1 Quad-Edge Construction Operators

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1 CS48: Computer Grphics Hndout # Geometric Modeling Originl Hndout #5 Stnford University Tuesdy, 8 December 99 Originl Lecture #5: 9 November 99 Topics: Mnipultions with Qud-Edge Dt Structures Scribe: Mike Lenz nd Michel Goldwsser In the previous lecture we introduced the qud-edge dt structure, in which ech edge of surfce is represented by n edge record. The record contins pointers to the four neighboring edges e Onext, e Lnext, e Dnext, nd e Rnext. Vertices nd fces re represented by circulr, linked lists of edges. Qud-Edge Construction Opertors We introduce two opertors for creting edge records nd combining them to form representtions of complex shpes. Both opertors hve the property tht they lwys return vlid qud-edge structures when given vlid inputs. A vlid qud-edge structure is one in which the xiom e Rot Onext Rot Onext = e holds for ech edge. The MkeEdge opertor cretes n edge record; it tkes no rguments. When new edge is creted, ech of its four pointers must point to some edge, for the structure to be vlid; since there is no other edge vilble, they must in fct point bck t the newly-creted edge. Figure shows the two possible rrngements of pointers tht rise in vlid qud-edge dt structures with single edge. The subdivision tht ech rrngement represents is shown in Figure. In the left-hnd pictures, we hve two vertices connected by priml edge, nd single fce. In the right-hnd pictures, single vertex is connected to itself by priml edge tht is loop, seprting the plne into two fces. Ech digrm is the dul of the other. By convention, the first digrm represents the result of cll to MkeEdge: topologicl sphere contining two points with n edge between them. The second opertor, Splice, is used to connect qud-edge structures together, to split them prt, nd to rerrnge them in other wys. Remember tht vertices nd fces re represented by circulr lists which must be mintined for the structure to be vlid. A group of circulr lists hs the property tht, if you swp the outgoing pointers of two nodes, the result is gin group of circulr lists. In prticulr, performing Swp[,b] on nodes nd b gives the following result. If the nodes were in the sme list, tht list is split into two circulr lists, one contining nd the other contining b. If the nodes were in different lists, those two lists re combined into single circulr list. In either cse, second swp restores the originl structure. The requirement for pplying Splice[,b] is tht edges nd b must either both be priml, or both be dul. The lgorithm for cll to Splice is s follows:. α := Onext Rot. β := bonext Rot

2 CS48: Hndout # Dnext Dnext Lnext Lnext Onext Rnext Rnext Onext Figure : Two possible results of cll to MkeEdge Dul Edge Priml Edge b Priml Edge Dul Edge Figure : Geometric interprettions of ech edge structure. Swp[Onext,bOnext] 4. Swp[α Onext,β Onext] The first exchnge swps the next edges in the origin rings of nd b. The second swp hs similr effect on the edges left-fce rings. To see wht this does, consider the four possible cses. Cse. Edges nd b shre the sme origin but hve different left fces. After splicing, the origin is split into two vertices, one now the origin of nd the other the origin of b. The edges two left fces, on the other hnd, re merged into single fce. See Figure. Cse. Edges nd b shre the sme left fce but hve different origins. The two origins re merged into single vertex, but the fce is split into two seprte fces. This is lso shown in Figure, but reding from right to left.

3 CS48: Hndout # lph = F u G 7 bet 6 Splice[,5] lph u F=G u = 7 bet 4 5=b 4 5=b 6 Figure : Effects of Splice[,b] on single origin ring Cse. Edges nd b re in different origin rings nd different left-fce rings. After splicing, they shre the sme origin s well s the sme left fce. See Figure 4, in which the shpe on the bottom is something like brbell or telephone receiver. The topologicl effect of this type of splice depends upon whether the edges nd b strted out in the sme connected component or not. If they strted out in seprte components, doing this splice pstes those two components together into single component. If they strted out in the sme component, doing this splice dds hndle to tht component, hence incresing the number of holes tht component hs. For exmple, if nd b both strted out belonging to the sme sphere-like component, tht component will be torus-like fter the splice. Cse 4. Edges nd b shre the sme origin nd the sme left fce. After splicing, both their origins nd their left fces re distinct. This is lso shown in Figure 4, but reding from bottom to top. Topologiclly, this type of splice cn either split one connected component into two or else reduce by one the number of hndles tht some component hs. Topology Review The Euler Chrcteristic, χ, of surfce is the number of vertices, minus the number of edges, plus the number of fces. It is lso twice the difference between the number of connected components nd the totl genus of the surfce. χ = V E + F = (c g) Consider the result of cll to MkeEdge, topologicl sphere with two vertices, one edge, nd one fce. Thus χ is, which is consistent with connected component of genus 0. Ech

4 4 CS48: Hndout # F G u 4= 5 v 7=b 6 Splice[4,7] u=v F=G 7=b 4= 5 6 Figure 4: Effects of Splice[,b] on seprte rings cll to MkeEdge increses the totl Euler Chrcteristic by, in effect dding connected component ( new sphere) to the surfce. In the cse of torus, we my hve pir of edges from single vertex tht still prtition the surfce into only one fce. In this cse χ is 0, since the genus is, indicting surfce contining one hole or hndle. Note tht orientble surfces hve even, but possibly negtive, vlues of χ; non-orientble surfces hve odd vlues of χ. For cll to Splice, cse increments the number V of vertices nd decrements the number F of fces; cse does the opposite. In either cse, the Euler Chrcteristic is unchnged. In cse, both V nd F re decremented, so χ is decresed by. There re two wys this cn hppen: either two connected components re combined into one, or the genus is incresed by one by dding hndle to the surfce. In cse 4, both V nd F re incremented, so the Euler Chrcteristic is incresed by. There re two wys tht this cn hppen lso: either

5 CS48: Hndout # 5 c b d c b Figure 5: Two simple polygons with directed edges one connected component is split into two, or the genus is decresed by one by removing one hndle. Polygons in the Plne We turn to the problem of finding the interior of simple (not self-intersecting) polygon in the plne. For ech directed edge e, let us designte the hlf-plne on its left side by e. Consider the first polygon in Figure 5, mde up of three convex corners (tht is, t ech corner, the polygon mkes counterclockwise turn). Its interior is bc, where multipliction represents logicl nd, tht is, intersection. The second polygon hs concve (clockwise) corner, nd its interior is (b + c)d, where ddition stnds for logicl or, tht is, union. Suppose we wnt to find generl formul for the interior of simple polygon. The criteri re tht we use ech literl only once, the vrious sub-expressions being combined by either ddition or multipliction. The tricky prt is determining how to prenthesize the resulting expression. To do this, we recursively build formul by splitting the boundry of the polygon into smller nd smller polygonl chins. When combining the formuls for two smller chins to get formul for lrger chin, we use either nd or or s the opertor, depending upon whether the joint between the two subchins is convex or concve. For more bout this, see Hndout. Homework 4 dels with different formul for the interior of simple polygon. Tht formul is more complicted in tht it uses complement, intersection, nd exclusive-or s its opertors, insted of the two opertors intersection nd union. But it is simpler in tht there re no tricky issues bout prentheses. 4 How does the surfce give you solid? In discussing the surfced-bsed txonomy of solid-modeling, we hve left out n importnt question. Given representtion of surfce, how do we decide wht points re in the solid

6 6 CS48: Hndout # - Figure 6: Winding numbers represented by the surfce. We hve n intuitive notion of the solid being the interior of the surfce, nd this notion is formlized in two-dimensionl topology by the Jordn Curve Theorem, which sttes tht every closed boundry divides the spce into pieces, the unbounded exterior, the boundry itself, nd the bounded interior. It is this interior, tht we wnt to use s the solid defined by the boundry. For simplicity, we consider this problem in lower dimension. The concepts we discuss re esily generlized. Thus the question is, given closed curve, C, in the plne, nd point p, how do we know if the point is inside the curve? We introduce the concept of the winding number of point p with respect to curve C, denoted by ω(p,c). An intuitive notion of winding number is s follows: A viewer stnding t point p, wtches cr trce the curve from some rbitrry strting point until it reches tht point gin. The winding number of p with respect to the curve is the number of full revolutions the viewer hs mde t the end of the process. For instnce, let C be circle. If you re t point inside the circle, nd wtch the cr round, you will end up mking one full turn, nd so your winding number will be or depending on the orienttion of the circle. However if you re t point outside of the circle, the winding number is 0. You might turn to the right or left s you wtch, but t the end of the process, you hve not mde ny full rottions from when you strted. Figure 6 shows few curves, nd the winding number of the regions creted by the curves. Although we hve defined this notion of winding number, we still hve not nswered the question of how to define the interior of the curve. There re two seprte conventions for this. (p is inside C)if(ω(p,C) 0) (p is inside C)if(ω(p,C) is odd) Notice tht in the first nd third curves of Figure 6, these conventions gree completely, however in the second curve, the region with winding number is treted differently. The clssifiction of this region is determined by the convention chosen. Now tht we hve use for the winding number, given point nd curve, how do we efficiently compute the vlue. One school of thought is to divide up the curve into smll

7 CS48: Hndout # 7 w(p,c) = + +/ Figure 7: Ry Shooting to clculte winding numbers pieces, nd then for ech piece, ctully clculte the difference in ngles between p nd the two endpoints. The winding number is clculted by summing these ngles for ll the pieces. (To use this method, the pieces must be smll enough so tht no single piece winds completely round the point) A second, nd more populr, method of computing the winding number is by shooting out hlf-infinite ry from the point, nd then crefully counting the crossings between tht ry nd the curve. By counting the crossings in n oriented wy, the result is the winding number. Whenever the ry crosses the curve, we dd or subtrct one from the vlue, with the sign bsed on whether we cross the curve from left-to-right, or from right-to-left. As with most geometric lgorithms, degenercies cuse some concern. We must be creful in cses where our ry crosses curve t the intersection of two segments. If our ry hits the endpoint between two segments, does the ry intersect both segments? Does it intersect either segment? Fortuntely, there is solution to this problem. (Unlike most geometric lgorithms). If we shoot horizontl ry out from one side of the point, then we ignore ll of the horizontl segments in the curve. For ll of the other segments, we will sy tht every segment contins only the higher endpoint. (This notion of higher cn esily be fixed reltive to the horizontl ry) By using this method, we correctly compute the winding number, despite ny degenercies cused by our ry pssing through endpoints. Figure 7 gives n exmple of using ry shooting to compute the winding number of point.