Neat Stuff from Vector Calculus & Related Subjects Chris Hecker [email protected] definition six, inc. & Maxis.

Neat Stuff from Vector CalculusNeat Stuff from Vector Calculus&&

Related SubjectsRelated Subjects

Chris [email protected]

definition six, inc. & Maxis

Related SubjectsRelated Subjects

vectorcalculus

scalarcalculus

linearalgebra

differentialgeometryclassical

mechanics,dynamics

optimization &constraints

differential &integral calculus

Intro & PrerequisitesIntro & Prerequisites

• this is a total hodge-podge, not a gentle introduction– lack of rationale for caring about this stuff, trust me? :)

• stuff that I found non-intuitive or hard to figure out• tour of fairly basic examples building on themselves to get

firm grounding and intuition in calculus concepts for the kinds of math games do– touch on lots of different areas during tour, many sidestreets

• comfortable with algebra, linear algebra• need to understand scalar calc, at least at the plug’n’chug

level of differentiating functions– f(x) = ax2+bx+c f’(x) = 2ax + b

• a function maps values from the domain to the range uniquely– can be multivalued in range, but not domain

What is a Function?What is a Function?

)(xfy

sin

cos)( f

y

x01),( 22 yxyxf

mnf RR:

)(xfy -1

0

1

2 -1

0

1

2

0

2

4

6

8

-1

0

1

2

1),( 22 yxyxfz

),( vufy

x

• derivative is another function that maps changes in the domain to changes in the range, “normalized”

What is a Derivative?What is a Derivative?

)( xxfyy

)(xfy y

x

x

xfxxf

x

y

)()(

• the key insight of calculus: the change is so small that you can ignore it anytime it’s multiplied by itself...so, you can treat any continuous function as linear if you’re zoomed in far enough (to 1st order)– “continuity” keeps us from dividing by zero– normalization makes the numbers finite

Derivatives are LinearDerivatives are Linear

202

1000 )()()()( xxfxxfxfxxf

)()()(

000 xf

x

xfxxf

)(xfy

x0

)(xfy

x0

boom!

Random but Nifty Example of Ignoring InfinitessimalsRandom but Nifty Example of Ignoring Infinitessimals(nothing up my sleeve!)(nothing up my sleeve!)

• prove infinitessimal rotations in 3D are vectors (add, commute, etc.)– angular velocity is a vector, cross product differentiates rotating vectors

11)1)(1( TT T

infinitessimal rotations are vectors:

3x3 skew symmetric matrices are isomorphic to vectors

rrrdt

drv dt ~ 1

rrrrdr )( vec ' angular velocity differentiates with cross product:

1 ' RRRrr Tfinite rotation: rrrr )1('infinitessimal rotation:

21212121 11)1)(1( infinitessimal rotations add and commute:

Scalar Derivatives: a lineScalar Derivatives: a line

• how does y (or f(x)) change for a change in x?– for lines, the change (derivative) is constant

everywhere

– drawn as red vector, but actually a scalar, “slope”

)(xfy

a

1

baxxf )(axf )('

adxdf

RR: f

RR: df

adx

df

Scalar Derivatives: a curveScalar Derivatives: a curve

• for curves, the derivative is position dependent– derivative is a function itself, mapping change in

domain to change in range

– both direction and magnitude, but still a scalar

cbxaxxf 2)(

)(xfy

RR: f

RR: df

baxxf 2)('dxbaxdf )2(

• vectors are columns of numbers in this talk, not rows• we right-multiply matrices by column vectors: v’ = Mv

• matrix & vector ops “fit together” nicely• keeping the shapes right is the key to sanity with vector calc• “m by n * n by p = m by p”

• this is why “v’=vM” makes no sense for column vectors– either you’re using row vectors, or you’re confused– either way, you’re in for some pain when trying to do real math

• because all math books use columns for vectors and rows are special• early computer graphics books got this backwards, and hosed everybody

The Shape of Matrix OperationsThe Shape of Matrix Operations

131211 aaa

31

21

11

b

b

b

11c

ab = c = aTb = c

333231

232221

131211

aaa

aaa

aaa

31

21

11

b

b

b

31

21

11

c

c

c

Ab=c

232221

131211

aaa

aaa

34333231

24232221

14131211

bbbb

bbbb

bbbb

24232221

14131211

cccc

ccccm

n

n

p

AB=C

Shape of DerivativesShape of Derivativesdf needs to accept a df needs to accept a domain linearly (ie. right-multiplied vector or domain linearly (ie. right-multiplied vector or

scalar) to produce a scalar) to produce a rangerange

• scalar valued function of scalar– y=f(x) dy =dfdx

• vector valued function of scalar– v=f(t) dv = dfdt

• scalar valued function of vector– z=f(x,y)

• vector valued function of vector– p=f(u,v)

d

d

d

d

df

Shape of Derivatives (cont.)Shape of Derivatives (cont.)

• scalar valued function of vector: expands to row vector– the resulting range value depends on all the domain values– the differential needs a slot for a delta/change in each domain

dimension...so it must be a row vector, there’s no T or – z=f(x,y)– dz=dfdx

• vector valued function of vector: expands to matrix– p=f(u,v)

df = d

df = d

dy

dx

y

f

x

fdf

dyy

fdxx

fdz

xx

dz = dfdx = d

dp = dfdx = d

• for a change in the parameter (domain), how does the function (range) change?– in this case, differential is a vector

Derivatives: a parametric functionDerivatives: a parametric function“a vector function of a scalar”“a vector function of a scalar”

sin

cos

y

x

dd

d

dy

dx

cos

sin

cos

sin

2RR: f2RR: df

Derivatives: a scalar function of a vectorDerivatives: a scalar function of a vector“height field”“height field”

• differential is not in range of function

0 0.5 1 1.5 20

0.5

1

1.5

2

dcybxyaxyxfz 22),(

dycybxdxbyax )2()2(

dyy

fdxx

fdfdz

dy

dxcybxbyax 22

RR: 2 f

RR: 2 df

-1

0

1

2 -1

0

1

2

0

2

4

6

8

-1

0

1

2

row vector

column vector

-1

0

1

2 -1

0

1

2

0

2

4

6

8

-1

0

1

2

Derivatives: a scalar function of a vectorDerivatives: a scalar function of a vector

• view “height field” as implicit surface in 3d– write g(x,y,z) > 0 above surface

– differential is surface normal (not unit)

),(),,( yxfzzyxg

dz

dy

dx

cybxbyax

dzz

gdyy

gdxx

gdg

122

RR: 3 f

RR: 3 df

1

2

2

cybx

byaxg

gT

x


• example plane

),(),,( yxfzzyxg

dz

dy

dx

dzz

gdyy

gdxx

gdg

111

RR: 3 f

RR: 3 df

-1

-0.5

0

0.5

1 -1

-0.5

0

0.5

1

-2

-1

0

1

2

-1

-0.5

0

0.5

1

0 dpnz

y

x

pn ,

1

1

1

dyxyxfz

dzyx

),(

0

dyxzzyxg ),,(


• example sphere 1),,( 222 zyxzyxg

dz

dy

dx

zyx

dzz

gdyy

gdxx

gdg

222

RR: 3 f

RR: 3 df

A Surface Normal is Not a Vector!A Surface Normal is Not a Vector!this is why you need to keep shapes distinctthis is why you need to keep shapes distinct

• vector is a difference between two points

• normal is “really” a mapping from a vector to a scalar

a

b

n

v

'''

'''

'

'

abv

abTaTbTvv

abv

Tbb

Taa

points &vectors transform like this:

T

T

TT

T

T

T

TS

ST

SnTv

SnTv

TvvSnn

nv

nvnv

1

0

0)(

' ,'

0''

0

normals transform like this:

0.5 1 1.5 2

-1

-0.5

0.5

1

Normals and Vectors ExampleNormals and Vectors Example

• scaling an ellipse 1),( 22 yaxyxg

yaxg

22

x

2

4

11

41

21

xa

x values for y=0

Derivative of Vector MappingsDerivative of Vector Mappings

• barycentric coordinates in 2D triangle– vector function of a vector

– if square, can invert Jacobian to find du,dv given dp– function is linear in this case, but works generally

– determinant of Jacobian is how areas distort under function

e1

e2

p0

p1

p2

pdv

duee

dv

du

v

f

u

fv

f

u

f

ddyy

xx

21

uu

fp

22 RR: f22 RR: df

veuepvufp 210),(

Jacobian matrix

Barycentric Coordinates in 3D TriangleBarycentric Coordinates in 3D Triangle

• now jacobian is 3x2– can still find du,dv from dp with least squares

e1

e2

p0

p1

p2

p

dv

duee

dv

du

v

f

u

fv

f

u

fv

f

u

f

ddp

zz

yy

xx

21

uu

f

32 RR: f32 RR: df

veuepvufp 210),(

bAAxAbAx TT

dpe

e

dv

duee

e

eT

T

T

T

2

121

2

1

dv

du

eeee

eeee

dpe

dpeTT

TT

T

T

2212

2111

2

1

same as projecting dp down into triangle

Implicit FunctionsImplicit Functions• equalities (constraints) subtract off DOFs

• explicit to implicit is easy: z = f(x,y) g(x,y,z) = z – f(x,y) =0

• implicit to explicit is hard: f(x,y) = 0 y = f(x)• solving nonlinear equations, sometimes multiple or no solutions

• but, inverting it differentially is easy because of linearization:

0),,( zyxg

0),,( zyxf

21 xy

0

xxdf

0

dyy

fdxx

f

dxdy

dxdyy

fx

f

01),( 22 yxyxf dxdy yx

22

shape/rank of jacobian will tell you how constrained you are as well

f(x,y) = 0

dxdy 0

Product Rule for Vector DerivativesProduct Rule for Vector Derivatives

• for scalar multiplication:• differential of a scalar function of a scalar is a scalar

dbabdabad )(



• for dot product it’s a little wackier• if vectors in dot are functions of scalars, it’s the same

dbabdabad )(

dbabdabad )(




• if vectors in dot are functions of vectors, need to watch shape!– we know result must be row:

dbabdabad )(

dbabdabad )(

d





dbabdabad )(

dbabdabad )(

d

d d ???

dbabdabad TTT )(use transpose picture to reason about it:





dbabdabad )(

dbabdabad )(

dbadabbad TTT )(

d

d d ???

dbabdabad TTT )(

don’t want to use tensors, so we pull a fast one with the commutativity of dot (ab=ba)

d d

use transpose picture to reason about it:

Dot Product Derivative ExampleDot Product Derivative Example

• derivative of squared length of vector

qpr 22 )( qprrrrr T

)()(2 qpqpddr T

dprdpqpdr TT 2)(22

)()()()(2 qpdqpqpdqpdr TT

dbadabbad TTT )(

p

q

rdp

makes intuitive sense:if dp is orthogonal, length doesn’t change

Cross Product Works the Same WayCross Product Works the Same Way

• if cross is a vector function of vectors...• result must be a matrix

• use the skew symmetric picture of cross product

abbaba~~

d d ???

dbadabbad ~~)~(

dbabdabad )(

same problem with tensors, so we pull the same (skew-)commutativity trick

Cross Product Derivative ExampleCross Product Derivative Example

• differentiate a cross product of moving vector with constant vector

nr

dr

c

dc

drndc

nrc nrdnrddc ~)(

drndnrdrnnrd ~~~~

dbadabbad ~~)~(

again, makes intuitive sense:dc changes orthogonally to dr changes

ReferencesReferences

• Advanced Calculus of Several Variables– Edwards, Dover

• Calculus and Analytic Geometry– Thomas & Finney

• Classical Mechanics– Goldstein

Neat Stuff from Vector Calculus & Related Subjects Chris Hecker [email protected] definition six, inc. & Maxis.

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