Representations for Lines and Curves

Transcription

1 Computer Graphics Drawing algorithms for two-dimensional graphics primitives Ref: Computer Graphics, 2nd Edition, with C, by Hearn and Baker, Prentice-hall, Inc., Representations for Lines and Curves A preliminary step to drawing lines and curves is choosing a suitable representation for them. There are three possible choices which are potentially useful. 1. Explicit: y = f(x) Line: Circle: 2

2 Representations for Lines and Curves (cont.) 2. Parametric: Line: x = f(t), y = f(t) Circle: 3 Representations for Lines and Curves (cont.) 3. Implicit: f(x,y) = 0 Line: Circle: 4

3 Optimal Line Drawing Drawing lines on a raster grid implicitly involves approximation. The general process is called rasterization or scan-conversion conversion. What is the best way to draw a line from the pixel (x 1,y 1 ) to (x 2,y 2 )? Such a line should ideally have the following properties. straight pass through endpoints smooth independent of endpoint order uniform brightness brightness independent of slope 5 Scan Conversion: Lines ( DDA Algorithm ) DDA (Digital( Differential Analyzer) Algorithm Line equation: y = mx + b m = (y(2) - y(1)) / (x(2) - x(1)) b = y(1) - m x(1) Differential line equation: dy = m dx,, i.e., dy = y(k+1) - y(k) = m if dx=1 dx = x(k+1) - x(k) = 1/m if dy=1 If slope is less than 1, increment x by 1, and y(k+1) = y(k) + m If slope is greater than 1, increment y by 1, and x(k+1) = x(k) + 1/m Problems: round off error slow 6

4 Implementation of DDA Algorithm A Straightforward Implementation void DrawLine(int x1, int y1, int x2, int y2) { float y; int x; for (x = x1; x< = x2; x++) { y = y1 + (x - x1) * (y2 - y1)/(x2 - x1); SetPixel(hdc, x, Round(y) ); // end for x // end of DrawLine A Better Implementation DrawLine(int int x1, int y1, int x2, int y2) { float m,y; int dx, dy,, x; dx = x2 - x1; dy = y2 - y1; m = dy / dx; y = y ; for (x=x1; x<=x2; x++) { SetPixel(x, Floor(y) ); y = y + m; 7 Scan Conversion: Lines (Bresenham Algorithm) Bresenham s s Line Algorithm, 1965 (Midpoint Line Algorithm) Basic Concepts: Line equation 1: F=ax+ x+by+ y+c=0 (1) Line equation 2: if a point u(x,y) is on a line, F = u v = (x-x1 x1)dy - (y-y1 y1)dx = (dy)x x + (-dx)y y +(y1dx+ (y1dx-x1dy) x1dy) = 0 (2) Comparing Eqs.. (1) and (2) yields a = dy, b = -dx, c = const. Deviation d d =F -F= F= (ax +by +by +c) +c)-(ax+by+c) = a x + b y = dy x - dx y We also note that F > 0 if point (x,y) is on the right-hand hand side or below the line F = 0 if point (x,y) is on the line 8 F < 0 if point (x,y) is on the left-hand side or above the line v u(x,y)

5 Detail of the Midpoint Line Algorithm The midpoint algorithm is better than the DDA algorithm in that it uses only integer calculations. It treats line drawing as a sequence of decisions. For those lines with slope m<1 If a pixel is to be drawn, the next pixel, E or NE, can be chosen by computing F(x p +1, y p +0.5) and testing its sign Define: d=f(x p +1, y p +0.5) If d 0, midpoint is on the lhs of line, choose E If d > 0, midpoint is on the rhs of line, choose NE Case 1: if E is chosen, x=1, y=0 (d( 0) 9 d d = dy x - dx y d new = d old + d 即函數 d (=F) 實際上只須求一次 Detail of the Midpoint Line Algorithm (continued) Case 2: if NE is chosen, x x = y y = 1 d d = dy x - dx y For the first pixel at (xo,yo,yo), the first midpoint is at (x( o +1, y o +0.5). Thus, d=f(x,y) = (x-x1 x1)dy - (y-y1 y1)dx Note: the value of dx/2 may not be an integer Since we would like an integer-only algorithm, we can multiply the decision variable D by two as follows 10

6 11 Implementation of Bresenham s Line Algorithm void DrawLine(int x1, int y1, int x2, int y2, int color) { int dx = x2 - x1; int dy = y2 - y1; int d = 2 * dy dx; int x = x1; int y = y1; for (i = 0; i < dx ; i++) { SetPixel(x, y, color); If (d > 0) { y = y + 1; d = d -2* dx; // end if d > 0 x = x + 1; d = d + 2 * dy; //note: d = d 2 * dx + 2 * dy if d > 0 // end for I // end of DrawLine ***Only suitable for slope m 1*** d start =2 =2dy - dx d d = 2dy - 2dx ( if d > 0, NE is chosen ) d d = 2dy (if d 0, E is chosen) d new =d old old + d Polygons Different types of polygons: Triangles are often particularly nice to work with because they are always planar and simple convex. 12

7 Polygon Representations A variety of polygon representations can be used, one of the most common ones being an ordered list of references to a vertex list. This avoids redundant storage and redundant computations. We'll also be associating a variety of other information with vertices, such as normals,, colors, and texture coordinates. 13 A Simple Example: Cube 14

8 Scan Conversion: Polygon The job of scan conversion is to shade pixels lying within a closed polygon, and to do so efficiently. The fill color will in general depend on the lighting, texture,, and visibility of the polygon being scan-converted. converted. These will be ignored for the time being. It is assumed that the polygon is closed and has ordered edges. 15 Scan Conversion Algorithm for Polygons The scan conversion algorithm then works as follows: intersect each scan line with all edges sort intersections in x calculate parity of intersections to determine in/out fill the 'in' pixels Special cases: horizontal edges can be excluded vertices lying on scan lines! change in sign of slope: count twice! no change in sign: shorten edge by one scan line 16

9 The Coherence Between Adjacent Scan Lines Taking advantage of coherence ( 連貫性 ) between adjacent scan lines can eliminate many intersection tests. Edges that intersect scan line y are likely to intersect y+1 x changes predictably from scan line y to y+1 The following two data structures can be used by an efficient scan an- conversion algorithm: Edge Table and Active Edge List (AEL). 17 Procedure for Establishing An Edge Table Traverse edges Eliminate horizontal edges, e.g., edge e7 If not local extremum (minimum or maximum),, shorten upper vertex, e.g., upper vertex of edges e2 and e5 ( 比較三個連續點加以判斷 ) Add edge to linked-list list for the scan line according to the lower vertex (e.g., e2-e3, e3, e4-e5, e5, e1, e6),, and store the following data: y_upper: the last scan line to consider for an edge, e.g. 8 for edge e1 x_lower: the starting x coordinate for an edge, e.g. 9 for edge e1 1/m (= x/ x/ y= y= x) x): : will be used for incrementing x (compute before shortening) 18 y_upper, x_lower, 1/m

10 Active Edge List (AEL) & Scan Conversion Algorithm The AEL is a linked list of active edges on the current scan line, y. Each active edge has the following information: y_upper: the last scan line to consider x: edge's intersection with current y 1/m: for incrementing x The active edges are kept sorted by x y_upper, x_lower, 1/m Scan Conversion Algorithm Do for each scan line { Add edges in edge table to AEL; If (AEL!= NULL) { Sort AEL by x; Fill pixels between edge pairs; Delete finished edges; Update each edge's x; y_upper, x_lower, 1/m 0 wcpt2 pt[]={ 0.6, 0.8, 0.9, 0.5, 0.9, 0.1, 0.5, 0.5, 0.1, 0.2, 0.2, 0.7, 0.4, 0.8, ; 1

11 Assignment pts[0].x = 0.25*w; pts[1].x = 0.75*w; pts[2].x = 0.75 *w; pts[0].y = 0.25*h; pts[1].y = 0.75*h; pts[2].y = 0.25*h; pts[3].x = 0.25*w; pts[3].y = 0.75* h; 21 Struct A struct for defining a point in device coordinates typedef struct { int x; int y; dcpt; A struct for defining an edge typedef struct tedge { int yupper; float xintersect; float dxperscan; // dx/dy = 1/m struct tedge* next; Edge; data Edge next dcpt Int pts[7]; winwidth, winheight; // width and height of a window 22

12 Main Function: ScanFill void scanfill (int nop, dcpt * pts) {//cnt{ /cnt=7 Edge* active; int i, scan; Edge** edges = (Edge **) malloc ( winheight * sizeof (Edge*) ); /* pointer 'edges' will point to a pointer 'edges[i]', I=0,,, h */ 0 1 for (i=0; i<winheighti <winheight; ; i++) { edges[i] = (Edge *) malloc (sizeof (Edge)); edges[i]->next = NULL; buildedgelist (cnt,, pts, edges); active = (Edge *) malloc ( sizeof (Edge) ); active->next = NULL; for (scan=0; scan<winheight <winheight; ; scan++) { buildactivelist (scan, active, edges); if (active->next) { fillscan (scan, active); updateactivelist (scan, active); resortactivelist (active); 23 edges[7] edges Edge y_upper, x_lower, 1/m, next Build Edge List void buildedgelist (int nop, dcpt * pts, Edge * edges[]) { Edge* edge; Prev v1 dcpt v1, v2; v1 v2(i=0) int i, yprev = pts[cnt - 2].y; 6 0 Prev v1.x = pts[cnt [cnt-1].x; 5 v1.y = pts[cnt [cnt-1].y; 3 v2(i=1) for (i=0; i<cnti <cnt; ; i++) { 1 v1 v2 = pts[i];//check edge 6_0 if (v1.y!= v2.y) {//nonhorizontal{ /nonhorizontal line 4 2 edge = (Edge*) malloc (sizeof (Edge)); if (v1.y < v2.y) {//up{ //up-going edge v2(i=2) makeedgerec (v1, v2, ynext (i, nop,, pts),, edge, edges); else {//down{ //down-going edge Prev makeedgerec (v2, v1, yprev,, edge, edges); next v2(i=6) Next 6 yprev = v1.y; v2(i=5) 5 0 v1 = v2; 3 1 //end for v2(i=3) v1 4 2 Next 24 v1

13 Make Edge Record void makeedgerec(dcpt lower, dcpt upper, int ycomp,, Edge * edge, Edge * edges[]) { edge->dxperscan = (float) (upper.x - lower.x) / (upper.y - lower.y); edge->xintersect = lower.x; if (upper.y < ycomp) ) { edge->yupper = upper.y - 1;//shorten upper vertex else { edge->yupper = upper.y; upper comp upper insertedge (edges[lower.y], edge); //insert an edge into the list for the scan line passing through P0 Edge edges[7] comp lower lower 25 edges y_upper, x_lower, 1/m Inserts Edge Into List void insertedge (Edge* list, Edge * edge) { Edge* p; Edge* q = list; //insert an edge and sort p = q->next; q while (p!= NULL) { if (edge->xintersect < p->xintersectp >xintersect) ) { p = NULL; q=list else { q = p; p = p->next; p //end while edge->next = q->next; q q->next = edge; upper 1. p=q->next data next data next comp lower 3. q->next=edge 2. edge->next=q->next data next 26 y_upper, x_lower, 1/m

14 Insert a Node (General) list data next data next q=list 1. p=q->next data next data next // // 記住記住 q 原來所指之節點 p = q->next; // // 使欲插入之節點 edge edge 指向 q 原來所指之節點 edge->next = q->next; // // 使 q 指向插入之節點 q->next = edge; edge; 3. q->next=edgeq data next 2. edge->next=q >next=q->next edge 27 Insert a Node & Sort q=list data next data next data list next data p=q->next edge next q=p data next p=p->next data next data next p = q->next; q while (p!= NULL) { if (edge->xintersect< < p->xintersect) ) { p = NULL;//force to leave the loop else { q = p; p = p->next;//shift edge->next = q->next; q q->next = edge; 28 data next edge

15 Fill Polygon for (scan=0; scan<winheight <winheight; ; scan++) { buildactivelist (scan, active, edges); if (active->next) { fillscan (scan, active); updateactivelist (scan, active); resortactivelist (active); edges[7] Edge active edges y_upper, x_lower, 1/m 29 Build Active List void buildactivelist (int scan, Edge * active, Edge * edges[]) { Edge * p, * q; p = edges[scan]->next; while (p) { q = p->next; p insertedge (active, p);//add edge p into active list p = q; Edge edges[7] 30 edges y_upper, x_lower, 1/m

16 Fill A Scan Line void fillscan (int scan, Edge * active) { Edge * p1, * p2; int i; p1 = active->next; while (p1) { p2 = p1->next; for (i=p1->xintersect >xintersect; ; i<p2->xintersect >xintersect; ; i++) Draw_Point ((int( (int)) i, scan); p1 = p2->next; edges[7] Edge 31 edges y_upper, x_lower, 1/m 32 Update Active List Delete completed edges. Update 'xintersect' ' field for others void updateactivelist (int scan, Edge * active) { Edge * q = active, * p = active->next; while (p) if (scan >= p->yupperp >yupper) ) { p = p->next; p deleteafter (q); else { p->xintersect += p->dxperscanp >dxperscan; q = p; p = p->next; p //end if scan void deleteafter (Edge * q) { Edge* p = q->next; q->next = p->next; free (p); edges[7] edges q 2. q->next=p->next data next data next data next 1. p=q->next Delete a node Edge y_upper, x_lower, 1/m

17 Resort Active List void resortactivelist (Edge * active) { Edge * q, * p = active->next; active->next = NULL; while (p) { q = p->next; p insertedge (active, p); p = q; edges[7] Edge 33 edges y_upper, x_lower, 1/m Polygon Clipping The Sutherland-Hodgman algorithm can be used to clip any polygon (convex( or concave) ) against any convex clipping polygon. The algorithm clips against one edge at a time, producing a new vertex list each time. The following figure assumes the most common case, a rectangular clipping window. 34 Polyline Clipping

18 Pseudo Codes for the Algorithm The algorithm can be summarized as follows: for each side of clipping window for each edge of polygon output points based upon the following table 35 Example 36

19 Using Outcodes for Trivial Accept and Reject In many cases, two trivial tests can be used to quickly determine whether a polygon is completely inside or outside the viewing window. This then allows us to skip the above clipping procedure. The trivial tests require first computing outcodes. A vertex outcode consists of four bits: TBRL, where: T is set if y > top, B is set if y < bottom, R is set if x > right, and L is set if x < left. T L Trivial accept: all vertices are inside (all outcodes are 0000) Trivial reject: all vertices are outside with respect to any given side (bitwise AND is not 0000) 37 Main Procedures & Varibles #define N_EDGE 4 typedef enum { Left, Right, Bottom, Top Edge;// 列舉, Left=0, Right=1, #define N_PTS 6 wcpt2 pts[n_pts] = { 60, 20, 375, 80, 280, 280, 100, 280, 100, 100, 60, ; dcpt winmin = { 50, 50 ; dcpt winmax = { 350, 250 ; (100,280) (280,280) (350,250) int i, npts; wcpt2 clippedpts[256]; (375,80) void render(void) { (50,50) (60,20) /* Clip the polygon against the window, returning 'clippedpts' ' */ npts = clippolygon (winmin, winmax, N_PTS, pts, clippedpts); // end function 38

20 Function: inside Input a point p and an edge b to determine whether it is inside or not Ex: inside (p1, b, wmin, wmax) int inside (wcpt2 p,, Edge b, dcpt wmin, dcpt wmax) ) { switch (b)( ) { case Left: (wmax.x, wmax.y) if (p.x( < wmin.x) return (FALSE); break; case Right: if (p.x( > wmax.x) return (FALSE); break; case Bottom: if (p.y( < wmin.y) return (FALSE); (wmin.x, wmin.y) break; case Top: if (p.y( > wmax.y) return (FALSE); break; // end switch return (TRUE); 39// end function Function: cross Given: two points of a line segment and an clipping edge Return: whether the line cross an edge Ex: cross (p, s[b], b, wmin, wmax) int cross (wcpt2 p1,, wcpt2 p2,, Edge b, dcpt wmin, dcpt wmax) ) { if (inside( (p1, b, wmin, wmax) ) == inside (p2, b, wmin, wmax)) return (FALSE); //p1 and p2 are both inside or outside else return (TRUE); //p1 and p2 are on the different side of edge b // end function 40

21 Function: intersect EX: ipt = intersect (p, s[b], b, wmin, wmax); wcpt2 intersect (wcpt2 p1, wcpt2 p2, Edge b, dcpt wmin, dcpt wmax) ) { wcpt2 ipt;//intersection point, 交點 float m;//slope, 斜率 if (p1.x!= p2.x) m = (p1.y - p2.y) / (p1.x - p2.x); // 線段不平行 y 軸時才求斜率 switch (b) { case Left: ipt.x = wmin.x; ipt.y = p2.y + (wmin.x - p2.x) * m; break; case Right: ipt.x = wmax.x; ipt.y = p2.y + (wmax.x - p2.x) * m; break; case Bottom: ipt.y = wmin.y; if (p1.x!= p2.x) ipt.x = p2.x + (wmin.y - p2.y) / m; else ipt.x = p2.x; break; case Top: ipt.y = wmax.y; if (p1.x!= p2.x) ipt.x = p2.x + (wmax.y - p2.y) / m; else ipt.x = p2.x; break; (wmax.x, wmax.y) P, p1 return (ipt); );// 傳回交點之座標 // end function S, p2 41 (wmin.x, wmin.y) Function: clippolygon int clippolygon (dcpt wmin, dcpt wmax, int n,, wcpt2* pin,, wcpt2* pout) ) { // 'first' first' ' holds pointer to first point processed against a clip edge // 's'' ' holds most recent point processed against an edge wcpt2* first[n_edge] = { 0, 0, 0, 0 ; wcpt2 s[n_edge]; int i, cnt = 0; for (i=0; i<n; ; i++) clippoint (pin[i], Left, wmin, wmax, pout, &cnt, first, s); closeclip (wmin, wmax, pout, &cnt, first, s); return (cnt); // end function (100,280) (280,280) (350,250) 42 (50,50) (60,20) (375,80)

22 Function: clippoint Ex: for (i=0; i<n; ; i++) clippoint (pin[i], Left, wmin, wmax, pout, &cnt, first, s); void clippoint (wcpt2 p, Edge b, dcpt wmin, dcpt wmax,, wcpt2* pout, int* cnt,, wcpt2* first[], wcpt2* s) { //may replace wcpt2* first[] by wcpt2** first wcpt2 ipt; Enter with input vertex P if (!first[b]) { first[b] ] = &p; //first[b] is also a pointer, so first is a pointer s pointer else { No Yes First Point? if (cross( (p, s[b], b, wmin, wmax)) { ipt = intersect (p, s[b], b, wmin, wmax); Does line SP No if (b < Top) P"S cross a plane? clippoint (ipt, b+1, wmin, wmax, pout, cnt,, first, s); else Yes { pout[*cnt *cnt]] = ipt; ; (*cnt( *cnt)++; Compute intersection, I, // end if (cross ) of SP and the plane P"S // end if (!first ) P"F"S s[b] ] = p; if (inside( (p, b, wmin, wmax)) { if (b < Top) clippoint (p, b+1, wmin, wmax, pout, cnt,, first, s); else { pout[*cnt *cnt]] = p; (*cnt( *cnt)++; // end if (inside ) // end function 43 Debug Data (100,280) (280,280) (350,250) I (50,50) F,S(60,20) Output I P(375,80) No Is S on visible side of plane? Yes Output S Exit Function: closeclip void closeclip (dcpt wmin, dcpt wmax, wcpt2* pout, int* cnt, wcpt2* first[], wcpt2* s) { wcpt2 i; Edge b; No Close polygon entry Was there any output? for (b = Left; b <= Top; b++) { if (cross (s[b], *first[b], b, wmin, wmax)) { i = intersect (s[b], *first[b], b, wmin, wmax); No if (b < Top) clippoint (i, b+1, wmin, wmax, Reset pout, cnt,, first, s); first flag else { pout[*cnt *cnt]] = i; (*cnt( *cnt)++; Close // end if (b<top) next stage // end if cross() // end for b // end function Exit Yes Does link SF cross plane? Compute intersection, I, of edge SF and Yes the the plane Output I (100,280) (280,280) (350,250) (375,80) 44 (50,50) (60,20)

23 Debug Start!!! i=0, call clippoint... clipping edge=0 first=[ 60.0, 20.0], s=[ 60.0, 20.0] clipping edge=1 first=[ 60.0, 20.0], s=[ 60.0, 20.0] clipping edge=2 first=[ 60.0, 20.0], s=[ 60.0, 20.0] i=1, call clippoint... clipping edge=0 first=[375.0, 80.0], s=[375.0, 80.0] clipping edge=1 clipping edge=2 clipping edge=3 first=[217.5, 50.0], s=[217.5, 50.0] first=[350.0, 75.2], s=[350.0, 75.2] clipping edge=3 first=[350.0, 75.2], s=[350.0, 75.2] first=[375.0, 80.0], s=[375.0, 80.0] i=2, call clippoint... clipping edge=0 first=[280.0, 280.0], s=[280.0, 280.0] clipping edge=1 clipping edge=2 first=[350.0, 132.6], s=[350.0, 132.6] clipping edge=3 first=[350.0, 132.6], s=[350.0, 132.6] first=[280.0, 280.0], s=[280.0, 280.0] clipping edge=2 first=[280.0, 280.0], s=[280.0, 280.0] clipping edge=3 first=[280.0, 280.0], s=[280.0, 280.0] i=3, call clippoint... clipping edge=0 first=[100.0, 280.0], s=[100.0, 280.0] clipping edge=1 first=[100.0, 280.0], s=[100.0, 280.0] clipping edge=2 first=[100.0, 280.0], s=[100.0, 280.0] clipping edge=3 first=[100.0, 280.0], s=[100.0, 280.0] i=4, call clippoint... clipping edge=0 first=[100.0, 100.0], s=[100.0, 100.0] clipping edge=1 first=[100.0, 100.0], s=[100.0, 100.0] clipping edge=2 first=[100.0, 100.0], s=[100.0, 100.0] clipping edge=3 first=[100.0, 100.0], s=[100.0, 100.0] i=5, call clippoint... clipping edge=0 first=[ 60.0, 20.0], s=[ 60.0, 20.0] clipping edge=1 first=[ 60.0, 20.0], s=[ 60.0, 20.0] clipping edge=2 clipping edge=3 first=[ 75.0, 50.0], s=[ 75.0, 50.0] first=[ 60.0, 20.0], s=[ 60.0, 20.0] clipping edge=1 first=[ 0.0, 0.0], s=[ 50.0, 20.0] clipping edge=2 first=[ 0.0, 0.0], s=[ 50.0, 20.0] (50,50) (100,280) (60,20) (280,280) (350,250) (375,80) 45