1. CAP4730: Computational Structures in Computer Graphics Triangle Scan Conversion

2. Triangle Area Filling Algorithms • Why do we care about triangles? • Edge Equations • Edge Walking

4. We want something easier • It is easier to do 1 thing VERY fast than 2 things pretty fast. • Why? Think about how you code. • Scan conversion • Polygon • Circle • Clipping

5. Do something easier! • Instead of polygons, let’s do something easy! • TRIANGLES! Why? 1) All polygons can be broken into triangles 2) Easy to specify 3) Always convex 4) Going to 3D is MUCH easier

6. Polygons can be broken down Triangulate - Dividing a polygon into triangles. Is it always possible? Why?

7. Any object can be broken down into polygons

8. Specifying a model • For polygons, we had to worry about connectivity AND vertices. • How would you specify a triangle? (What is the minimum you need to draw one?) • Only vertices (x1,y1) (x2,y2) (x3,y3) • No ambiguity • Line equations A1x1+B1y1+C1=0 A2x2+B2y2+C2=0 A3x3+B3y3+C3=0

9. Triangles are always convex • What is a convex shape? An object is convex if and only if any line segment connecting two points on its boundary is contained entirely within the object or one of its boundaries. Think about scan lines again!

10. Scan Converting a Triangle • Recap what we are trying to do • Two main ways to rasterize a triangle • Edge Equations • A1x1+B1y1+C1=0 • A2x2+B2y2+C2=0 • A3x3+B3y3+C3=0 • Edge Walking

11. Types of Triangles What determines the spans? Can you think of an easy way to compute spans? What is the special vertex here?

12. Edge Walking • 1. Sort vertices in y and then x • 2. Determine the middle vertex • 3. Walk down edges from P0 • 4. Compute spans P0 P1 P2

13. Pros Fast Easy to implement in hardware Cons Special Cases Interpolation can be tricky Edge Walking Pros and Cons

14. Color Interpolating P0 (?, ?, ?) P1 P2 (?, ?, ?)

15. Edge Equations P0 • A1x1+B1y1+C1=0 • A2x2+B2y2+C2=0 • A3x3+B3y3+C3=0 • How do you go from: x1, y1 - x2, y2 to A1x1+B1y1+C1? P1 P2

16. Given 2 points, compute A,B,C C = x0y1 – x1y0 A = y0 –y1 B = x1 – x0

17. Edge Equations P0 • What does the edge equation mean? • A1x1+B1y1+C1=0 • Pt1[2,1], Pt2[6,11] • A=-10, B=4, C=16 • What is the value of the equation for the: • gray part • yellow part • the boundary line • What happens when we reverse P0 and P1? P1

18. Positive Interior • We add the C element from each edge • area = edge0.C + edge1.C + edge2.C • if (area>0) then inside points are in the positive half spaces • if (area<0) then what should we do? • What happens if area=0?

19. Combining all edge equations P0 1) Determine edge equations for all three edges 2) Find out if we should reverse the edges 3) Create a bounding box 4) Test all pixels in the bounding box whether they too reside on the same side P1 P2

20. Edge Equations: Interpolating Color • Given colors (and later, other parameters) at the vertices, how to interpolate across? • Idea: triangles are planar in any space: • This is the “redness” parameter space • Note:plane follows formz = Ax + By + C • Look familiar? Taken w/ permission from: David Luebke, UVA http://www.cs.virginia.edu/~gfx/Courses/2003/Intro.spring.03/lecture17.ppt

21. Edge Equations: Interpolating Color • Given redness at the 3 vertices, set up the linear system of equations: • The solution works out to:

22. Edge Equations:Interpolating Color • Notice that the columns in the matrix are exactly the coefficients of the edge equations! • So the setup cost per parameter is basically a matrix multiply • Per-pixel cost (the inner loop) cost equates to tracking another edge equation value (which is?) • A: 1 add

23. Pros If you have the right hardware (PixelPlanes) then it is very fast Fast tests Easy to interpolate colors Cons Can be expensive if you don’t have hardware 50% efficient Pros and Cons of Edge Equations

24. Recap P0 P0 P1 P1 P2 P2