Windowing and clipping

Windowing and clipping • Window - The rectangle defining the part of the world we wish to display. • Viewport - The rectangle on the raster graphics screen (or interface window for “window” displays) defining where the image will appear. (Default is usually entire screen or interface window.) Chapter 6

Windowing & clipping • Windowing is showing on the viewport parts of the real seen that appears from the window. • Clipping is not showing on the viewport parts of the real seen outside the window boundaries. • Clipping needs to be fast, so often implemented in hardware. There are techniques for clipping primitive operations. Chapter 6

Terminology • World Coordinate System (Object Space) - Representation of an object measured in some physical units. • Screen Coordinate System (Image Space) - The space within the image is displayed • Interface Window - The visual representation of the screen coordinate system for “window” displays (coordinate system moves with interface window). • Viewing Transformation - The process of going from a window in world coordinates to a viewport in screen coordinates. Chapter 6

Window Interface Window Viewport Information outside the viewport is clipped away Windows and Viewports Chapter 6

Translate to origin Choose Window in World Coordinates Clip to size of Window Scale to size of Viewport Translate to proper position on screen (Interface Window) Viewing Transformation Chapter 6

Notes on Viewing Transformation • Panning - Moving the window about the world • Zooming - Reducing the window size • As the window increases in size, the image in the viewport decreases in size and vice versa • Beware of aspect ratio. Chapter 6

(10, 30) (50, 30) (0, 1) (0.5, 1) Viewport wanted (0, 0.5) (0.5, 0.5) (10, 5) (50, 5) X scale = 0.5/40 = 1/80 (0,0) (1, 0) 1 0 0 0 1 0.5 0 0 1 1/80 0 0 0 1/50 0 0 0 1 1 0 -10 0 1 -5 0 0 1 3) Translate to proper coordinates 2) Scale to correct size 1) Translate window to origin Viewing Transformation Example Chapter 6

ymax xmax . Q P . ymin xmin Clipping Points to a Window Suppose the window has (xmin, ymin) bottom left vertex & (xmax,ymax) as its upper right vertex, then a point (x,y) is VISIBLE IF xmin < x < xmax; ymin < y < ymax Notice P is inside and q is outside, so they q will be clipped Chapter 6

E B D A H C J G F I Clipping Lines to a Window Can we quickly recognise lines which need clipping? Chapter 6

Clipping to a Window • Looking at end-points gives us a quick classification: • Both ends visible => line visible (AB) • One end visible, other invisible => line partly visible (CD) • Both ends invisible: • If both end-points lie to same side of window edge, line is invisible (EF) • Otherwise, line may be invisible (IJ) or partially visible (GH) Chapter 6

Line Clipping Algorithms (xmax, ymax) • Brute Force Method - Solve simultaneous equations for intersections of lines with window edges is impractical. (xmin, ymin) Chapter 6

Cohen-Sutherland Algorithm • Region Checks: Trivially reject or accept lines and points. • Fast for large windows (everything is inside) and for small windows (everything is outside). • Each vertex is assigned a four-bit outcode. Chapter 6

1001 1000 1010 0001 0000 0010 0101 0100 0110 Cohen-Sutherland Line Clipping Algorithm • Each end-point is coded according to its position relative to the window • Four-bit code assigned as follows: Bit 1 Set if x < xmin Bit 2 Set if x > xmax Bit 3 Set if y < ymin Bit 4 Set if y > ymax Chapter 6

Cohen-Sutherland Line Clipping Algorithm • Notice: if • Both end-point codes 0000 => VISIBLE (trivially accepted) • Logical AND = NOT 0000 => INVISIBLE (trivially rejected) • Logical AND = 0000 => INVISIBLE or PART VISIBLE • To clip P1P2: • Check if P1P2 totally visible or invisible • If not, for each edge in turn (left/right/bottom/top): (i) Is edge crossed ? (the corresponding bit is set for ONE of the points, say P1) (ii) If so, replace P1 with intersection with edge. Chapter 6

Example P1 Clip against left, right, bottom, top boundaries in turn. P1’ P1: 1001 P2: 0100 First clip to left edge, giving P1’P2 P2 x=xmin Chapter 6

Example P1’: 1000 P2 : 0100 P1’ P1’’ No need to clip against right edge Clip against bottom gives P1’P2’ Clip against top gives P1’’P2’ P2’ P2 x=xmin Chapter 6

Calculating the Intersection • To calculate intersection of P1P2 with, say left edge: Left edge: x = xmin Line : y - y2 = m (x-x2) where m = (y2 - y1) / (x2 -x1) Thus intersection is (xmin, y*) where y* = y2 + m (xmin - x2) P2 P1 Chapter 6

Other Line Clippers • Cohen-Sutherland is efficient for quick acceptance or rejection of lines. • Less so when many lines need clipping. • Other algorithms are: • Liang-Barsky • Nicholl-Lee-Nicholl see:Hearn and Baker for details Chapter 6

Line Parametric Equation • Parametric equation of a line between twp points (x1,y1) (x2,y2) x = x1 + u (x2−x1) & y = y1 + u (y2−y1) Where u Є [0, 1]. When u = 0 the point is the first (x1, y1) & if u =1 the point is (x2, y2) Chapter 6

Line Parametric Equation • Find the range of the parameter [u1, u2] so that: xmin <= x <= xmax −> xmin <= x1+u(x2−x1) <=xmax ymin <= y <= ymax −> ymin <= y1+u(y2−y1) <= ymax 0<= u <=1 • The above inequalities allow us to find the final range of the parameter u −> [u1,u2] !! • Liang−Barsky algorithm can do it fast. Chapter 6

Liang-Barsky Line Clipping I • Let dx = x2 − x1, dy = y2 − y1, Then the inequalities: x1+u(x2−x1)>= xmin => u*(−dx) <= x1 − xmin x1+u(x2−x1)<= xmax => u*( dx) <= xmax − x1 y1+u(y2−y1)>= ymin => u*(−dy) <= y1 − ymin y1+u(y2−y1)<= ymax => u*( dy) <= ymax − y1 • We can rewrite the above equations as: u. pk <= qk where k corresponds to window boundaries 0,2,3,4 (left, right, bottom, top) p0 = −dx, q0 = x1 − xmin p1 = dx, q1 = xmax − x1 p2 = −dy, q2 = y1 − ymin p3 = dy, q3 = ymax − y1 Chapter 6

Liang-Barsky Line Clipping II • If pk <0, u >= qk/pk , so we know the line proceeds from outside to inside of the window boundary k, thus ==> update u1 !! • if pk > 0 u <= qk / pk, so we know the line proceeds from inside to outside of the window boundary k, thus ==> update u2 !! • else if pk=0 (which means dx or dy = 0) The line is parallel to the window boundary if qk <0 trivially reject the line Chapter 6

Clipping Algorithm • Notice that u1 can’t be greater than u2, so after updating u1 or u2 if u1 > u2, reject the line. Cliptest(p, q, *u1, *u2) if p < 0 r = q / p; if r > u2 return false else if r > u1 *u1 = r else if p > 0 r = q / p; if r < u1 return false else if r < u2 *u2 = r else if q < 0 return false Return true Clipline(p1, p2) u1 = 0; u2 = 1; dx=p2.x–p1.x; dy =p2.y-p1.y; If cliptest(−dx, x1 − xmin, &u1, &u2) If cliptest(dx, xmax − x1, &u1, &u2) If cliptest(−dy, y1 − ymin, &u1, &u2) If cliptest(dy, ymax − y1, &u1, &u2) if u2 < 1 p2.x = p1.x + u2*dx p2.y = p1.y + u2*dy else u1 > 0 p1.x + = u1 * dx p1.y + = u1 * dy Return updated values of p1 & p2 Chapter 6

Polygon Clipping • Basic idea: clip each polygon side - but care needed to ensure clipped polygon is closed. B C A D E Chapter 6 F

Sutherland-Hodgman Algorithm • This algorithm clips a polygon against each edge of window in turn, ALWAYS keeping the polygon CLOSED • Points pass through as in a pipeline INPUT: List of polygon vertices OUTPUT: List of polygon vertices on visible side of window edge Chapter 6

Sutherland-Hodgman Algorithm Consider a polygon side: starting vertex S, end vertex P and window edge x = xmin. What vertices are output? xmin xmin xmin xmin P P P P I I S S S S OUTPUT: - I, P I P Chapter 6

Example - Sutherland-Hodgman Algorithm Take each edge in turn start with left edge. Take each point in turn: (i) Input point and call it P - thus P = A (ii) If P is first point: - store it as P1 - output if visible (not in this particular example) - let S = P B C A D E F Input: A B C D E F Chapter 6

Example - Sutherland-Hodgman Algorithm B (iii) If P not first point, then if SP crosses window edge: - compute intersection I - output I output P if visible (iv) let S = P A’ C A D E F Output: A’ B C D E F Input: A B C D E F Chapter 6

Example - Sutherland-Hodgman Algorithm B Finally, if some points have been output, then if SP1 crosses window edge: - compute intersection I - output I A’ C A D G E F Output: A’ B C D E F G Input: A B C D E F Chapter 6

Example - after clipping to left edge B The result of clipping against the left edge A’ C D G E F Chapter 6

Example - clip against right edge B B A’ A’ C C D E’ D E’ G G E E’’ E’’ F F INPUT: A’ B C D E F G OUTPUT: A’ B C D E’ E’’ F G Chapter 6

Example - clip against bottom edge B B A’ A’ C C D D E’ E’ G G E’’’ E’’’ E’’ F’ F’ F INPUT: A’ B C D E E’ E’’ F G OUTPUT: A’ B C D E’ E’’’ F’ G Chapter 6

Example - clip against top edge B A’’ B’ A’’ B’ A’ A’ C C D D E’ E’ G G E’’’ E’’’ F’ F’ INPUT: A’ B C D E E’ E’’’ F’ G OUTPUT: A’ A’’ B’ C D E’ E’’’ F’ G Chapter 6

Polygon clipping algorithm • typedef enum { Left, Right, Bottom, Top} Edge; • # define N_EDGE 4 int clipPolygon(dcpt wmin,dcpt wmax,int n, wcpt2 * pin,wcpt2 * pout) { wcpt2 * first [N_EDGE] = {0,0,0,0}, // first point clipped 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; } Chapter 6

void clippoint (wcPt2 p, Edge b, dcPt wMin, dcPt wMax, wcPt2 * pOut, int * cnt, wcPt2 * first[]., wcPt2 * s) { wcPt2 ipt; // If no previous point exists for this edge, save this point. if (!first [b]) first[b] = &p; Else // Previous point exists. If 'p' and previous point cross edge, find intersection. Clip against next boundary, if any. If no more edges, add intersection to output list if (cross (p, s[b]., b, wMin, wMax)) { iPt = intersect (p, s[b], b, wMin, wMax); if (b < Top) clipPoint (iPt, b+l, wMin, wMax, pOut, cnt, first, s); else{ pOut [*cnt] = iPt; (*cnt)++; } } s[b] = p; // Save 'p' as most recent point for this edge // For all, if point is ‘inside' proceed to next clip edge, if any. If no more edges, add intersection to output list if (inside (p, b, wMin, wMax)) if (b < Top) clippoint (p, b+l, wMin, wMax, pOut, cnt, first, s); else{ pOut [*cnt]= p; (*cnt(++; } } Clip point function Chapter 6

Inside int inside (wcPt2 p, Edge b, dCPt wMin, dcPt wMax) { switch (b) { case Left: 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); break; case Top: if (p.y > wMax.y) return (FALSE); break; } return (TRUE(; } Chapter 6

Cross: Line p1 p2 with edge b int cross (wcPt2 pI, wcPt2 p2, Edge b, dCPt wMin, dcPt wMax( { // both end points are inside edge b if (inside (pI, b, wMin, wMax) == inside (p2, b, wMin, wMax)) return (FALSE(; else return (TRUE(; } Chapter 6

wcPt2 intersect (wcPt2 pI, wcPt2 p2, Edge b, dcPt wMin, dcPt wMax( { wcPt2 iPt; float m; if (pl. x != p2.x) m = (pl.y -p2.y) / (pl. x -p2.x); 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 (pl.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 (pl.x != p2.x) iPt..x = p2.x + (wMax.y - p2.y) / m; else iPt.x = p2.x; break; } return (iPt); } Point of Intersection Chapter 6

Close clip void closeClip (dcPt wMin, dcPt wMax, wcPt2 * pOUt,int * cnt, wcPt2 * first[], wcPt2 * s) { wcPt2 I; Edge b; for (b = Left; b <= Top; b++){ if (cross (s[b], *first[b], b, wMin, wMax)){ i = intersect (s [b], *first [b], b, wMin, wMax); if (b < Top( clippoint (i, b+l, wMin, wMax,POut, cnt, first, s); else{ pOut [*cnt] = i; (*cnt)++; } } } } Chapter 6

Text Clipping • All or none string clipping • All or none character clipping • Clip the components of the individual characters Chapter 6

Windowing and clipping