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- 1. CSC 406: Applied Computer Graphics LECTURE 8: Clipping
- 2. Lecture 8: Scope: Clipping Clipping algorithms
- 3. Clipping We have talked about 2D scan conversion of line-segments and polygons What if endpoints of line segments or vertices of polygons lie outside the visible device region? Need clipping! Clipping .pdf notes
- 4. Clipping Clipping: Remove points outside a region of interest – clip window. Want to discard everything that’s outside of our window... Primitives: point, line-segment, and polygon. Point clipping: Remove points outside window. A point is either entirely inside the region or not. Line-segment clipping: Remove portion of line segment outside window. Line segments can straddle the region boundary. Polygon clipping: Remove portion of polygon outside window
- 5. Review: Parametric representation of line or equivalently A and B are non-coincident endpoints. For , L(t) defines an infinite line. For , L(t) defines a line segment from A to B. Good for generating points on a line. Not so good for testing if a given point is on a line. A B
- 6. Review: Implicit line Equation representation of line: •P is a point on the line. • is a vector perpendicular to the line. • gives us the signed distance from any point Q to the line. •The sign of tells us if Q is on the left or right of the line, relative to the direction of . •If is zero, then Q is on the line. •Use same form for the implicit representation of a half space.
- 7. Clipping a point to a half space Represent window edges implicitly. Use the implicit form of a line to classify a point Q. Must choose a convention for the normal: points to the inside. Check the sign of : If > 0, then Q is inside. Otherwise clip (discard) Q: It is on the edge or outside.
- 8. Clipping a line segment to a half space There are three cases: The line segment is entirely inside: Keep it. The line segment is entirely outside: Discard it. The line segment is partially inside and partially outside: Generate new line to represent part inside.
- 9. Input Specification •Window edge: implicit, •Line segment: parametric, L(t) = A + t(B-A).
- 10. Do the easy stuff first Do the easy stuff first: We can devise easy (and fast!) tests for the first two cases: AND Outside AND Inside Need to decide whether “on the boundary” is inside or outside? Trivial tests are important in computer graphics: Particularly if the trivial case is the most common one. Particularly if we can reuse the computation for the non- trivial case
- 11. Do the hard stuff only if we have to If line segment partially inside and partially outside, need to clip it: •Line segment from A to B in parametric form: •When t=0, L(t)=A. When t=1, L(t)=B. •We now have the following:
- 12. Find the Intersection •We want t such that : •Solving for t gives us •NOTE: The values we use for our simple test can be reused to compute t:
- 13. Clipping a line segment to a window Just clip to each of four half spaces in turn. given: line segment (A, B), clip in-place: for each edge (P,n) wecA = (A-P) . n wecB = (B-P) . n if ( wecA < 0 AND wecB < 0 ) then reject if ( wecA > 0 AND wecB > 0 ) then next t = wecA / (wecA - wecB) if ( wecA < 0 ) then A = A + t*(B-A) else B = A + t*(B-A) endif endfor Note: This Algorithm can clip lines to any convex window. Optimizations can be made for the special case of horizontal and vertical window edges
- 14. Optimization (Cohen- Sutherland) Q: Solving for the intersection point is expensive. Can we avoid evaluation of a clip on one edge if we know the line segment is out on another? A: Yes. Do all trivial tests first. Combine results using Boolean operations to determine Trivial accept on window: all edges have trivial accept Trivial reject on window: any edge has trivial reject Do full clip only if no trivial accept/reject on window
- 15. Cohen-Sutherland Algorithm 0 1 2 3 0000 00010101 1001 0100 1000 00100110 1010 (x1, y1) (x2, y2) (x2, y1) (x1, y2) Half space code (x < x2) | (x > x1) | (y > y1) | (y < y2) Outcodes
- 16. Cohen-Sutherland Algorithm Computing the outcodes for a point is trivial Just use comparison Trivial rejection is performed using the logical and of the two endpoints A line segment is rejected if the result of and operation > 0. Why?
- 17. Using Outcodes Consider the 5 cases below AB: outcode(A) = outcode(B) = 0 Accept line segment
- 18. Using Outcodes CD: outcode (C) = 0, outcode(D) ≠ 0 Compute intersection Location of 1 in outcode(D) determines which edge to intersect with Note if there were a segment from A to a point in a region with 2 ones in outcode, we might have to do two interesections
- 19. Using Outcodes EF: outcode(E) logically ANDed with outcode(F) (bitwise) ≠ 0 Both outcodes have a 1 bit in the same place Line segment is outside of corresponding side of clipping window reject
- 20. Using Outcodes GH and IJ: same outcodes, neither zero but logical AND yields zero Shorten line segment by intersecting with one of sides of window Compute outcode of intersection (new endpoint of shortened line segment) Re-execute algorithm
- 21. Cohen-Sutherland Algorithm Now we can efficiently reject lines completely to the left, right, top, or bottom of the rectangle. If the line cannot be trivially rejected (what cases?), the line is split in half at a clip line. Not that about one half of the line can be rejected trivially This method is efficient for large or small windows.
- 22. Cohen-Sutherland Algorithm clip (int Ax, int Ay, int Bx, int By) { int cA = code(Ax, Ay); int cB = code(Bx, By); while (cA | cB) { // not completely inside the window if(cA & cB) return; // rejected if(cA) { update Ax, Ay to the clip line depending on which outer region the point is in cA = code(Ax, Ay); } else { update Bx, By to the clip line depending on which outer region the point is in cB = code(Bx, By); } } drawLine(Ax, Ay, Bx, By); }
- 23. 3D Line-clip Algorithm • Half-space now lies on one side of a plane. • Implicit formula for plane in 3D is same as that for line in 2D. • Parametric formula for line to be clipped is unchanged. – Use 6-bit outcodes – When needed, clip line segment against planes
- 24. Polygon Clipping Not as simple as line segment clipping Clipping a line segment yields at most one line segment Clipping a polygon can yield multiple polygons However, clipping a convex polygon can yield at most one other polygon
- 25. Tessellation and Convexity One strategy is to replace nonconvex (concave) polygons with a set of triangular polygons (a tessellation) Also makes fill easier Tessellation code in GLU library
- 26. Pipeline Clipping of Polygons Three dimensions: add front and back clippers
- 27. Polygon Clipping (Sutherland-Hodgman): Window must be a convex polygon Polygon to be clipped can be convex or not Approach: Polygon to be clipped is given as a sequence of vertices Each polygon edge is a pair Don't forget wrap around; is also an edge Process all polygon edges in succession against a window edge Polygon in - polygon out Repeat on resulting polygon with next sequential window edge Sutherland-Hodgman algorithm deals with concave polygons efficiently Built upon efficient line segment clipping Sutherland-Hodgman Polygon Clipping Algorithm
- 28. Four Cases There are four cases to consider Some notations: s = vj is the polygon edge starting vertex p = vj+1 is the polygon edge ending vertex i is a polygon-edge/window-edge intersection point wj is the next polygon vertex to be output
- 29. Case 1 Polygon edge is entirely inside the window edge •p is next vertex of resulting polygon
- 30. Case 2 Polygon edge crosses window edge going out • Intersection point i is next vertex of resulting polygon •
- 31. Case 3 Polygon edge is entirely outside the window edge •No output
- 32. Case 4 Polygon edge crosses window edge going in • Intersection point i and p are next two vertices of resulting polygon •
- 33. Summary of cases
- 34. An Example with a non-convex polygon
- 35. Summary of Clipping Point Clipping Simple: in or out Line Clipping: Based on point clipping Compute the line segment inside the window Need to compute intersections between the line segment and window edges. Consider trivial cases first to reduce computation. outcodes algorithm Polygon Clipping: More complex. A convex polygon may be clipped into multiple polygons. Based on line clipping Clip the polygon against window edges in a sequence

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