

Q

Q

QsQ

sQ

sQ

= s3

Q

Q

Q

(det T)2

What Stays Constant?

Q

Q

L L

Q

Q

Q

Zeroness

Sign

0

0

0

0

0

2 2 22ACF BED D C E A B F

Odd number

of nodes

Even number

of nodes

Where do we go from here

Other Dimensions

2 2, , ...f x y w Dx Eyw Fw Curves in P2

3D algebra

Surfaces in P3

4D algebra 2, , , ...f x y z w G x H yw J zw

Polynomials in P1

2D algebra 2 2,f x w Ax Bxw Cw

The Grid 2D=P1

Point sets on

line

3D=P2

Curves in

plane

4D=P3

Surfaces in

space

LINEAR

QUADRATIC

CUBIC

QUARTIC

etc

1

Order of traversal?

Other Dimensions

P L

2 :D ax bw

3 :D ax by cw

4 :D ax by cz dw

Same Across Dimensionality

P L = 0

P Q P = 0

P = 0C

P

P

= C'CT*

T*

T*

F

P

PP

P

P Q = L

Dimensionality and Epsilon

ijk

i

k

j

l

ijklijk

i

kj

ij

i j

3D algebra

2D geometry

2D algebra

1D geometry

4D algebra

3D geometry

Previews of Coming Attractions

Discriminant Surface

2 2 3 3 2 26 4 4 3 0A D ABCD AC B D B C

Resultants

Q

Q

C CQ C Q C QQ 8 ,Q C 4 3

C

Q

Theorem of Pascal

B

A

FE

D

CP1

P2

P3

B D

A

D

F

E

C

B

EA

C F

= 0

5 Points Determine a Quadratic

B

A C

D

CA

B D

E

E

= Q

D

B A

C

AB

D C

E

E

B

A

D

C

E

Intersecting Two Quadratic Curves

Q R

Þ

QR QR

QR QR

P1

P2

P3

P4

QR

QR

QR

QRRank

Q R

Analyzing Cubic Curves

KL

M

N

C

LM

N

KK

K

Parametric Curves

= x,y,w

p

Q

p

x,y,wQx,y,w x,y,wx,y,w

Q Q

Q Q

=

Q

Q

Q = 0

Q

Q

Implicit

Degeneracy:

Base Point if

Parametric

Group Structure of Cubic

=

a b

a b

C

C

C

C

c

ab

c

u

a+b

=

c u

c u

C

C

C

C

a+b

Three Dimensional Projective

Geometry

= E

P

S

T

=S

T

L

Q

Q

Q

Q

3 Points = A Plane

2 Points = A Line

Discriminant of

Quadric

Three Skew Lines

M

L

K

Q

K

L

M

Q KL

M ML

K

Steiner Surfaces

f

f

f

F1

p

p

fParametric

F1F1

F1 F1

F4

Tangent

Implicit

Tensor Diagrams

• Keep Track of CoVariant/ContraVariant

Pairings

• Represent Higher Order Curves Nicely

• Express Only Invariant Quantities

• Allow for Algebra on These Quantities

• Are coordinate free

• Allow us to feel really cool at sharing

notation with Einstein and Feynman

More Information

www.JimBlinn.com

http://www.jimblinn.com/

algebraic geometry - university of washington€¦ · algebraic geometry a personal view cse 590b...

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