Computer Graphics and Image Processing Geometry IPart 1 – Lecture 2

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Today’s Outline Points, Vectors and Matrices The Dot Product The Cross Product

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POINTS, VECTORSAND MATRICES

3

vyux

Points and Vectors Cartesian coordinate system:

orthogonal axes with numbers (coordinates)Origin: where all coordinates are zero (center)

Point: a position in space, e.g. AucklandGiven as pair (x,y) of coordinate values

Vector: a displacement / differencebetween two points (direction + length)

Example: Where is Hamilton? Point:

37.43S Latitude, 175.19E Longitude Vector:

120km to the south-south-westof Auckland

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Representing Points and Vectors Points are represented as tuples

2D: 2-tuples (x, y) with x and y coordinates 3D: 3-tuples (x, y, z) with x, y, z coordinates

Vectors are also represented as tuples,but usually written as a column instead of a row

x componentand y component (in 3D also z component)

Position vector of a point: vector from origin to the point(often convenient to use vectors instead of points)

Our notation: Points are written in capital letters, e.g. P Vectors in small bold letters, e.g. position vector of P is p 5

yx

v

x

y

z

Right-handed 3Dcoordinate system

Operations on Points and VectorsVectors Add, subtract Scale (change length)

Points Subtracting one point from another gives a vector (displacement) Cannot add two points (Auckland + Hamilton = ???),

but can add and subtract their position vectors

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a

b

a+b0.5 a

a

2 a

O

P

Q

P-Q=p-q

O

P

Q

p+q

Basic Operations on Vectors Addition

Represents the combined displacement Add corresponding components

Subtraction Subtract the corresponding components Same as addition of a negated vector,

i.e. one in opposite direction Scaling

Changing the length Defined such that v + v = 2v Multiply all components by the scalar

Magnitude of a vector Calculating its "length" or 2-norm

Normalization Scaling vector so that it has length 1 (unit vector) Scale by reciprocal of magnitude

1 1 1 1

2 2 2 2

u v u vu v u v

u v

1 1

2 2

u s u

s su s u

u

1 1 1 1

2 2 2 2

u v u vu v u v

u v

2 21 2 , u u s s u u u

ˆ uuu

u1

u2u

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MatricesMatrix: several vectors stuck together mn matrix has m rows and n columns Like m row vectors or n column vectors

Operations Addition/Subtraction:

like adding/subtracting several vectors at the same time

Scaling: like scaling several vectors at the same time

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11 12 13

21 22 23

r r rr r r

R

row

column

11 12 11 12 11 11 12 12

21 22 21 22 21 21 22 22

m m n n m n m nm m n n m n m n

M N

11 12

21 22

s m s ms

s m s m

M

Matrix MultiplicationMultiplying an l m matrix B and an m n matrix C

gives an l n matrix A with

“Rows times columns” with the products summed up

Elements of A are the dot products of the row vectors of B and column vectors of C

Can be used to transform several vectors simultaneously

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11 12 11 12 11 11 12 21 11 12 12 22

21 22 21 22 21 11 22 21 21 12 22 22

b b c c b c b c b c b cb b c c b c b c b c b c

A B C

1 11

...m

ij i j im mj ik kjk

a b c b c b c

21

11

21

11Baa

cc

22

12

22

12Baa

cc

Identity and InverseIdentity matrix I: Neutral element of matrix

multiplication, i.e. for all M:I M = M I = M

Inverse matrix M-1 of a square matrix M (does not always exist):

M M-1 = M-1 M = I and (M-1) -1 = M

Inverse of a 2x2 matrix:

Exists only if m11 m22 – m12 m21 ≠ 010

111 12 22 12

21 22 21 1111 22 12 21

1m m m mm m m mm m m m

1001

I

The Transpose Operation TMake rows out of columns (or vice versa) Transpose of a row vector is a

column vector (and vice versa) For matrix M:

mij and mji are swappedfor all i=1..m, j=1..n

Rules:(MT)T=M(sM)T=s(MT)(M+N)T=MT+NT

(MN)T=NTMT

11

11 2

2

Tuu u

u

u u

11 2111 12 13

12 2221 22 23

13 23

T

m mm m m

m mm m m

m m

M M

THE DOT PRODUCT

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)cos(vuvuvu T

The Dot Product •Takes two vectors u and v, the result is a scalar (a single number)(therefore also known as scalar product)

( is the angle between u and v)

Rules:a•b=b•a (Symmetry)(a+b)•c=a•c+b•c (Linearity)(sa)•b=s(a•b) (Homogeneity)|b|2= b•b

Example: |a - b|2 = (a - b) • (a - b) = a • a - 2a • b + b • b13

vuvu T

2211

2

1

2

1 vuvuvv

uu

)cos(vu

u

v

The Angle between Two VectorsMost important application of dot product:find angle between two vectors (or two intersecting lines)

Two non-zero vectors b and c with common start point are:less than 90o apart if b•c>0exactly 90o apart if b•c=0 [b and c are orthogonal (perpendicular)]more than 90o apart if b•c<0

c b

b

c

x

y

cos

cos

sinsincoscoshence

sincos

,sincos

cb

cb

cbcb

cc

bb

bc

cbcb

c

c

b

b

cb

cb

cos

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Orthogonal Projection of a VectorProjecting a vector b onto another vector a: L is a line through A with direction a b is the vector from A to BGiven: a and bWanted: ba (orthogonal projection of b onto a)Solution:1. Length of ba: because of def. •

2. Vector ba: because

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ababba

)cos(

aaaba

aa

ababa

)cos(baba

2aaa

c

a

b

abL

B

A

Distance from a Line to a Point L is a line through A with direction a b is the vector from A to B

Given: a and bWanted: Solution:

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c

aaababbbc a

c

a

b

abL

B

A

THE CROSS PRODUCT

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nbaba )sin(

1221

3113

2332

babababababa

The Cross Product Takes two 3D vectors u and v, the result is a 3D vector(therefore also known as vector product)

is the angle between a and b n is unit normal vector (|n|=1) orthogonal to a and b Hard to remember?Rules:

(a+b)c = ac + bc (Linearity)(sa)b = s(ab) (Homogeneity)(no symmetry)

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nbaba )sin(

1221

3113

2332

babababababa

sin

Properties of the Cross Product1. a b is a vector perpendicular/orthogonal to both a and b2. Direction of a b is given by "right-hand rule"3. a b = – b a4. |a b| is area of parallelogram defined by a and b5. Area of triangle defined by a and b is 0.5 * |a b|

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nbaba )sin(

a

b

a

b

c

SUMMARY

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Summary1. Vectors:

addition, subtraction, scaling, magnitude, normalization2. Matrices:

addition, subtraction, scaling, multiplication3. Dot product:

4. Cross product:

References: Vectors: Hill, Chapter 4.2 Dot Product: Hill, Chapter 4.3 Cross Product: Hill, Chapter 4.4 21

)cos(vuvuvu T

nbaba )sin(

1221

3113

2332

babababababa

Quiz

1. Calculate: a+b, |b| a, Ma, MN, ab, ab2. What can you tell about the angle between a and b?3. What is the projection of b onto a?4. What is the distance between the point given by b and the line

going through theorigin along a?

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321

a

212

b

203012101

M

131111010

N