Basic Perspective ProjectionWatt Section 5.2, some typos

• Define a focal distance, d, and shift the origin to be at that distance (note d is negative)

xv

yv

-zvd

P(xv,yv,zv)P(xs,ys)

Basic Case

• Similar triangles gives:z

x

d

x

z

y

d

y

yv

-zv

P(xv,yv,zv)P(xs,ys)

View Plane

d

Homogeneous Screen Coordinates

• Using homogeneous coordinates we can write:

dzz

y

x

d

y

x

PP

0100

0100

0010

0001

d

Clipping Planes

• The image has a finite width and height• Edges define clipping planes that cut off geometry

outside the view• Also define near and far clipping planes

– Clip things behind the camera and very distant

– Far plane can be problematic: Use tricks like “fog” to blend it out

Clipping Planes

xv

-zv

ImagePlane

Near Clip Plane

Far Clip PlaneView

Volume

Left ClipPlane

Right ClipPlane

f

OpenGL Basic Case

• gluPerspective(…)– Field of view (determines focal distance)

– Aspect ratio (should match window aspect ratio)

– Near and far clipping planes

• Assumes that the image is centered in the image plane

Screen Space

• Image corners should be (-1,-1), (1,1) instead of (-w,-h), (w,h)

• We need depth information to decide what’s in front

• Useful properties:– Points on the image plane should map to zs=0

– Points on the far clip plane should map to zs=1

– Intersections of lines and planes in view space should map to their intersections in screen space

• Straight lines should transform to straight lines

• Planes should transform to planes

Computing Screen Depth

• Intersections pts maintained if: zs=A+B/zv

• Desired mapping of image plane and far plane gives constraints. Solving equation gives:

v

vs wz

xdx

v

vs hz

ydy

)(

)1(

df

zdfz v

s

Homogeneous Screen Coords

• Using homogeneous coordinates we can write:

• Note: ws is homogeneous coordinate, w is window width

10100

)()(00

000

000

v

v

v

s

s

s

s

z

y

x

dfdfdff

hd

wd

w

z

y

x

Decomposing the Transformation

• The transformation taking view to screen space can be decomposed into two components:– One scales the space to make the side clipping planes

of the form x=z, y=z

– The other deforms space to take the frustum to a box, with the focal point at -

Small Complication

• We really want the near clip plane to map to zs=0

• Change focal dist to near clip dist, modify image size (or specify field of view and derive image size)

ImagePlane New Image

Plane

General Case

• Previous case assumed that view window was centered with corners (-w,-h), (w,h)

• General case uses arbitrary area on image plane (xmin,ymin), (xmax,ymax)

• OpenGL: glFrustum(...)– Corners of frustum in image plane

– Near and far clip planes

General to Basic Case

• Shear the volume so that the central axis lies on the n-axis– This is a shear, because rectangles on planes n=constant

must stay rectangles

• Shear takes old window midpoint to (0, 0, d) - this means that matrix is:

1000

01002

010

2001

maxmin

maxmin

d

yyd

xx

u

v

n

u

v

n

General to Basic Case

Near/Far and Depth Resolution

• It may seem sensible to specify a very near clipping plane and a very far clipping plane– Sure to contain entire scene

• But, a bad idea:– OpenGL only has a finite number of bits to store screen

depth– Too large a range reduces resolution in depth - wrong

thing may be considered “in front”

• Distant stuff is very small anyway!

Screen to Window Space

• Points in screen space are in homogeneous form– Clipping (described next) must be done in this form

• “Perspective divide”, converts homogeneous points into 3D screen points– x,y range from –1 to 1, z from 0 to 1

– Do lighting here

• Viewport transformation scales and translates x,y to fill the window in the screen: glViewport(…)

Viewing Transformation Summary

• Convert world to view: Translation and rotation• Convert view to screen: Translation, possibly

shearing, scaling and perspective• Convert screen to window: Scale and translate• All managed by OpenGL

– You just give the parameters

Clipping

• Parts of the geometry to be rendered may lie outside the view volume– View volume maps to memory addresses

– Out-of-view geometry generates invalid addresses

• Clipping removes parts of the geometry that are outside the view

• Best done in screen space before perspective divide

Clipping (2)

• Points are trivial to clip - just check which side of the clip planes they are on (dot product)

• Many algorithms for clipping lines exist– Next lecture

• Two main algorithms for clipping polygons exist– Sutherland-Hodgman (today)

– Weiler (next lecture)

Polygon-Rectangle Clipping (2D)

• Task: Clip a polygon to a rectangle• Easy cases:

• Task: Clip a polygon to a rectangle• Easy cases:

• Hard cases:

Sutherland-Hodgman Clip (1)

• Clip the polygon against each edge of the clip region in turn– Clip polygon each time to line containing edge

– Only works for convex clip regions (Why?)

Sutherland-Hodgman Clip (2)

• To clip a polygon to a line:– Consider the polygon as a list of vertices

– One side of the line is inside the clip region, the other outside

– Think of the process as rewriting the polygon, one vertex at a time

– Check start vertex: if “inside”, emit it, otherwise ignore it

– Process vertex list proceeding as follows…

Sutherland-Hodgman (3)

• Look at the next vertex in the list:– polygon edge crosses clip edge going from out to in:

output crossing point, next vertex

– polygon edge crosses clip edge going from in to out: output crossing

– polygon edge goes from out to out: output nothing

– polygon edge goes from in to in: output next vertex

Sutherland-Hodgman (4)Inside Outside

s

p

Output p

Inside Outside

sp

Output i

Inside Outside

s

p

No output

Inside Outside

sp

Output i,p

i

i

Sutherland-Hodgman (5)

• In 3D, clip against planes instead of lines– Six planes to clip against

– Inside/Outside test still works

• Suitable for hardware implementation– Only need the clip edge, the endpoints of the current

edge, and the last output point

– Polygon edges are output as they are found, and passed right on to the next clip region edge