Graphics Graphics Lab @ Korea University Mathematics for Computer Graphics Graphics Laboratory...

Graphics

Graphics Lab @ Korea University

http://kucg.korea.ac.kr

Mathematics for Computer Graphics

Graphics LaboratoryGraphics Laboratory

Korea UniversityKorea University


KUCG


Contents

Coordinate Systems

Points and Vectors

Matrices

Parametric vs. Nonparametric Representations


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Coordinate Systems

Rectangular x, y, z axes

Typical coordinate system

Right/left-hand system

Polar

Cylindrical

Spherical


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2D Rectangular Coordinate System

Coordinate origin at the lower-left screen corner

y

x

y

x

Coordinate origin at the upper-left screen corner

<Window Coordinate System><Window Coordinate System><Screen Coordinate System><Screen Coordinate System>


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3D Rectangular Coordinate System

Right-hand system Standard in most graphics packages

Left-hand system Easy to know the distance from the

viewer Video monitor coordinate system


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Polar Coordinate System

sin,cos ryrx

xyyxr /tan, 122

rs

rr ss

xx

yy

p(r,p(r,))


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Why Polar Coordinates in Circles?

xx

yy

dddd

Irregularly Distributed Adjacent PointsIrregularly Distributed Adjacent PointsConstant Distance Constant Distance

among the Adjacent Pointsamong the Adjacent Points

Polar Coordinate SystemRectangular Coordinate System

222 ryx

sin

,cos

ry

rx

In rectangular system Irregular distribution of continuous points

xx

yy

dxdxdxdx


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Cylindrical / Spherical System

cosrx sinry

zz

cossinrx sinsinry

cosrz

zz

yy

xx

rr zz

p(r,p(r,,z),z)

22 yxr )/(tan 1 xy

yy

xx

zz

rr

p(r,p(r,, , ))

222 zyxr )/(tan 1 xy)/(cos 1 rz

Spherical Coordinate SystemCylindrical Coordinate System


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Point: location, position

Vector: direction from one point to another

Represented by using magnitude and unit direction

),(),( 121212 yx VVyyxxPPV

VV

PP22

PP11

xx11 xx22

yy11

yy22

22yx VVV

x

y

V

V1tan

Points and Vectors

xx

yy


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Vectors

3D Vector

Vector addition and scalar multiplication

222zyx VVVV

||cos,

||cos,

||cos

V

V

V

V

V

V zyx

1coscoscos 222

VV

xx

zz

yy

),,( 21212121 zzyyxx VVVVVVVV

),,( zyx VVVV


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Scalar Product

Definition

Properties Commutative

Distributive

0,cos|||| 2121 VVVV

Dot Product / Inner Product

zzyyxx VVVVVV 212121

1221 VVVV

3121321 )( VVVVVVV

|V|V22|cos|cos

VV22

VV11


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Vector Product

Definition

Properties Anti-commutative Not associative Distributive

0,sin|||| 2121 VVVV u

),,( 212121212121 xyyxzxxzyzzy VVVVVVVVVVVV

Cross Product / Outer Product

)( 1221 VVVV

321321 )()( VVVVVV )()()( 3121321 VVVVVVV

VV11

VV22

VV11 V V22

uu

Careful for its direction!!!Careful for its direction!!!


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Plane Normal Calculation

Frequently used in Back face detection Shading function

Vector productVector product between Two edges of the target polygon

VV11

VV22

N = VN = V11 ×× V V22

PP00

PP11PP22

PP33

PP44


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Back Face Detection

Not drawing the back faces to be culled Can make the drawing speed faster

Scalar productScalar product between Eye direction DDeyeeye and face normal vector NNii

If DDeyeeyeNNii > 0 FFii is back face


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Back Face Example

EyeEye Eye Direction DEye Direction Deyeeye(1,0)(1,0)

NN11

(-0.9, -0.1)(-0.9, -0.1)

NN22

(-0.8, 0.2)(-0.8, 0.2)

NN33

(-0.2, 0.8)(-0.2, 0.8)

NN44

(0.3, 0.7)(0.3, 0.7)

NN55

(0.8, 0.2)(0.8, 0.2)

FF11

FF22

FF33FF44

FF55


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Back Face Calculation

F1

DeyeN1 = (1,0)(-0.9, -0.1) = -0.9 F1 is a front face F2

DeyeN2 = (1,0)(-0.8, 0.2) = -0.8 F2 is a front face F3

DeyeN3 = (1,0)(-0.2, 0.8) = -0.2 F3 is a front face F4

DeyeN4 = (1,0)(0.3, 0.7) = 0.3 F4 is a back face F5

DeyeN5 = (1,0)(0.8, 0.2) = 0.8 F5 is a back face


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Back Face Culled Result

EyeEye Eye Direction DEye Direction Deyeeye(1,0)(1,0)

NN11

(-0.9, -0.1)(-0.9, -0.1)

NN22

(-0.8, 0.2)(-0.8, 0.2)

NN33

(-0.2, 0.8)(-0.2, 0.8)

NN44

(0.3, 0.7)(0.3, 0.7)

NN55

(0.8, 0.2)(0.8, 0.2)

FF11

FF22

FF33FF44

FF55


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Shading Function

The amount of illumination depends on coscos If the incoming light IIinin is perpendicular to the surface

IIsurfsurf is maximum, so the surface is fully illuminated

= 0, coscos = 1

NN

LL

cosinsurf kII LNkI in

IIsurfsurf: intensity of the surface: intensity of the surfaceIIinin: intensity of the incident light: intensity of the incident lightkk: surface reflection coefficient: surface reflection coefficientLL: direction from the surface to a light source: direction from the surface to a light source

<Simple Shading Function><Simple Shading Function>


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Matrices

Definition A rectangular array of quantities

Scalar multiplication and matrix addition

mnmm

n

n

aaa

aaa

aaa

A

...

:::

...

...

21

22221

11211

2221

1211

2221

1211 ,bb

bbB

aa

aaA

22222121

12121111

baba

babaBA

2221

1211

kaka

kakakA


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Matrix Multiplication

Definition

Properties Not commutative Associative Distributive Scalar multiplication

× = (i,j)

j-th column

i-th row

m l

nnm

l

n

kkjikij bac

1

ABC

BAAB )()( BCACAB BCABCBA )(

)()()( ABkkBABkA


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Matrix Transpose

Definition Interchanging rows and columns

Transpose of matrix product

c

b

a

cba T

T

,

63

52

41

654

321

TTT ABAB


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Determinant of Matrix

Definition For a square matrix

Combining the matrix elements to product a single number

2 2 matrix

Determinant of nn matrix A (n 2)

n

jjkjk

kj a1

det)1(det AA

211222112221

1211 aaaaaa

aa


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Inverse Matrix

Definition

Non-singular matrix If and only if the determinant of the matrix is non-zero

2 2 matrix

Properties

IAAIAA 11

TT )A()A(AB)AB(A)A( 1111111

ac

bd

bcadA

11

dc

baA


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Parametric vs. Nonparametric Representations

Circle example in computer graphics radius 2, centered at the origin

Parametric expression: x = 2cos, y = 2sin Nonparametric expression

Implicit: , explicit: 422 yx 24 xy

xx

yy

xx

yy

00 11 22-2-2 -1-1

Parametric ExpressionParametric ExpressionInterval of Interval of : : /4/4

Nonparametric ExpressionNonparametric ExpressionInterval of x: 1Interval of x: 1

Which one is balanced?Which one is balanced?

22-2-2


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Parametric Representation

Easy to draw the shape of an object smoothly Just increase one parameter ex)

The other parameters are automatically calculated by Especially for symmetric objects

Circle, sphere, ellipsoid, etc.

Preferred in computer graphics Nonparametric representation is used mainly in

numerical analysis

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