CAP 4703CAP 4703Computer Graphic Computer Graphic

MethodsMethodsProf. Roy LevowProf. Roy Levow

Chapter 5Chapter 5

ViewingViewing

Classical viewingClassical viewing– Need to be able to reproduce classical Need to be able to reproduce classical

views for a variety of applicationsviews for a variety of applications– IsometricIsometric

Preserves measurementsPreserves measurements

– ElevationElevationView a face of objectView a face of object

– PerspectivePerspectiveReflects size change of distant objectsReflects size change of distant objects

Viewing TerminologyViewing Terminology

Center of Projection (COP)Center of Projection (COP)– Point where all projectors meetPoint where all projectors meet– Center of camera or eye lensCenter of camera or eye lens– Origin of synthetic camera frameOrigin of synthetic camera frame

Direction of Projection (DOP)Direction of Projection (DOP)– Direction of projectors when COP is Direction of projectors when COP is

moved to infinitymoved to infinity

Types of ViewingTypes of Viewing

Classical (many)Classical (many)– OrthographicOrthographic– One-, two-, and 3-point perspectivesOne-, two-, and 3-point perspectives

ComputerComputer– OrthographicOrthographic– PerspectivePerspective

Classical Viewing ConceptsClassical Viewing Concepts

Principal facePrincipal face– Primary surface of viewPrimary surface of view

Based on rectangular solid structuresBased on rectangular solid structures

Orthographic ProjectionOrthographic Projection– Single viewSingle view– MultiviewMultiview

Show three orthogonal viewsShow three orthogonal views

Axonometric ProjectionsAxonometric Projections

Projectors are orthogonal to the Projectors are orthogonal to the projection plane but plane can be at projection plane but plane can be at angle to principal faceangle to principal face– Isometric – symmetric with all three Isometric – symmetric with all three

axesaxes– Dimetric – symmetric with two axesDimetric – symmetric with two axes– Trimetric – general caseTrimetric – general case– Produces foreshortening of distancesProduces foreshortening of distances

Oblique ProjectionOblique Projection

Most general parallel viewMost general parallel view– Projectors make arbitrary angle with Projectors make arbitrary angle with

projection planeprojection plane

In computer graphics system, In computer graphics system, isometric, axonometric and oblique isometric, axonometric and oblique projections are all variations on one projections are all variations on one casecase

PerspectivePerspective

Characterized by diminution of size Characterized by diminution of size of more distant objectsof more distant objects

Classically, viewer is symmetrical Classically, viewer is symmetrical with respect to the projection planewith respect to the projection plane

One-, two-, and three-point One-, two-, and three-point perspectives depending on number perspectives depending on number of vanishing pointsof vanishing points

Computer ViewingComputer Viewing

Choose parallel or perspective viewChoose parallel or perspective view No separation of perspective degreesNo separation of perspective degrees

Two key elementsTwo key elements– Position cameraPosition camera– Apply projection transformatonApply projection transformaton

OpenGL CameraOpenGL Camera

DefaultDefault– Camera at originCamera at origin– Pointing in negative z directionPointing in negative z direction– Orthogonal viewOrthogonal view– Viewing volume is cubeViewing volume is cube

Centered at originCentered at originSide of length 2Side of length 2

Default CameraDefault Camera

Positioning Camera FramePositioning Camera Frame

Can construct camera frame through Can construct camera frame through translation and rotation of model translation and rotation of model view to get camera to desired view to get camera to desired viewing position from default positionviewing position from default position

For orthographic view, this does not For orthographic view, this does not change clipping volume set by change clipping volume set by glOrtho()glOrtho()– Size is unchangedSize is unchanged– What is seen will changeWhat is seen will change

View as Change of FramesView as Change of Frames

Set View-Reference Point (VRP)Set View-Reference Point (VRP)– Center of cameraCenter of camera

View plane defined byView plane defined by– View-plane normal (VPN)View-plane normal (VPN)– View-up vector (VUP)View-up vector (VUP)– Project VUP onto projection plane to get Project VUP onto projection plane to get

up direction (vup direction (vupup)) Construct new frame with basis for Construct new frame with basis for

view plane, u-v and normal n; u-v-nview plane, u-v and normal n; u-v-n

Look-At PositioningLook-At Positioning

OpenGL Look-AtOpenGL Look-At

OpenGL simplifies camera positioning as OpenGL simplifies camera positioning as followsfollows– e = eye pointe = eye point– a = look-at pointa = look-at point– Determines vpn as e – aDetermines vpn as e – a

glMatrixMode(GL_MODELVIEW);glMatrixMode(GL_MODELVIEW);

glLoadIdentity();glLoadIdentity();

gluLookAt(eyex, eyey, eyez, gluLookAt(eyex, eyey, eyez,

atx, aty, atz, upx, upy, upz);atx, aty, atz, upx, upy, upz);

/* define objects here *//* define objects here */

Perspective ProjectionPerspective Projection

Note: x/z = xNote: x/z = xpp/d or x/d or xpp= x/(z/d)= x/(z/d)– Similarly for ySimilarly for y– Provides non-uniform foreshorteningProvides non-uniform foreshortening

Perspective TransformationPerspective Transformation– (x, y, z) -> (x(x, y, z) -> (xpp, y, ypp, z, zpp) )

Perspective Projection (cont)Perspective Projection (cont)

Moving to 4 dimensions, considerMoving to 4 dimensions, consider(wx, wy, wz, w)(wx, wy, wz, w)TT

Perspective transformation matrixPerspective transformation matrix11 0 0 00 0 0 takestakes0 1 0 00 1 0 0 (x, y, z, 1) (x, y, z, 1) TT

0 0 1 0 0 0 1 0 toto0 0 1/d 00 0 1/d 0 (x, y, z, z/d) (x, y, z, z/d) TT

Division by last coordinate givesDivision by last coordinate gives

(x(xpp, y, ypp, d, 1), the projection, d, 1), the projection

Orthogonal Projection in Orthogonal Projection in OpenGLOpenGL

Simply map z into zeroSimply map z into zero

Projection in OpenGLProjection in OpenGL

Clipping region is a frustum, a Clipping region is a frustum, a truncated pyramidtruncated pyramid

OpenGL FrustumOpenGL Frustum

glFrustum(xmin, xmax, ymin, ymax,glFrustum(xmin, xmax, ymin, ymax,near, far);near, far);

// left, right, top, bottom, near, far// left, right, top, bottom, near, far

Parallel Viewing in OpenGLParallel Viewing in OpenGL

glOrtho(xmin, xmax, ymin, ymax,glOrtho(xmin, xmax, ymin, ymax,near, far);near, far);

Hidden Surface RemovalHidden Surface Removal

Remove hidden surfacesRemove hidden surfaces Different viewDifferent view

– Visible-surface algoritimsVisible-surface algoritims Identify visible surfacesIdentify visible surfaces

Work in eitherWork in either– Object spaceObject space– Image spaceImage space

Z-Buffer AlgorithmZ-Buffer Algorithm

Image SpaceImage Space Requires depth buffer, called z-buffer to Requires depth buffer, called z-buffer to

store depthstore depth Depth resolution usually 16, 24, or 32 bitsDepth resolution usually 16, 24, or 32 bits As polygon is rasterized, depth is As polygon is rasterized, depth is

computed and compared with current z-computed and compared with current z-buffer value; only nearer values updatebuffer value; only nearer values update

Very efficientVery efficient

OpenGL ImplementationOpenGL Implementation

Enable withEnable with– glutInitDisplayMode(GLUT_DEPTH…);glutInitDisplayMode(GLUT_DEPTH…);– glEnabel(GL_DEPTH_TEST);glEnabel(GL_DEPTH_TEST);

Clear withClear with– glClear(GL_DEPTH_BUFFER_BIT);glClear(GL_DEPTH_BUFFER_BIT);

Scene WalkthroughScene Walkthrough

Move camera to view color cube from Move camera to view color cube from different locationsdifferent locations– cubeview.ccubeview.c

Parallel-Projection MatricesParallel-Projection Matrices

Previous projection matrices do not Previous projection matrices do not cover all possibilitiescover all possibilities

Projection NormalizationProjection Normalization– convert all projections to orthogonalconvert all projections to orthogonal– by distorting objectsby distorting objects– distortion is called normalizationdistortion is called normalization

Map viewing volume into 2x2x2 cube Map viewing volume into 2x2x2 cube at origin; translate and then scaleat origin; translate and then scale

Oblique ProjectionOblique Projection

Projectors do not need to be Projectors do not need to be orthogonal to projection plane as in orthogonal to projection plane as in standard OpenGL projectionstandard OpenGL projection

Equivalent to a shear transformation Equivalent to a shear transformation of the objectsof the objects

Perspective-Projection MatricesPerspective-Projection Matrices

Again, distort objectAgain, distort object Skipping detailsSkipping details OpenGL Perspective TransformationsOpenGL Perspective Transformations

2z/(X-x) 0 (X+x)/(X-x) 02z/(X-x) 0 (X+x)/(X-x) 0

0 2z/(Y-y) (Y+y)/(Y-y) 00 2z/(Y-y) (Y+y)/(Y-y) 0

0 0 -(f+n)/(f-n) -2fn/(f-n)0 0 -(f+n)/(f-n) -2fn/(f-n)

0 0 -1 00 0 -1 0

Projections and ShadowsProjections and Shadows

ShadowShadow– projection of original polygon onto projection of original polygon onto

surfacesurface– center of projection center of projection

is light sourceis light source– shadow-polygonshadow-polygon– shadow.cshadow.c