Geometric Objects 2001. 7. 6 Computer Graphics Lab. Sun-Jeong Kim.

Geometric Objects

2001. 7. 6Computer Graphics Lab.Sun-Jeong Kim

Korea UniversityComputer Graphics-2-

PointsSingle Coordinate Position

Set the bit value(color code) corresponding to a specified screen position within the frame buffer

x

ysetPixel (x, y)


Lines Intermediate Positions between Two End

points DDA, Bresenham’s line algorithms

Jaggies= Aliasing


DDA Algorithm Digital Differential Analyzer

Slope >= 1 Unit x interval = 1

0 < Slope < 1 Unit y interval = 1

Slope <= -1 Unit x interval = -1

-1 < Slope < 0 Unit y interval = -1

x1

y1

x2

y2

myy kk 1

x1

y1

x2

y2

mxx kk

11

x1

y1

x2

y2

myy kk 1

x1

y1

x2

y2

mxx kk

11


Bresenham’s Line AlgorithmMidpoint Line Algorithm

Decision variable

NE

MQ

cybxa

yxF

MFd

pp

pp

2

11

2

1,1

d > 0 : choose NE : dnew= dold+a

d <= 0 : choose E : dnew= dold+a+b

2

1,2 ppnew yxFd

2

3,2 ppnew yxFd

P(xp, yp) E


Bresenham’s Algorithm(cont.) Initial Value of d

Update d

bayxF

cybxayxF

2

1,

2

11

2

1,1

00

0000

cbyaxyxF 2,

bad 2

2

,

,

then ,0 if

bad

y

x

d 2

, then ,0 if

ad

xd

0, cbyaxyxF

Bxdx

dyy

0, dxBydxxdyyxF

dxBcdxbdya , ,


PolygonsFilling Polygons

Scan-line fill algorithm Inside-Outside tests

Boundary fill algorithm


Scan-Line Polygon Fill Topological Difference between 2 Scan

lines y : intersection edges are opposite sides y’ : intersection edges are same side

y

y’


Scan-Line Polygon Fill (cont.)Edge Sorted Table

C

C’

B

D

E

A

01

yA

yD

yC

Scan-Line Number

yE xA1/mAE yB xA

1/mAB

yC’ xD1/mDC yE xD

1/mDE

yB xC1/mCB


Inside-Outside TestsSelf-Intersections

Odd-Even ruleNonzero winding number rule

exterior

interior


Boundary-Fill AlgorithmProceed to Neighboring Pixels

4-Connected8-Connected


AntialiasingAliasing

Undersampling: Low-frequency sampling

Nyquist sampling frequency:Nyquist sampling interval: max2 ff s

2cyclex

xs


Antialiasing (cont.)Supersampling (Postfiltering)

Pixel-weighting masksArea Sampling (Prefiltering)Pixel Phasing

Shift the display location of pixel areasMicropositioning the electron beam in relati

on to object geometry


SupersamplingSubpixels

Increase resolution

10 11 12

20

21

22

(10, 20): Maximum Intensity

(11, 21): Next Highest Intensity

(11, 20): Lowest Intensity


Pixel-Weighting MasksGive More Weight to Supixels Near the C

enter of a Pixel Area

1 2 1

2 4 4

1 2 1


Area SamplingSet Each Pixel Intensity

Proportional to the Area of Overlap of Pixel2 Adjacent vertical (or horizontal)

screen grid lines trapezoid

10 11 12

20

21

22

(10, 20): 90%

(10, 21): 15%


Filtering TechniquesFilter Functions (Weighting

Surface)

Box Filter Cone Filter Gaussian Filter

Mathematics for CG


Coordinate Reference Frames2D Cartesian Reference Frames

x

y

x

y


2D Polar Coordinate Reference Frame

r

r

x

y

x

yyxr

ryrx

122 tan,

sin,cos

s

rP

radianr

rr

sradian

22

360


3D Cartesian Reference FrameRight-Handed v.s. Left-Handed

y

xz

Right-handed Left-handed

y

x

z


3D Curvilinear Coordinate SystemsGeneral Curvilinear Reference

FrameOrthogonal coordinate system

Each coordinate surfaces intersects at right angles

axisX1

axisX 2

axisX 3

33 constX11 constX

22 constX


Cylindrical-Coordinate

: radius of vertical cylinder

: vertical plane containing z-axis

: horizontal plane parallel to xy-plane

z

constant

zz

y

x

sin

cos

Transform to Cartesian coordinator

z

x axis

y axis

z axis

),,( zP


Spherical-Coordinate

: radius of sphere

: vertical plane containing z-axis

: cone with the apex at the origin

constant

cos

sinsin

sincos

rz

ry

rx

Transform to Cartesian coordinator

z

x axis

y axis

z axis

),,( rP

r


Solid Angle3D Angle Defined on a Sphere

Steradian

r

A2r

ASteradian :

Total solid angle :

44

2

2

2

r

r

r

Asteradian


Points & VectorsPoint

Position in some reference frameDistance from the origin depends on

the reference frame

PFrame B

Frame Ax

y

AO

BO


Points & Vectors (cont.)Vector

Difference between two point positions

Properties : Magnitude & direction Same properties within a single

coordinate system Magnitude is independent from

coordinate frames

yx VV

yyxx

PP

,

, 1212

12

V 22yx VV VMagnitude :

x

y

V

V1tanDirection :


3D VectorMagnitude

Directional angle

222zyx VVV V

x

y

z

VVVzyx VVV

cos,cos,cos

1coscoscos 222


Vector Addition &Scalar MultiplicationAddition

Scalar multiplication

),,( 21212121 zzyyxx VVVVVV VV

1V

2V 2V

1V

21 VV

),,( zyx aVaVaVa V


Vector MultiplicationScalar Product(Inner Product)

zzyyxx VVVVVV 212121

2121 cos

VVVV Commutative :

Distributive :

1221 VVVV

3121321 )( VVVVVVV Orthogonal : 021 VV

1V

2V

cos2V


Vector Multiplication (cont.)Vector Product(Cross Product)

zyx

zyx

zyx

xyyxzxxzyzzy

VVV

VVV

VVVVVVVVVVVV

222

111

212121212121

2121

,,

sin

uuu

VVuVV

Noncommutative :

Nonassociative :

Distributive :

1221 VVVV

3121321 )( VVVVVVV

321321 )()( VVVVVV

21 VV 2V

1V

uRight-handed rule!

Geometric Objects 2001. 7. 6 Computer Graphics Lab. Sun-Jeong Kim.

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