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Clipping Rasterization

Clipping Rasterization. CS 445/645 Introduction to Computer Graphics David Luebke, Spring 2003. Admin. Homework 1 Not quite graded yet (by Wed or Fri latest—sorry!) Peer evaluations All very positive so far

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Clipping Rasterization

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  1. ClippingRasterization CS 445/645Introduction to Computer Graphics David Luebke, Spring 2003

  2. Admin • Homework 1 • Not quite graded yet (by Wed or Fri latest—sorry!) • Peer evaluations • All very positive so far • Remember that honor code applies: be frank, honest, unbiased, don’t casually give either bad or good evaluations • Homework 2 • Out today David Luebke 26/7/2014

  3. Recap:Sutherland-Hodgman Clipping • Sutherland-Hodgman basic routine: • Go around polygon one vertex at a time • Current vertex has position p • Previous vertex had position s, and it has been added to the output if appropriate David Luebke 36/7/2014

  4. inside outside inside outside inside outside inside outside p s p s p s p s p output i output no output i outputp output Recap:Sutherland-Hodgman Clipping • Edge from s to ptakes one of four cases: (Purple line can be a line or a plane) David Luebke 46/7/2014

  5. q n p P Recap:Point-to-Plane test • A very general test to determine if a point p is “inside” a plane P, defined by q and n: (p - q) • n < 0: p inside P (p - q) • n = 0: p on P (p - q) • n > 0: p outside P q q n n p p P P David Luebke 56/7/2014

  6. Recap:Point-to-Plane Test • Dot product is relatively expensive • 3 multiplies • 5 additions • 1 comparison (to 0, in this case) • Can often optimize or special-case this David Luebke 66/7/2014

  7. Recap:Line-Plane Intersections • Use parametric definition of edge: E(t) = s + t(p - s) • If t = 0 then E(t) = s • If t = 1 then E(t) = p • Otherwise, E(t) is part way from s to p • Edge intersects plane P where E(t) is on P • q is a point on P • n is normal to P (E(t) - q) • n = 0 t = [(q - s) • n] / [(p - s) • n] • The intersection point i = E(t)for this value of t David Luebke 76/7/2014

  8. Recap: Perspective Projection • Recall the matrix: • Or, in 3-D coordinates: David Luebke 86/7/2014

  9. Clipping Under Perspective • Problem: after multiplying by a perspective matrix and performing the homogeneous divide, a point at (-8, -2, -10) looks the same as a point at (8, 2, 10). • Solution A: clip before multiplying the point by the projection matrix • I.e., clip in camera coordinates • Solution B: clip after the projection matrix but before the homogeneous divide • I.e., clip in homogeneous screen coordinates David Luebke 96/7/2014

  10. Clipping Under Perspective • We will talk first about solution A: Clippedworldcoordinates Canonicalscreencoordinates Clip againstview volume Apply projectionmatrix andhomogeneousdivide Transform intoviewport for2-D display 3-D world coordinateprimitives 2-D devicecoordinates David Luebke 106/7/2014

  11. Recap: Perspective Projection • The typical view volume is a frustumor truncated pyramid x or y z David Luebke 116/7/2014

  12. Perspective Projection • The viewing frustum consists of six planes • The Sutherland-Hodgeman algorithm (clipping polygons to a region one plane at a time) generalizes to 3-D • Clip polygons against six planes of view frustum • So what’s the problem? David Luebke 126/7/2014

  13. Perspective Projection • The viewing frustum consists of six planes • The Sutherland-Cohen algorithm (clipping polygons to a region one plane at a time) generalizes to 3-D • Clip polygons against six planes of view frustum • So what’s the problem? • The problem: clipping a line segment to an arbitrary plane is relatively expensive • Dot products and such David Luebke 136/7/2014

  14. Back or yon plane Front or hither plane Perspective Projection • In fact, for simplicity we prefer to use the canonical view frustum: x or y 1 z -1 Why is this going to besimpler? -1 David Luebke 146/7/2014

  15. Back or yon plane Front or hither plane Perspective Projection • In fact, for simplicity we prefer to use the canonical view frustum: x or y 1 z -1 Why is the yon planeat z = -1, not z = 1? -1 David Luebke 156/7/2014

  16. Clipping Under Perspective • So we have to refine our pipeline model: • Note that this model forces us to separate projection from modeling & viewing transforms Applynormalizingtransformation projectionmatrix;homogeneousdivide Transform intoviewport for2-D display Clip against canonical view volume 3-D world coordinateprimitives 2-D devicecoordinates David Luebke 166/7/2014

  17. Apply projectionmatrix Clipagainst view volume Homogeneousdivide Transform intoviewport for2-D display 3-D world coordinateprimitives 2-D devicecoordinates Clipping Homogeneous Coords • Another option is to clip the homogeneous coordinates directly. • This allows us to clip after perspective projection: • What are the advantages? David Luebke 176/7/2014

  18. Clipping Homogeneous Coords • Other advantages: • Can transform the canonical view volume for perspective projections to the canonical view volume for parallel projections • Clip in the latter (only works in homogeneous coords) • Allows an optimized (hardware) implementation • Some primitives will have w  1 • For example, polygons that result from tesselating splines • Without clipping in homogeneous coords, must perform divide twice on such primitives David Luebke 186/7/2014

  19. Clipping Homogeneous Coords • So how do we clip homogeneous coordinates? • Briefly, thus: • Remember that we have applied a transform to normalized device coordinates • x, y [-1, 1] • z  [0, 1] • When clipping to (say) right side of the screen (x = 1), instead clip to (x = w) • Can find details in book or on web • Assignment 2: little bit of extra credit for clipping in homogeneous coords David Luebke 196/7/2014

  20. Clipping: The Real World • In some renderers, a common shortcut used to be: • But in today’s hardware, everybody just clips in homogeneous coordinates Projectionmatrix;homogeneousdivide Clip in 2-D screen coordinates Clip against hither andyon planes Transform intoscreen coordinates David Luebke 206/7/2014

  21. Next Topic: Drawing Lines

  22. Line Algorithms • Drawing 2-D lines: a case study in optimization • “One good thief is worth ten good scholars” • Sometimes somebody does something so well there is no point in doing it over • We will use Professor Leonard McMillan’s web-based lecture notes (then at MIT, now at UNC) • http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture5 • His examples are in Java, but the correspondences are obvious. • Raster: his implementation of a framebuffer class David Luebke 226/7/2014

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