OpenGLGeometric Transformations

OpenGL Matrix Stacks

• Stack processing– The top of the stack is the “current” matrix– glPushMatrix(); // Duplicate the current matrix at the top

– glPopMatrix(); // Remove the matrix at the top

로보트 팔 만들기

Hierarchical Modeling

• A hierarchical model is created by nesting the descriptions of subparts into one another to form a tree organization

Homework #2

• 다각형을 이용하여 움직이는 2 차원 아름다운 애니메이션 만들기

• Push/Pop Matrix 를 사용하여 2 단계 이상의 계층적인 움직임 디자인을 할 것– Ex): 태양계 시스템 ( 태양지구달 )

사람의 움직임 ( 몸어깨팔꿈치손목손 )

• 숙제제출 : 이메일 제출 (Screenshot, code, report)• 숙제마감 : 4 월 28 일 수요일 23 시 59 분까지

Viewing Part I: Two-Dimensional Viewing

Sang Il Park

Sejong University

Lots of slides are stolen from Jehee Lee’s

Two-Dimensional Viewing

• Two dimensional viewing transformation– From world coordinate scene description to device

(screen) coordinates

Normalization and Viewport Transformation

• World coordinate clipping window• Normalization square: usually [-1,1]x[-1,1]• Device coordinate viewport

OpenGL 2D Viewing

• Projection Mode– glMatrixMode(GL_PROJECTION);

• GLU clipping-window function– gluOrtho2D(xwmin,xwmax,ywmin,ywmax);– Normalized to [-1,1]x[-1,1]

• OpenGL viewport function– glViewport(xvmin,xvmax,yvmin,yvmax);

GLUT window Functions:

• glutInitWindowPosition• glutInitWindowSize• glutCreateWindow• glutDestroyWindow• glutSetWindow/glutGetWindow• ….. And more!

• See the text book pp 346~354

Clipping

• Remove portion of output primitives outside clipping window

• Two approaches– Clip during scan conversion: Per-pixel bounds check– Clip analytically, then scan-convert the

modified primitives

Two-Dimensional Clipping

• Point clipping – trivial• Line clipping

– Cohen-Sutherland– Cyrus-beck– Liang-Barsky

• Fill-area clipping– Sutherland-Hodgeman– Weiler-Atherton

• Curve clipping• Text clipping

Line Clipping

• Basic calculations:– Is an endpoint inside or outside the clipping window?– Find the point of intersection, if any, between a line

segment and an edge of the clipping window.

Both endpoints inside:

trivial accept

One inside: find

intersection and clip

Both outside: either

clip or reject

Cohen-Sutherland Line Clipping

• One of the earliest algorithms for fast line clipping• Identify trivial accepts and rejects by bit operations

< Region code for each endpoint >

above below right leftBit 4 3 2 1

0000

10001001

0001

0101 0100 0110

0010

1010Clipping window

Cohen-Sutherland Line Clipping

• Compute region codes for two endpoints• If (both codes = 0000 ) trivially accepted• If (bitwise AND of both codes 0000) trivially rejected• Otherwise, divide line into two segments

– test intersection edges in a fixed order.

(e.g., top-to-bottom, right-to-left)

0000

10001001

0001

0101 0100 0110

0010

1010Clipping window

Cohen-Sutherland Line Clipping

• Fixed order testing and clipping cause needless clipping (external intersection)

Cohen-Sutherland Line Clipping

• This algorithm can be very efficient if it can accept and reject primitives trivially– Clip window is much larger than scene data

• Most primitives are accepted trivially– Clip window is much smaller than scene data

• Most primitives are rejected trivially

• Good for hardware implementation

Cyrus-Beck Line Clipping

• Use a parametric line equation

• Reduce the number of calculating intersections by exploiting the parametric form

• Notations– Ei : edge of the clipping window

– Ni : outward normal of Ei

– An arbitrary point PEi on edge Ei

10 ),()( 010 tPPtPtP

Cyrus-Beck Line Clipping

halfplane outside in thepoint a0))((

edge thecontaining line on thepoint a0))((

halfplane inside in thepoint a0))((

i

i

i

Ei

Ei

Ei

PtPN

PtPN

PtPN

Cyrus-Beck Line Clipping

• Solve for the value of t at the intersection of P0P1 with the edge– Ni · [P(t) - PEi] = 0 and P(t) = P0 + t(P1 - P0)

– letting D = (P1 - P0),

– Where

• Ni 0

• D 0 (that is, P0 P1)

• Ni · D 0 (if not, no intersection)

DN

PPNt

i

Eii

][ 0

Cyrus-Beck Line Clipping• Given a line segment P0P1, find intersection

points against four edges– Discard an intersection point if t [0,1] – Label each intersection point either PE

(potentially entering) or PL (potentially leaving)– Choose the smallest (PE, PL) pair that defines the

clipped line

PL0

PE0

10

10

PPN

PPN

i

i

Liang-Barsky Line Clipping

• Liang-Barsky optimized Cyrus-Beck for upright rectangular clip windows

),(1

),(0

22

11

yxQt

yxPt

tdyyyytyy

tdxxxxtxx

1121

1121

)(

)(

P(x1,y1)

Q(x2.y2)

tB

tT

tR

tL

Liang-Barsky Line Clipping

T

B

RL

tT

tB

Cyrus-Beck Line Clipping

• Cyrus-Beck is efficient when many line segments need to be clipped

• Can be extended easily to convex polygon (rather than upright rectangle) clip windows

Nicholl-Lee-Nicholl Line Clipping

• Divide more cases and reduce the computation.

(1)

(2)(3)

General Clipping Window

• Line clipping using nonrectangular polygon clip windows– Convex polygon

• Cyrus-Beck algorithm can be readily extended– Concave polygon

• Split the concave polygon into convex polygons

Vector Method for Concave Splitting

• Calculate edge-vector cross products in a counterclockwise order

• If any z component turns out to be negative, the polygon is concave

Polygon Fill-Area Clipping

• Polyline vs polygon fill-area

• Early rejection is useful

Clipping Window

Bounding box of polygon fill area

Sutherland-Hodgman Polygon Clipping

• Clip against 4 infinite clip edges in succession

Sutherland-Hodgman Polygon Clipping

• Accept a series of vertices (polygon) and outputs another series of vertices

• Four possible outputs

Sutherland-Hodgman Polygon Clipping

• The algorithm correctly clips convex polygons, but may display extraneous lines for concave polygons

Weiler-Atherton Polygon Clipping

• For an outside-to-inside pair of vertices, follow the polygon boundary

• For an inside-to-outside pair of vertices, follow the window boundary in a clockwise direction

Weiler-Atherton Polygon Clipping

• Polygon clipping using nonrectangular polygon clip windows

Text Clipping

• All-or-none text clipping– Using boundary box for the entire text

• All-or-non character clipping– Using boundary box for each individual character

• Character clipping– Vector font: Clip boundary polygons or curves– Bitmap font: Clip individual pixels

Three-Dimensional Viewing

Sang Il ParkSejong University

Lots of slides are stolen from Jehee Lee’s

Viewing Pipeline

Virtual Camera Model

• Viewing Transformation– The camera position and orientation is determined

• Projection Transformation– The selected view of a 3D scene is projected onto a

view plane

General 3D Viewing Pipeline

• Modeling coordinates (MC)• World coordinates (WC)• Viewing coordinates (VC)• Projection coordinates (PC)• Normalized coordinates (NC)• Device coordinates (DC)

Viewing-Coordinate Parameters

• View point (eye point or viewing position)

• View-plane normal vector N

),,( 0000 zyxP

Viewing-Coordinate Parameters

• Look-at point Pref

• View-up vector V– N and V are specified in the world coordinates– V should be perpendicular to N, but it can be difficult

to a direction for V that is precisely perpendicular to N

refPPN 0

Viewing-Coordinate Reference Frame

• The camera orientation is determined by the uvn reference frame

),,(

),,(

),,(

zyx

zyx

zyx

vvv

uuu

nnn

unv

V

nVu

N

Nn

u

nv

World-to-Viewing Transformation

• Transformation from world to viewing coordinates– Translate the viewing-coordinate origin to the world-

coordinate origin– Apply rotations to align the u, v, n axes with the world

xw, yw, zw axes, respectively

u

nv

World-to-Viewing Transformation

1000

100

010

001

1000

0

0

0

0

0

0

z

y

x

nnn

vvv

uuu

zyx

zyx

zyx

TR

10000

0

0

, Pn

Pv

Pu

TRMzyx

zyx

zyx

vcwc nnn

vvv

uuu

Perspective Projection

• Pin-hold camera model– Put the optical center (Center Of Projection) at the origin– Put the image plane (Projection Plane) in front of the COP– The camera looks down the negative z axis

• we need this if we want right-handed-coordinates

Parallel Projection

• Special case of perspective projection– Distance from the COP to the PP is infinite

– Also called “parallel projection”

– What’s the projection matrix?

Image World

Slide by Steve Seitz

Taxonomy of Geometric Projections

geometric projections

parallel perspective

orthographic axonometric oblique

trimetric

dimetric

isometric

cavalier cabinet

single-point two-point three-point

Orthographic Transformation

• Preserves relative dimension• The center of projection at infinity• The direction of projection is parallel to a principle axis• Architectural and engineering drawings

Axonometric Transformation

• Orthogonal projection that displays more than one face of an object– Projection plane is not normal to a principal axis, but

DOP is perpendicular to the projection plane– Isometric, dimetric, trimetric

Oblique Parallel Projections

• Projection plane is not normal to a principal axis, but DOP is perpendicular to the projection plane

• Only faces of the object parallel to the projection plane are shown true size and shape

Oblique Parallel Projections

www.maptopia.com

Oblique Parallel Projections

Oblique Parallel Projections

• Typically, is either 30˚ or 45˚• L1 is the length of the projected side edge

– Cavalier projections• L1 is the same as the original length

– Cabinet projections• L1 is the half of the original length

Oblique Parallel Projections

• Cavalier projections

• Cabinet projections

Perspective Projection

• Pin-hold camera model– Put the optical center (Center Of Projection) at the origin– Put the image plane (Projection Plane) in front of the COP– The camera looks down the negative z axis

• we need this if we want right-handed-coordinates

Perspective Projection

• Projection equations– Compute intersection with PP of ray from (x,y,z) to COP– Derived using similar triangles (on board)

– We get the projection by throwing out the last coordinate:

Homogeneous coordinates

• Is this a linear transformation?

Homogeneous coordinates

• Trick: add one more coordinate:

• Converting from homogeneous coordinates

homogeneous projection coordinates

homogeneous viewing coordinates

Perspective Projection• Projection is a matrix multiply using homogeneous coordinates:

divide by third coordinate

• This is known as perspective projection– The matrix is the projection matrix– Can also formulate as a 4x4

divide by fourth coordinate

Perspective Projection

• The projection matrix can be much involved, if the COP is different from the origin of the uvn coordinates– See the textbook for the detailed matrix

Traditional Classification of Projections

• Three principle axes of the object is assumed– The front, top, and side face of the scene is apparent

Traditional Classification of Projections

Perspective-Projection View Volume

• Viewing frustum– Why do we need near and far clipping plane ?

Normalizing Transformation

• Transform an arbitrary perspective-projection view volume into the canonical view volume

• Step 1: from frustum to parallelepiped

Normalizing Transformation

• Transform an arbitrary perspective-projection view volume into the canonical view volume

• Step 2: from parallelepiped to normalized

OpenGL 3D Viewing Functions

• Viewing-transformation function– glMatrixMode(GL_MODELVIEW);– gluLookAt(x0,y0,z0,xref,yref,zref,vx,vy,vz);– Default: gluLookAt(0,0,0, 0,0,-1, 0,1,0);

• OpenGL orthogonal-projection function– glMatrixMode(GL_PROJECTION);– gluOrtho(xwmin,xwmax, ywmin,ywmax, dnear,dfar);– Default: gluOrtho(-1,1, -1,1, -1,1);– Note that

• dnear and dfar must be assigned positive values• znear=-dnear and zfar=-dfar• The near clipping plane is the view plane

OpenGL 3D Viewing Functions

• OpenGL perspective-projection function– The projection reference point is the viewing-coordinate origin– The near clipping plane is the view plane– Symmetric: gluPerspective(theta,aspect,dnear,dfar)– General: glFrustum(xwmin,xwmax,ywmin,ywmax,dnear,dfar)