Computer GraphicsComputer Graphics

Zhen JiangZhen Jiang

West Chester UniversityWest Chester University

TopicsTopics

• Projection

• Visibility

ProjectionProjection

Film Plane

Pinhole

Multiple raysof projection

ProjectionProjection

Film Plane

Pinhole

One rayof projection

ProjectionProjection

• Field of View• Field of View

Film Plane

Field of ViewPinhole

FocalLength

ProjectionProjection

Film Plane

Field of ViewPinhole

FocalLength

ProjectionProjection• Varying distance to film plane

• What does this do?

• Varying distance to film plane

• What does this do?

Film Plane

d1 Field of ViewPinhole

d2

ProjectionProjection• We use

– Center of Projection (COP)– Projection Plane

• We use – Center of Projection (COP)– Projection Plane

COP

ProjectionPlane

ProjectionProjection• Perspective vs. Orthographic Views• Perspective vs. Orthographic Views

Perspective

When COP at infinity, Orthographic View

ProjectionProjection• One-point Perpective

– One Vanishing Point

• Two-point Perspective– Two Vanishing Points

• One-point Perpective– One Vanishing Point

• Two-point Perspective– Two Vanishing Points

http://www.sanford-artedventures.com/create/tech_2pt_perspective.html

ProjectionProjection

• Our camera must model perspective

• Our camera must model perspective

ProjectionProjection

How tall shouldthis bunny be?

COP

ProjectionPlane

ProjectionProjection

d

P (x, y, z)X

Z

Viewplane

(0,0,0) x’ = ?

ProjectionProjection• Desired result for a point [x, y, z, 1]T projected

onto the view plane:

• What could a matrix look like to do this?

• Desired result for a point [x, y, z, 1]T projected onto the view plane:

• What could a matrix look like to do this?

dzdz

y

z

ydy

dz

x

z

xdx

z

y

d

y

z

x

d

x

,','

',

'

ProjectionProjection

0100

0100

0010

0001

d

M eperspectiv

ProjectionProjection• Example:

• Or, in 3-D coordinates:

• Example:

• Or, in 3-D coordinates:

10100

0100

0010

0001

z

y

x

ddz

z

y

x

d

dz

y

dz

x,,

ProjectionProjection

• Orthographic Camera ProjectionOrthographic Camera Projection

11000

0000

0010

0001

1

z

y

x

z

y

x

p

p

p

ProjectionProjection

• glFrustum – for perspective projections– xmin– xmax– ymin– ymax– near – far

• glFrustum – for perspective projections– xmin– xmax– ymin– ymax– near – far

– Camera looks along –z– min/max need not be

symmetric about any axis– near and far planes are parallel

to plane z=0

– Camera looks along –z– min/max need not be

symmetric about any axis– near and far planes are parallel

to plane z=0

ProjectionProjection

• gluPerspective – for perspective projections– fovy– aspect– near– far

• gluPerspective – for perspective projections– fovy– aspect– near– far

– fovy is the angle between top and bottom of viewing volume

– aspect is ratio of width over height

– This volume is symmetrical– View plane is parallel to

camera

– fovy is the angle between top and bottom of viewing volume

– aspect is ratio of width over height

– This volume is symmetrical– View plane is parallel to

camera

ProjectionProjection

• glOrtho – for orthographic projections– left– right– bottom– top– near– far

• glOrtho – for orthographic projections– left– right– bottom– top– near– far

– (left, bottom) and (right, top) define dimensions of projection plane

– near and far used to clip

– (left, bottom) and (right, top) define dimensions of projection plane

– near and far used to clip

VisibilityVisibility• Why might a polygon be invisible?

– Polygon outside the field of view– Polygon is backfacing– Polygon is occluded by object(s) nearer the

viewpoint

• For efficiency reasons, we want to avoid spending work on polygons outside field of view or backfacing

• For efficiency and correctness reasons, we need to know when polygons are occluded

• Why might a polygon be invisible?– Polygon outside the field of view– Polygon is backfacing– Polygon is occluded by object(s) nearer the

viewpoint

• For efficiency reasons, we want to avoid spending work on polygons outside field of view or backfacing

• For efficiency and correctness reasons, we need to know when polygons are occluded

VisibilityVisibility

• Remove polygons entirely outside frustum– Note that this includes polygons “behind” eye

(actually behind near plane)

• Pass through polygons entirely inside frustum

• Modify remaining polygonsto include only portions intersecting view frustum

• Remove polygons entirely outside frustum– Note that this includes polygons “behind” eye

(actually behind near plane)

• Pass through polygons entirely inside frustum

• Modify remaining polygonsto include only portions intersecting view frustum

VisibilityVisibility• Most objects in scene are typically

“solid”• More rigorously: compact,

orientable manifolds– Must not cut through itself– Must have two distinct sides

• A sphere is orientable since it has two sides, 'inside' and 'outside'.

• A Mobius strip or a Klein bottle is not orientable

– Cannot “walk” from one side to the other

• A sphere is a closed manifold whereas a plane is not

• Most objects in scene are typically “solid”

• More rigorously: compact, orientable manifolds– Must not cut through itself– Must have two distinct sides

• A sphere is orientable since it has two sides, 'inside' and 'outside'.

• A Mobius strip or a Klein bottle is not orientable

– Cannot “walk” from one side to the other

• A sphere is a closed manifold whereas a plane is not

VisibilityVisibility• For most interesting scenes, some polygons will

overlap:

• To show the correct image, we need to determine which polygons occlude which

• For most interesting scenes, some polygons will overlap:

• To show the correct image, we need to determine which polygons occlude which

VisibilityVisibility

• Simple approach: display the polygons from back to front, “painting over” previous polygons:

– Draw blue, then green, then orange

• Will this work in the general case?

• Simple approach: display the polygons from back to front, “painting over” previous polygons:

– Draw blue, then green, then orange

• Will this work in the general case?

VisibilityVisibility• Intersecting polygons present a problem

• Even non-intersecting polygons can form a cycle with no valid visibility order:

• Intersecting polygons present a problem

• Even non-intersecting polygons can form a cycle with no valid visibility order:

VisibilityVisibility• Early visibility algorithms computed the set of

visible polygon fragments directly, then showed the fragments to a display:

• Early visibility algorithms computed the set of visible polygon fragments directly, then showed the fragments to a display:

VisibilityVisibility• So, for about a decade (late 60s to

late 70s) there was intense interest in finding efficient algorithms for hidden surface removal

• Examples: – Binary Space-Partition (BSP) Trees

http://symbolcraft.com/graphics/bsp– The Z-buffer Algorithm

• So, for about a decade (late 60s to late 70s) there was intense interest in finding efficient algorithms for hidden surface removal

• Examples: – Binary Space-Partition (BSP) Trees

http://symbolcraft.com/graphics/bsp– The Z-buffer Algorithm

VisibilityVisibility

• We know how to rasterize polygons into an image discretized into pixels:

• We know how to rasterize polygons into an image discretized into pixels:

VisibilityVisibility

• What happens if multiple primitives occupy the same pixel on the screen? Which is allowed to paint the pixel?

• What happens if multiple primitives occupy the same pixel on the screen? Which is allowed to paint the pixel?

VisibilityVisibility

• Simple!!!

• Easy to implement in hardware

• Polygons can be processed in arbitrary order

• Easily handles polygon interpenetration

• Simple!!!

• Easy to implement in hardware

• Polygons can be processed in arbitrary order

• Easily handles polygon interpenetration

VisibilityVisibility

• Lots of memory (e.g. 1280x1024x32 bits)– With 16 bits cannot discern millimeter differences in

objects at 1 km distance

• Read-Modify-Write in inner loop requires fast memory• Hard to do analysisi

– We don’t know which polygon to map pixel back to

• Shared edges are handled inconsistently– Ordering dependent

• Hard to simulate translucent polygons– We throw away color of polygons behind closest one

• Lots of memory (e.g. 1280x1024x32 bits)– With 16 bits cannot discern millimeter differences in

objects at 1 km distance

• Read-Modify-Write in inner loop requires fast memory• Hard to do analysisi

– We don’t know which polygon to map pixel back to

• Shared edges are handled inconsistently– Ordering dependent

• Hard to simulate translucent polygons– We throw away color of polygons behind closest one