Introduction to Computer Graphics5. Viewing in 3D (A)

National Chiao Tung Univ, TaiwanBy: I-Chen Lin, Assistant Professor

Textbook: Hearn and Baker, Computer Graphics, 3rd Ed., Prentice HallRef: E.Angel, Interactive Computer Graphics, 4th Ed., Addison Wesley

OutlineClassical views

Computer viewing

Projection matrices

Classical ViewingViewing requires three basic elements

One or more objectsA viewer with a projection surfaceProjectors that go from the object(s) to the projection surface

Each object is assumed to constructed from flat principal faces

Buildings, polyhedra, manufactured objects

Planar Geometric ProjectionsStandard projections project onto a plane.

Projectors are lines that eitherconverge at a center of projectionare parallel

Such projections preserve linesbut not necessarily angles

When do we need non-planar projections?

Classical Projections

Perspective vs ParallelClassical viewing developed different techniques for drawing each type of projection

Mathematically parallel viewing is the limit of perspective viewing

Computer graphics treats all projections the same and implements them with a single pipeline

Taxonomy of Planar Geometric Projections

parallel perspective

axonometricmultivieworthographic

oblique2 point1 point 3 point

planar geometric projections

isometric dimetric trimetric

Perspective Projection

Parallel Projection

Orthographic ProjectionProjectors are orthogonal to projection surface

Multiview Orthographic ProjectionProjection plane parallel to principal facesUsually form front, top, side views

isometric (not multivieworthographic view) front

side

in CAD and architecture, we often display three multiviews plus isometric

top

Advantages and DisadvantagesPreserves both distances and angles

Shapes preservedCan be used for measurements

Building plansManuals

Cannot see what object really looks like because many surfaces hidden from view

Often we add the isometric

Axonometric ProjectionsAllow projection plane to move relative to object

classify by how many angles ofa corner of a projected cube are the same

none: trimetrictwo: dimetricthree: isometric

θ 1θ 3θ 2

Types of Axonometric Projections

Advantages and DisadvantagesLines are scaled (foreshortened) but can find scaling factors

Lines preserved but angles are notProjection of a circle in a plane not parallel to the projection plane is an ellipse

Does not look real because far objects are scaled the same as near objects

Used in CAD applications

Oblique ProjectionArbitrary relationship between projectors and projection plane

Perspective ProjectionProjectors coverage at center of projection

Vanishing PointsParallel lines (not parallel to the projection plan):

converge at a single point in the projection (the vanishing point)

Drawing simple perspectives by hand uses these vanishing point(s)

vanishing point

Three-Point PerspectiveNo principal face parallel to projection planeThree vanishing points for cube

Two-Point PerspectiveOn principal direction parallel to projection planeTwo vanishing points for cube

One-Point PerspectiveOne principal face parallel to projection planeOne vanishing point for cube

Advantages and DisadvantagesDiminution:

Objects further from viewer are projected smaller (Looks realistic)

Nonuniform foreshortening:Equal distances along a line are not projected into equal distances

Angles preserved only in planes parallel to the projection plane

More difficult to construct by hand than parallel projections

Computer Viewing

Pipeline View

ModelingTransformation

ModelingTransformation

ViewingTransformation

ViewingTransformation

ProjectionTransformation

ProjectionTransformation

Normalizationand

clipping

Normalizationand

clipping

ViewportTransformation

ViewportTransformation

MC WC VC

PC NC DC

Let’s skip the clipping temporarily !

Computer ViewingThree aspects of the viewing process implemented in the pipeline:

Positioning the cameraSetting the model-view matrix

Selecting a lensSetting the projection matrix

ClippingSetting the view volume

The OpenGL CameraIn OpenGL, initially the object and camera frames are the same

Default model-view matrix is an identity

The camera is located at origin and points in the negative z direction

OpenGL also specifies a default view volume that is a cube with sides of length 2 centered at the origin

Default projection matrix is an identity

Moving the Camera FrameIf we want to visualize object with both positive and negative z values we can either

Move the camera in the positive z directionTranslate the camera frame

Move the objects in the negative z directionTranslate the world frame

Both of these views are equivalent and are determined by the model-view matrix

Want a translation (glTranslatef(0.0,0.0,-d);)d > 0

Moving Camera back from Origin

default frames

frames after translation by –dd > 0

Moving the CameraWe can move the camera to any desired position by a sequence of rotations and translations

Example: side viewRotate the cameraMove it away from originModel-view matrix C = TR

gluLookAtglLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz)

Projections and NormalizationThe default projection in the eye (camera) frame is orthogonal For points within the default view volume

xp = xyp = yzp = 0

Homogeneous Coordinate Representation

default orthographic projectionxp = xyp = yzp = 0wp = 1

pp = Mp

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1000000000100001

M =

In practice, we can let M = I and setthe z term to zero later

Simple PerspectiveCenter of projection at the originProjection plane z = d, d < 0

Perspective EquationsTop view Side view

dzx/

dzx/ dz

y/

xp = yp = zp = d

Homogeneous Coordinate Form

M =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0/100010000100001

d

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

dzzyx

/

consider q = Mp where

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1zyx

⇒ q = p =

Perspective DivisionHowever w ≠ 1, so we must divide by w to return from homogeneous coordinatesThis perspective division yields

the desired perspective equations

dzx/ dz

y/

xp = yp = zp = d

Viewport TransformationFrom the working coordinate to the coordinate of display device.

(1,1)

(-1,-1)

(0,0)

(0,0)

(600,600)

By 2D scaling and translation

Next: clipping and normalization !