6/16/2015©Zachary Wartell 2D Coordinate Systems, Change of Coordinates, and Matrices Revision:...

04/18/23©Zachary Wartell

2D Coordinate Systems,Change of Coordinates,

and Matrices

Revision: 1/8/2008 6:14:11 PMCopyright Zachary Wartell, University of North Carolina

All Rights Reserved

Textbook:Chapter 5 and Appendix


Overview

• points, vectors, alignments & coordinates thereof• coordinate systems & coordinates thereof• change of coordinates for pt’s, vt’s


Geometric Point

• Point – a location in space


• Consider points in 1D space, drawn as a dot

• We also have 1D distance, drawn as a line segment

– Note, any line segment of the right length is a valid drawing of a given distance

Distance (in 1D space)

0 1 ….

d=2 d=2

l0=[0,2]0 1 ….

l1=[4.5,6.5]

d=2 d=2

l0=[0,2]0 1 ….

l1=[4.5,6.5]

d=2

l2=[5.5,7.5]


• Displacement in 1D space is a “signed” distance, drawn as an directed line segment (also an ‘arrow’)

– Note, any directed line segment of the correct length and direction is a valid drawing of a given displacement

Displacement (in 1D space)

d=2 1/3 d=-1.75

l2=[6,4.25]l2=[0,2 1/3]

d=1.75

l1=[2,3.75]0 1

d=-1.75

l2=[6,4.25]

d=-1.75

l1=[3.75,2]0 1

d=-1.75

l3=[2.25,1.5]


What about 2D Displacements? (Step #1)

• Step #1, need to define an alignment. – “Parallel lines all share a common alignment”

or– “An alignment, is what is common in a set of

parallel lines”• An alignment is drawn by drawing any line

parallel to the alignment

W/E alignment?/? alignmentNE/SW alignmentN/S alignment

3 different lines but all have the same “alignment”


• geometric vectors are “2D displacements” or “direction with sign and magnitude” -- drawn by an (2D) directed line segment (‘arrow’)– Note, any arrow of the right length and

alignment is valid drawing of a given geometric vector

2D Displacements are called ‘geometric vectors’

Length=5

“Move North 5 meters”

Length=7“Move NW 7 meters” “Move SE 7 meters”

Leng

th=7

3 different arrowsbut all representativeof the same vector


Geometric Vector versus Alignment

∙ a geometric vector gives more information than an alignment

Alignment is N/SVector is N at magnitude l

l this vector has same alignment as this one but different magnitude


Please draw point (5,3)

?


Point (5,3)A

(5,3)

A

“regular” recti-linear coordinate system (syn. Cartesian coordinate system)


Point (5,3)B

(5,3)

(5,3)


(5,3)

Point (5,3)C

(5,3)

(5,3)


Point (5,3)D

(5,3)

(5,3)

(5,3)(5,3)

oblique (!) recti-linear coordinatesystem


Please draw vector (3,2)

?


Vector (3,2)A

(3,2)


Vector (3,2)B

(3,2)


Review: Points, Vectors, Arrows, Alignments

p: (5,3)

v: (1,3)

v: (1,3)

v: (1,3)

a0: ((0,0),(1,3))

a1: ((2,1),(3,4))

a3: ((-2,-1),(-1,2))

al0: (2k,1k), k∈R, k≠0

all lines with slope ½ have

alignment al 0


DOF’s of 2D Elements

Element Degrees of freedom

Point 2 DOF

Vector 2 DOF

Arrow 4 DOF

Alignment 1 DOF


Operation on Points & Vectors (2D or 3D)

• Points: pi - pj = v pi ± v = pj (p + 0 = p) tpi + (1-t)pj = pk - “affine combination”

• Vectors: vi ±vj = vk , vj = a ∙ vk (a R) or avi ± bvj = vk - ”linear combination”

length(v), |v| =

normalize(v), v'= (1/|v|) ∙ v dot product: vivj =

2 2 2x y z

cos

j

i i i j i j i j i j

j

i j

x

x y z y x x y y z z

z

v v


Notation

(5,3)A

A,B,C – capital script letter is a coordinate system

(x,y)B or (xB,yB) – is a coordinate with respect to coordinate system B

A

Coordinate System is a 3-tuple, (o,x,y)o : a point called the originx : a vector called x-basis vectory : a vector called y-basis vector

o x

y


Different Point/Coordinate & Same Coordinate/Point

• So far we’ve looked at different points with the same coordinate but conversely we can considered the same point with different coordinates! (5,3)

(5,3)

(5,3)(5,3)

(5,3)

(5,3)

(5,3)(5,3)


(3,3)B, (1.33,0.33)A

Coordinates of point p

B

A

p


Coordinates of Coordinate System A relative to B

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

Equivalent Questions:What is MB←A? -- “A to B"Where is A (as measured) in B? How do I superimpose B onto A?

(3,3)B, (1.33,0.33)Ap


What is Minches←feet?

1D linear transform:

12 1/12

12 1/12

s s

M M

p M p p M p

p p

inches feet inches feet feet inches feet inches

inches inches feet feet feet feet inches inches

feet inches

p with coordinates (0.4)feet or(4.8)inches0inches unit vector

feet unit vector

Where is Feet in Inches? How to superimpose Inches onto Feet?


Where is Celsius in Fahrenheit?How to superimpose Fahrenheit onto Celsius?

1D affine transform:

9 / 5 32 5 / 9 17.7

s t s t

M p M p

p p p p

far cel far cel cel far cel cel far cel far cel cel far

far cel cel far

p with coordinates (-8.81)cel or (16)far0far

fahrenheit unit vector & origin

celsius unit vector & origin

0cel,32far

scale by 9/5add 32 ‘s

What is Mfar←cel?


Review: Matrices (1)

11 12 1

21 2

1 2

11

11 1

1

is a by matrix

, is a 'column matrix', is a 'row matrix'

Matrix Addition

is by , is by , is by

c

c

r r rc

c

r

ij ij

m m m

m mr c

m m m

v

v v

v

m n m n m n

c a b

M M

V U V U

C A B C A B

1

Matrix Multiplication:

is by , is by , is by

ij

n

ij ik kjk

m q m n p q

n p

c a b

C A BC AB



T

T T T

Property: Non-commutative:

Property: Distributive

Operation: Transpose

: where is by and is by

Property:

=

Operation: Determinant

det : , w

ij ji

m n n m

a b

AB BA

A B C AB AC

A B A B

AB B A

A A R

11 1211 22 21 12

21 22

11 12 13

21 22 231

31 32 33

1

here is by

det

det 1 det Ex:

1 det

nj k

jk jkj

nj k

jk jkk

n n

a aa a a a

a a

a a a

a a a a

a a a

a

A

A

A A

A



-1

1 1

1

Property:

det ( ) det det

Definition: Identity

1 0 0 0

0 1 0 0,

0 0 0

0 0 0 1

Operation: Inverse

: , where , are by and det 0 ( is 'non-singular')

1 det

det

j k

jkkj

n n

m

AB A B

I M I M

A B B A A B B B

M M M M I

M

M



Given two or more functions we 'compose' them by applying the result of one as

the input of the other

G( ), F( )

Then we might have

G(F( )), or F(G( )),

y x y x

x x

Operation : Composition (General)

F

or F(F(G(F( )))), etc...

We can use matrices to represent functions on column (or row) vectors.

G( ), where ,

F( ), where ,

Then we mi

x

G

Operation : Composition (Matrix)

y x y M x

y x y M x

F

F G

F G G F

ght have:

G(F( ))=

F(G( ))=

F(G(G(F( ))=

....etc...

Gx M M x

x M M x

x M M M M x


B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

Where is A in B?

A.x = ( 1/√2, 1/√2)A.y = (-1/√2, 1/√2)A.o = (2.5, 1.5)

Where is A in B? = What is MB←A?

1 2 1 2 2.5:

1.51 2 1 2

((0,0) ) (2.5,1.5)

((1,2) ) (1.79,3.62)

p

p

p

x x

y y

M

M

M

B A

B A

B A

B A

B A

A B

A B

(1,2)A, (1.79,3.62)B

p for “point”


B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

1 2 1 2 2.5:

1.51 2 1 2

((1,0) ) (1 2 ,1 2)

((0,1) ) ( 1 2 ,1 2)

V

V

V

x x

y y

M

M

M

B AB A

B A

B A A B

B A A B

What about the vectors?

Where is A in B?

A.x = ( 1/√2, 1/√2)A.y = (-1/√2, 1/√2)A.o = (2.5, 1.5)

Where is A in B? = What is MB←A? - vectors

V for “vector”


More Compact:

1 2 1 2 2.5

: 1 2 1 2 1.5

0 0 1

where 1 for a point

0 for a vector

x x

y y

w w

w

w

MB A

B A B A

(1.79,3.62)

Where is A in B?

A.y = ( 1/√2, 1/√2)A.x = (-1/√2, 1/√2)A.o = (2.5, 1.5)

Compact Representation

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B


GeneralizationIn General:

- point

1 0 0 1 1:

- vector

0 0 0 1 0

x x x

y y y

x x x

y y y

x x

y y

x x

y y

.x .y .o

.x .y .o

M.x .y .o

.x .y .o

B A

B A

B A

B A

B A

A A A

A A A

A A A

A A A B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

If I were being absurdly pedantic what sub-sub-script should go under the sub-scripts x,y,z? Answer:

x.xB

A


Where is B in A? = What is MA←B?

More Compact:

1 2 1 2 2.76

: 1 2 1 2 1.73

0 0 1

11 2 1 2 2.5 1 2 1 2 2.76

!! 1 2 1 2 1.5 1 2 1 2 1.73

0 0 1 0 0 1

x x

y y

w w

MA B

A B A B

Where is B in A?

B.x = ( 1/√2, -1/√2)B.y = ( 1/√2, 1/√2)B.o = (-2.76, 1.73)

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B


What is relationship of MA←B and MB←A?

1 1

11 2 1 2 2.5 1 2 1 2 2.76

1 2 1 2 1.5 1 2 1 2 1.73

0 0 1 0 0 1

In general:

,

M M M MA B B A A B B A -generally inverse computation requires expensive Gaussian elimination. But in various specific cases (translate, scale, rotate) inverse computation can be quick!

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B

B

A(2.5,1.5)B

(1/√2,1/√2)B(-1/√2,1/√2)B


Revisions1.1

1. Added slide 132. Added Overview Slide3. Added explicit slide referencing “ITCS 4120-2D Transforms.ppt”4. re-ordered slides to better suit insertion of “ITCS 4120-2D Transforms.ppt”5. added 1D change of coordinates slides

1.2 -typos1.3 -added Matrix: Review slides1.4 -moved affine transform slides to other .ppt file -added DOF slide

-added ‘superimpose’ terminology to coordinate transform slides


Alternative Convention MB←A and M A ←B

1 0 0

0 1 0

0 0 1 0 0 1 0 0 1

If and basis vectors have same interior angle (i.e. they are not sheared relative to e

x x x x x x

y y y yy y

.x .y .o .o .x .y

M .x .y .o .o .x .yB A

A A A A A A

A A A A A A

A B

11

ach other), then

1 0 0 0/ / 0

0 1 0 00 0 1

0 0 1 0 0 1

, where , and are a translation, rotation, and scale.

Now,

( )

x

y

.o .x.x .x .y .y

.o .y

T R S T R S

M M T R S

S

B A B A B A

B A A B A B A B A B

A B

A AA A A A

A A

1 1 1

1 1 1

Similarly we could write:

Note that here we've written by taking S,R, &T component matrices from . This is an alternative approach

used in some text

R T

M S R T

M M

A B A B

A B B A B A B A

A B B A

's (Hearn&Baker). This convention can be somewhat counter-intuitive. Why build by using inverse S,R,T

components . Is it not more intuitive build from it's own S,R,T components? So while

M

MA B

B A

1 1 1

1 1 1

the following are both correct

the T,R,S expression feels more intuitive than the S,R,T expression:

M T R S

S R T

M T R S

S R T

B A B A B A B A

A B A B A B

A B A B A B A B

B A B A B A

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