Basic Perspective Projection Watt Section 5.2, some typos

1. 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) P(xv,yv,zv) P(xs,ys) yv d -zv xv

2. Basic Case • Similar triangles gives: yv P(xv,yv,zv) P(xs,ys) d -zv View Plane

3. Homogeneous Screen Coordinates • Using homogeneous coordinates we can write:

4. 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

5. Clipping Planes Left Clip Plane Near Clip Plane xv Far Clip Plane View Volume -zv f Image Plane Right Clip Plane

6. 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

7. 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

8. Computing Screen Depth • Intersections pts maintained if: zs=A+B/zv • Desired mapping of image plane and far plane gives constraints. Solving equation gives:

9. Homogeneous Screen Coords • Using homogeneous coordinates we can write: • Note: ws is homogeneous coordinate, w is window width

10. 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 -

11. 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) Image Plane New Image Plane

12. 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

13. 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:

14. General to Basic Case

15. 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!

16. 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(…)

17. 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

18. 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

19. 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)

20. Polygon-Rectangle Clipping (2D) • Task: Clip a polygon to a rectangle • Easy cases: • Task: Clip a polygon to a rectangle • Easy cases: • Hard cases:

21. 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?)

22. 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…

23. 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

24. Sutherland-Hodgman (4) Inside Outside Inside Outside Inside Outside Inside Outside p s i s p p s i p s Output p Output i No output Output i,p

25. 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