Curve clipping involves using polygon clipping to test if a curved object's bounding rectangle overlaps a clipping window. If there is no overlap, the object is discarded. If there is overlap, the simultaneous curve and boundary equations are solved to find intersection points. Special cases like circles are considered, such as discarding a circle if its center is outside the clipping window plus/minus the radius. Bezier and spline curves can also be clipped by approximating them as polylines or using their convex hull properties.

1. Curve clipping

2. The bounding rectangle for a curved object can be used first to test for overlap with a rectangular clip window (we can use polygon clipping) XMAX , Y MAX XMIN , Y MAX XMIN , Y MIN XMAX , Y MIN Object

3. Case 1 If the bounding rectangle for the object is completely inside the window, we save the object. Clipping window Bounding rectangle

4. Case 2 If the rectangle is determined to be completely outside the window, we discard the object Object

5. Case 3 If the two regions overlap, we will need to solve the simultaneous line-curve equations to obtain the clipping intersection points.

8. Circle clipping cont.. If XC - R > Xright Then the circle is discarded R X right Xc -R XC

9. Circle clipping cont.. If YC - R >Ytop Then the circle is discarded Yc -R Y top Clipping window

10. Circle clipping cont.. If YC +R <Ybottom Then the circle is discarded Y bottom Yc + R

11. Circle clippingcont.. If all the four previous conditions are false then the circle is saved

12. Circle clipping cont.. Intersection conditions: With right edge: Xc+R>Xright With left edge: Xc-R<Xleft With top edge : Yc+R>Ytop With bottom edge: Yc-R<Ybottom

13. Circle clipping cont.. Getting intersection points : Example : The intersection with the right edge 1- Simply Cos α = Xright-Xc /R 2- Get α 3- y=R sin α 4- the segment from angle 0 to angle α is discarded 5- the segment from angle α to angle 360-α is considered 6- the segment from angle 360-α to angle 360 is considered First intersection angle=α α Start (angle=0) α Second intersection X right Xc

14. Other techniques Clip individual point : for point plotted curves , may consume time if number of points is great. Curves approximated to poly lines: clip individual line segments , if segment is not small enough no accurate result , if it is small more than enough , it will be time consuming for linear segments

15. Spline curve : definition A Spline Curve : Any Composite curve formed with polynomial sections satisfying specified continuity conditions at the boundary of the pieces.

16. Specifying Splines

17. Example : Third order spline In order to assure C1 continuity at two extremities, our functions must be of at least degree 3

18. 18 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Convex Hull Property Bezier curves lie in the convex hull of their control points Hence, even though we do not interpolate all the data, we cannot be too far away p1 p2 convex hull Bezier curve p3 p0

19. Bézier Clipping Problem Given polynomial p withdegree n Find all roots within an interval Algorithm Bézier representation Intersect the convex hull with t-axis Obtain a new interval

20. Bézier Clipping

21. Bézier Clipping

22. Bézier Clipping

23. Bézier Clipping

24. Bézier Clipping

25. The Approximated Roots A sequence of intervals that bound the root of p If the width of interval is smaller than the given tolerance, return the root (interval).

26. Convergence Rates A sequence of intervals that converge to the root: How fast does the sequence converge?

27. 2. Quadratic Clipping

28. Quadratic Clipping Idea Use quadratic bounds Motivation To improve the convergence rate

29. Quadratic Bounds Upper bound Lower bound

30. Quadratic Bounds But, how to compute the quadratic bounds efficiently?

31. Quadratic Bounds Approximated quadratic bounds Upper bound Best quadratic approximant Lower bound

32. Quadratic Clipping The same type of algorithm as Bézier clipping Convex hull  Quadratic bounds Find the best quadratic approximant q of p in L2 norm Compute error bound of p and q Construct quadratic functions: upper bound M, lower bound m Compute roots of M and m

33. Quadratic Clipping

34. Quadratic Clipping

35. Quadratic Clipping

36. Quadratic Clipping

37. Quadratic Clipping

38. Quadratic Clipping

39. Quadratic Clipping

40. Quadratic Clipping

41. The Approximated Root A sequence of intervals that bound the root of p If the width of interval is smaller than the given tolerance, return the root (interval).