CPCS 391 Computer Graphics 1 Lecture 5: Polygon Filling

1. CPCS 391 Computer Graphics 1Lecture 5: Polygon Filling Instructor: Dr. Sahar Shabanah

2. 3-D Graphics Rendering Pipeline Primitives Object space Modeling Transformation World space Viewing Transformation Camera space (I added this stepto the diagram) Culling Lighting & Shading Clipping Projection Today Normalized view space Scan conversion, Hiding Image space, Device coordinates Image

3. 2D Object Definition • Lines and Polylines: lines drawn between ordered points • Same first and last point make closed polyline or polygon • If it does not intersect itself, called simple polygon

4. triangle square rectangle 2D Object Definition • Convex: For every pair of points in the polygon, the line between them is fully contained in the polygon. • Concave: Not convex: some two points in the polygon are joined by a line not fully contained in the polygon. • Special Polygons

5. 2D Object Definition • Circles • Consist of all points equidistant from one predetermined point (the center) • (radius) r = c, where c is a constant • On a Cartesian grid with center of circle at origin equation is r2 = x2 + y2 • Circle as polygon • A circle can be approximated by a polygon with many sides (>15) P1 y r r P0 0 x

6. 6 6 5 5 4 4 3 3 2 2 1 1 0 0 1 2 3 4 5 6 7 8 9 10 1 3 4 5 6 7 8 9 10 2 2D Object Definition • (Aligned) Ellipses • A circle scaled along the x or y axis Example: height, on y-axis, remains 3, while length, on x-axis, changes from 3 to 6

7. 2D Shape Representation: Vertex-Edge Table • General purpose, simple overhead, reasonable efficiency • Each vertex listed once • Each edge is ordered pair of indices into vertex table • Sufficient information to draw shape and perform simple operations. • Order does not matter, convention is edges listed in counterclockwise order.

8. 2D Shape Representation: Splines • How they work: Parametric curves governed by control points • Mathematically: Several representations to choose from. More complicated than vertex lists. • Simple parametric representation: • Advantage: Smooth with just a few control points • Disadvantage: Can be hard to control • Uses: • representation of smooth shapes. Either as outlines in 2D or with Patches or Subdivision Surfaces in 3D • animation Paths for tweening • approximation of truncated Gaussian Filters

9. 2D to 3D Object Definition Vertices in motion (“Generative object description”) • Line • is drawn by tracing path of a point as it moves (one dimensional entity) • Square • drawn by tracing vertices of a line as it moves perpendicularly to itself (two dimensional entity) • Cube • drawn by tracing paths of vertices of a square as it moves perpendicularly to itself (three-dimensional entity) • Circle • drawn by swinging a point at a fixed length around a center point

10. Building 3D Primitives • Triangles and tri-meshes • Parametric polynomials, like the aforementioned splines used to define surface patches. Curves Patches Boundaries are Polynomial curves In 3D

11. Triangle Meshes • Most common representation of shape in three dimensions • All vertices of triangle are guaranteed to lie in one plane (unlike quadrilaterals or other polygons) • Uniformity makes it easy to perform mesh operations: subdivision, simplification, etc. • Many different ways to represent triangular meshes: http://en.wikipedia.org/wiki/Polygon_mesh

12. 3D Shape Representation Vertex-Triangle Tables • Each vertex gets listed once • Each triangle is ordered triple of indices into the vertex table • Edges between vertices inferred from triangles • Only need the triangular mesh representations to draw the shapes; • Counterclockwise ordering of vertices for normals

13. Polygon Fill Algorithms • A standard output primitive in general graphics package is a solid color or patterned polygon area. • There are two basic approaches to filling on raster systems. • Scan line polygon fill: determine overlap Intervals for scan lines that cross that area, mostly used in general graphics packages • Seed fill: start from a given interior point and paint outward from this point until we encounter the boundary, used in applications having complex boundaries and interactive painting systems

14. Seed Fill Algorithm • These algorithms assume that at least one pixel interior to a polygon or region is known • Regions maybe interior or boundary defined Interior-defined region Interior-defined region

15. Scan Line Polygon Fill Algorithm • For each scan line crossing a polygon are then sorted from left to right, and the corresponding frame buffer positions between each intersection pair are set to the specified color. • These intersection points are then sorted from left to right , and the corresponding frame buffer positions between each intersection pair are set to specified color • In the example, four pixel intersections define stretches from x=10 to x=14 and x=18 to x=24 Interior pixels along a scan line passing through a polygon area 10 14 18 24 Xk+1,yk+1 Scan Line yk+1 Scan Line yk Xk, yk

16. Scan Line Polygon Fill Algorithm • Maintaining a data structure of all intersections of polygons with scan lines • Sort by scan line • Fill each span • For each scan line: • Find the intersections of the scan line with all edges of the polygon. • Sort the intersections by increasing x- coordinate. • Fill in all pixels between pairs of intersections. • Problem: • Calculating intersections is slow. • Solution: • Incremental computation / coherence

17. Scan Line Polygon Fill Algorithm Edge Coherence • Observation: • Not all edges intersect each scanline. • Many edges intersected by scanlinei will also be intersected by scanline i+1 • Formula for scanlines is y = s, • for an edge is y = mx + b • Their intersection is • s = mxs+ bxs= (s-b)/m • For scanlines + 1, • xs+1= (s+1 - b)/m= xs+ 1/m Incremental calculation: xs+1 = xs+ 1/m

18. Scan Line Polygon Fill Algorithm • Problem of Scan-Line intersections at polygon vertices: • A scan Line passing through a vertex intersects two polygon edges at that position, adding two points to the list of intersections for the scan Line • In the example, scan Line y intersects 5 polygon edges and the scan Line y‘ intersects 4 edges although it also passes through a vertex • y‘ correctly identifies internal pixel spans, but y need some extra processing

19. Scan Line Polygon Fill Algorithm • Solution 1: • shorten some polygon edges to split those vertices that should be counted as one intersection • When the end point y coordinates of the two edges are increasing, the y value of the upper endpoint for the current edge is decreased by 1 • When the endpoint y values are monotonically decreasing, we decrease the y coordinate of the upper endpoint of the edge following the current edge (a) (b)

20. Scan Line Polygon Fill Algorithm • When a scan line intersects an edge endpoint (vertex), it intersects two edges. • Two cases: • Case A: edges are monotonically increasing or decreasing • Case B: edges reverse direction at endpoint • 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 B Case A

21. Scan Line Polygon Fill Algorithm • The Algorithm Steps: • Intersect each scanline with all edges using the iterative coherence calculations to obtain edge intersections quickly • Sort intersections in x • Calculate parity of intersections to determine in/out • Fill the “in” pixels • Special cases to be handled: • Horizontal edges should be excluded • For vertices lying on scanlines, • count twice for a change in slope. • Shorten edge by one scanline for no change in slope

22. Scan Line Polygon Fill Algorithm • Need 2 data structures: Edge Table and Active Edge Table • Traverse Edges to construct an Edge Table • In this table, there is an entry for each scan-line. • Add only the non-horizontal edges into the table. • For each edge, we add it to the scan-line that it begins with (that is, the scan-line equal to its lowest y-value). • Each scan-line entry thus contains a sorted list of edges. The edges are sorted left to right. (To maintain sorted order, just use insertion based on their x value.) • Add edge to linked-list for the scan line corresponding to the lower vertex. • Store the following: • y_upper: last scanline to consider • x_lower: starting x coordinate for edge • 1/m: for incrementing x; compute

23. Active Edge List • Process the scan lines from bottom to top to construct Active Edge Table during scan conversion. • Maintain an active edge list for the current scan-line. • When the current scan line reaches the lower / upper endpoint of an edge it becomes active. • When the current scan line moves above the upper / below the lower endpoint, the edge becomes inactive • Use iterative coherence calculations to obtain edge intersections quickly. • AEL is a linked list of active edges on the current scanline, y. • Each active edge line has the following information • y_upper: last scanline to consider • x_lower: edge’s intersection with current y • 1/m: x increment • The active edges are kept sorted by x

24. Scan Line Polygon Fill Algorithm • Set yto the smallest y coordinate that has an entry in the ET; i.e, y for the first nonempty bucket. • Initialize the AET to be empty. • Repeat until the AET and ET are empty: • Move from ET bucket yto the AET those edges whose y_min = y (entering edges). • Remove from the AET those entries for which y = y_max (edges not involved in the next scanline), the sort the AET on x (made easier because ET is presorted • Fill in desired pixel values on scanliney by using pairs of xcoordinates from AET. • Increment y by 1 (to the coordinate of the next scanline). • For each nonvertical edge remaining in the AET, update x for the new y.

25. Polygon fill example • Set y to smallest y with entry in ET, i.e., y for the first non-empty bucket • Init Active Edge Table (AET) to be empty • Repeat until AET and ET are empty: • Move form ET bucket y to the AET those edges whose ymin=y (entering edges) • Remove from AET those edges for which y=ymax (not involved in next scan line), then sort AET (remember: ET is presorted) • Fill desired pixel values on scan line y by using pairs of x-coords from AET Increment y by 1 (next scan line) • For each nonvertical edge remaining in AET, update x for new y

26. Polygon fill example