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Sat Oct 4 23:33:09 1997 UTC (23 years, 9 months ago) by monnier
File size: 5234 byte(s)
Sat Oct 4 23:33:09 1997 UTC (23 years, 9 months ago) by monnier
File size: 5234 byte(s)
Initial revision
(* spline.sml * * COPYRIGHT (c) 1991 by AT&T Bell Laboratories See COPYRIGHT file for details. *) structure Spline : SPLINE = struct structure G = Geometry open G fun round x = if x > 0.0 then floor (x+0.5) else ~1*floor(~x+0.5) fun addSeg ([], x0,y0,x1,y1) = [PT{x = round x0, y = round y0}, PT{x = round x1, y = round y1}] | addSeg (l, x0, y0, _,_) = (PT{x = round x0, y = round y0})::l (* isFlat: * Returns true if the polygon determined by the four points * is flat enough. Flatness is measured by the maximum distance * of (x1,y1) and (x2,y2) from the line determined by (x0,y0) * and (x3,y3). In addition, check that p1, p2 are close to the * line segment. To do this, make sure they are roughly within * the circle with center (p0+p3)/2 and radius = |p3-p0|/2+flatness *) fun isFlat {x0, y0, x1, y1, x2, y2, x3, y3} = let fun sqr x = x * x val dx = x3 - x0 and dy = y3 - y0 val midx = 0.5*dx and midy = 0.5*dy and dist2 = sqr dy + sqr dx val flatness2 = sqr 1.0 * dist2 and halfd2 = 0.25*dist2 fun inFlatRange (x,y) = sqr(dy * x - dx * y) <= flatness2 andalso let val d = sqr(x-midx) + sqr(y-midy) in d<=halfd2 orelse sqr(d-halfd2)<=flatness2 end in inFlatRange (x1-x0, y1-y0) andalso inFlatRange (x2-x0, y2-y0) end (* bezier: * Recursively compute a Bezier cubic section. If the points * determine a polygon flat enough to be represented as a line * segment, the segment is added to the list. Otherwise, the * the curve is bisected and each part is recursively computed, * with the lists concatenated. * * From "The Beta2-split: A special case of the Beta-spline Curve and * Surface Representation." B. A. Barsky and A. D. DeRose. IEEE, 1985, * as adapted by Crispin Goswell for xps. *) fun bezier (arg as {x0,y0,x1,y1,x2,y2,x3,y3}, l) = if (isFlat arg) then addSeg(l, x0, y0, x3, y3) else let val mid_x = (x0 + x3) / 8.0 + 3.0 * (x1 + x2) / 8.0 val mid_y = (y0 + y3) / 8.0 + 3.0 * (y1 + y2) / 8.0 val l' = bezier ({ x0 = mid_x, y0 = mid_y, x1 = (x1+x3) / 4.0 + x2 / 2.0, y1 = (y1+y3) / 4.0 + y2 / 2.0, x2 = (x2+x3) / 2.0, y2 = (y2+y3) / 2.0, x3 = x3, y3 = y3 }, l) in bezier ({ x0 = x0, y0 = y0, x1 = (x0+x1) / 2.0, y1 = (y0+y1) / 2.0, x2 = (x0+x2) / 4.0 + x1 / 2.0, y2 = (y0+y2) / 4.0 + y1 / 2.0, x3 = mid_x, y3 = mid_y }, l') end (* curve: * Given four points [p0,p1,p2,p3], return a list of points corresponding to * to a Bezier cubic section, starting at p0, ending at p3, with p1, p2 as * control points. * *) fun curve (PT{x=x0,y=y0}, PT{x=x1,y=y1}, PT{x=x2,y=y2}, PT{x=x3,y=y3}) = bezier ({ x0 = real x0, y0 = real y0, x1 = real x1, y1 = real y1, x2 = real x2, y2 = real y2, x3 = real x3, y3 = real y3 }, []) (* doSpline: * Given four points (p0,p1,p2,p3), return a list of points corresponding to * to the B-spline curve section, accumulating the results on the argument list. * We compute the curve by determining the corresponding Bezier control points, * and then use the Bezier routines above. * *) fun doSpline (PT{x=x0,y=y0}, PT{x=x1,y=y1}, PT{x=x2,y=y2}, PT{x=x3,y=y3}, l) = let val x0 = real x0 and y0 = real y0 val x1 = real x1 and y1 = real y1 val x2 = real x2 and y2 = real y2 val x3 = real x3 and y3 = real y3 in bezier ({ x0 = (x0 + 4.0*x1 + x2)/6.0, y0 = (y0+ 4.0*y1 + y2)/6.0, x1 = (2.0*x1 + x2)/3.0, y1 = (2.0*y1 + y2)/3.0, x2 = (x1 + 2.0*x2)/3.0, y2 = (y1 + 2.0*y2)/3.0, x3 = (x1 + 4.0*x2 + x3)/6.0, y3 = (y1 + 4.0*y2 + y3)/6.0 }, l) end (* loopSpline: * Given a list of spline control points, generate the corresponding * spline. Since we accumulate on the head of the list, we assume * the calling function has reversed the list of control points. * The loop continues as long as there are 4 control points left. *) fun loopSpline (p3, p2, p1, p0::tl, l) = loopSpline (p2, p1, p0, tl, doSpline(p0, p1, p2, p3, l)) | loopSpline (_, _, _, [], l) = l (* simpleBSpline: * Compute a simple B-spline with the given control points. *) fun simpleBSpline arg = (case (rev arg) of (p3::p2::p1::tl) => loopSpline (p3,p2,p1,tl,[]) | _ => arg (* end case *)) (* bSpline: * Compute a B-spline using the given control points. * In addition, we constrain the resultant spline to connect the * first and last points by adding copies of these points. *) fun bSpline (arg as (p0::_::_::_)) = let val (pn::tl) = rev (p0::p0::arg) in loopSpline (pn, pn, pn, tl, []) end | bSpline l = l (* closedBSpline: * Compute a closed B-spline. This is done by repeating the first * three points at the end of the list. * Note that the first and last points of the result are the same. *) fun closedBSpline (arg as (p0::p1::p2::_)) = loopSpline(p2,p1,p0,rev arg,[]) | closedBSpline l = l end (* Spline *)
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