Agenda. Polygon Terminology Types of polygons Inside Test Polygon Filling Algorithms. Scan-Line Polygon Fill Algorithm Flood-Fill Algorithm

Transcription

1 Polygons UNIT - III

2 Agenda Polygon Terminology Types of polygons Inside Test Polygon Filling Algorithms Scan-Line Polygon Fill Algorithm Flood-Fill Algorithm

3 A Polygon Vertex = point in space (2D or 3D) Polygon = ordered list of vertices Each vertex connected with the next in the list Last is connected with the first May contain self-intersections Simple polygon no self-intersections These are of most interest in CG

4 Polygons in graphics The main geometric object used for interactive graphics. Different types of Polygons Simple Convex Simple Concave Non-simple : self-intersecting Convex Concave Self-intersecting

5 Polygon Representation and Entering 5

6 Polygon Representation Some graphics packages store polygon as a whole unit. Some graphics packages provide trapezoid primitive: means polygons are drawn in the form of trapezoids. And polygon is considered as a collection of trapezoids. Sometimes we need to get information of vertices and polygon is drawn by using series of lines. Here polygon information is stored in the Display File. The information is in the form of commands. Opcode 1 move 2 line above 3 for no. of vertices of polygon

7 Inside Test Why? Want to fill in (color) only pixels inside a polygon What is inside of a polygon? Methods 1. Even odd method 2. Winding no. method

8 Even odd method Case :1 Case:2 P P Q Q P P Odd Q even Q Count number of intersections with polygon edges If N is odd, point is inside If N is even, point is outside

9 Winding no. Method Every side has given a no. called winding no. Total of this winding no. is called net winding. If net winding is zero then point is outside otherwise it is inside. +1-1

10 Polygon Filling Seed Fill Approaches 2 algorithms: Boundary Fill and Flood Fill works at the pixel level suitable for interactive painting apllications Scanline Fill Approaches works at the polygon level better performance 10

11 Seed Fill Algorithms: Connectedness 4-connected region: From a given pixel, the region that you can get to by a series of 4 way moves (N, S, E and W) 8-connected region: From a given pixel, the region that you can get to by a series of 8 way moves (N, S, E, W, NE, NW, SE, and SW) 4-connected 8-connected 11

12 Boundary Fill Algorithm Boundary-defined region Start at a point inside a region Paint the interior outward to the edge The edge must be specified in a single color Fill the 4-connected or 8-connected region 4-connected fill is faster, but can have problems: 12

13 Boundary Fill Algorithm (cont.) void BoundaryFill(int x, int y, newcolor, edgecolor) { int current; current = ReadPixel(x, y); if(current!= edgecolor && current!= newcolor) { putpixel(x,y, newcolor); BoundaryFill(x+1, y, newcolor, edgecolor); BoundaryFill(x-1, y, newcolor, edgecolor); BoundaryFill(x, y+1, newcolor, edgecolor); BoundaryFill(x, y-1, newcolor, edgecolor); } } 13

14 Flood Fill Algorithm Interior-defined region Used when an area defined with multiple color boundaries Start at a point inside a region Replace a specified interior color (old color) with fill color Fill the 4-connected or 8-connected region until all interior points being replaced 14

15 Flood Fill Algorithm (cont.) void FloodFill(int x, int y, newcolor, oldcolor) { if(readpixel(x, y) == oldcolor) { putpixel(x,y, newcolor); FloodFill(x+1, y, newcolor, oldcolor); FloodFill(x-1, y, newcolor, oldcolor); FloodFill(x, y+1, newcolor, oldcolor); FloodFill(x, y-1, newcolor, oldcolor); } } 15

16 Problems with Fill Algorithm Recursive seed-fill algorithm may not fill regions correctly if some interior pixels are already displayed in the fill color. This occurs because the algorithm checks next pixels both for boundary color and for fill color. Encountering a pixel with the fill color can cause a recursive branch to terminate, leaving other interior pixels unfilled. This procedure requires considerable stacking of neighboring points, more efficient methods are generally employed.

17 Scan line polygon Filling Interior Pixel Convention Pixels that lie in the interior of a polygon belong to that polygon, and can be filled. Pixels that have centers that fall outside the polygon, are said to be exterior and should not be drawn. Exploit coherence: pixels that are nearby each other tend to share common attributes (color, illumination, normal vectors, texture, etc.). Span coherence: Pixels in the same scan line tend to be similar. Scan-line coherence: Pixels in adjacent scan line tend to be similar.

18 Example The boundary of a polygon: (In practice, a polygon is defined as a list of vertices.)

19 Basic Scan-Fill Algorithm For each scan line: 1.Find the intersections of the scan line with all edges of the polygon. 2. Sort the intersections by increasing x-coordinate. 3. Fill in all pixels between pairs of intersections.

20 Edge Coherence Computing the intersections between scan lines and edges can be costly Use a method similar to the midpoint algorithm y = mx + b, x = (y b) / m At y = s, xs = (s b ) / m At y = s + 1, xs+1 = (s+1 - b) / m = xs + 1 / m Incremental calculation: xs+1 = xs + 1 / m

21 What happens at edge endpoint? Edge endpoint is duplicated. In other words, when a scan line intersects an edge endpoint, it intersects two edges. Two cases: Case A: edges are on opposite side of the scan line Case B: edges are on the same side of current scan line In Case A, we should consider this as only ONE edge intersection In Case B, we should consider this as TWO edge intersections Scan-line Scan-line Case A Case B

22 Spacial Handling (cont.) Case 1 Intersection points: (p0, p1, p2)??? ->(p0,p1,p1,p2) so we can still fill pairwise ->In fact, if we compute the intersection of the scanline with edge e1 and e2 separately, we will get the intersection point p1 twice. Keep both of the p1. 22

23 Spacial Handling (cont.) Case 2 However, in this case we count p1 once (p0,p1,p2,p3),

24 To increase the efficiency of the algorithm special data structures are maintained in Scan line Algorithm. Active Edge list and Active Edge Table. Each non-horizontal edge occupies one row/record. The rows are sorted according to ymin.

25 Active Edges Polygon edges are sorted according to their minimum Y. When the current scan line y value matches the ymin of an edge that edge becomes active. When the current scan line moves above the upper endpoint, the edge becomes inactive. Only Active edges are involved in intersection computation.

26 Active Edge Table Active edges are sorted according to increasing X.

27 v2 E1 E2 v3 v7 E6 E7 v6 E3 E5 v4 E4 v5 Edge Ymin Ymax X-coor of vertex with y=ymin E1 Y1 Y2-1 X1 E7 Y1 Y7 X1 E4 Y5 Y4-1 X5 E6 Y6 Y7 X6 E2 Y2 Y3 X2 E3 Y4 Y3 X4 An Edge List v1

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