Lighting and Triangulation: Understanding Convex Sets and Orientation

1. front and back lighting • convex sets, convex hull • triangulation • why triangulation matters • orientation • drawing curves Class 18

2. Lighting: front, back, or both litTriangle.cpp glLightModel*(GL_LIGHT_MODEL_TWO_SIDED, GL_TRUE); • a) use the GL_BACK (or GL_FRONT_AND_BACK) parameter values to color back-facing polygons • b) reverse the specified vertex normal for back-facing polygons. • front: specify vertices counterclockwise

3. litTriangle.cpp • try setting glColor3f( 1, 1, 0); //yellow • try disabling lighting

4. Chapter 7Convex set S If P and Q are points of S, then the segment connecting P and Q is also in S Does a triangle have to be convex? Does a quadrilateral have to be convex?

5. Convex set S

6. Convex Hull The smallest convex set containing a set S in the convex hull of S.

7. The Intersection of convex sets is convex.The Union of convex sets is NOT necessarily convex

8. If S={P1,P2, ... , Pk},then the convex hull of S is the set of all convex combinations of S.

9. Chapter 8Triangulation Suppose T is a collection of triangles and an object X is the union of these triangles. Let s and t be any two triangles from this collection. Good: s & t don't intersect s & t intersect at a vertex of each s & t intersect at a whole edge of each

10. Examples: ✓ ✓ X ✓ X

11. What's wrong with a bad "triangulation"?

12. Answer: The rendering depends on the order you render the triangles.

13. If an object is made from a good triangulation then the image is independent of the order in which the triangles are rendered.

14. More good triangles gives better lighting

15. Trouble with non-convex polygons instead of triangles. Makes a fan around the first vertex. Which is first vertex?

16. Make sure polygons are convex (and coplanar)! • GL_QUAD • GL_POLYGON • GL_QUAD_STRIP

17. Chapter 9Orientation Front VS Back

18. THU triangle on transparency • What do you see from each side? T H U

19. What does the viewer see? • For each primitive, does the viewer see the vertices : • CCW - counter clockwise: FRONT facing • CW - clockwise: BACK facing • (CCW is default FRONT, can be changed with glFrontFace(GL_CW) )

20. Should the triangle be drawn red or green?

21. Orientation of P,Q,R as viewed from the origin D = det PX QX RX PY QY RY PZ QZ RZ If D>0 then PQR looks Clockwise If D<0 then PQR looks Counter Clockwise

22. Consistent Orientation of a Triangulation: Any two triangles that share an edge order the edge oppositely. consistent inconsistent

23. squareOfWalls.cppconsistent

24. Run threeQuarterSphere.cpp • one hemisphere, solid outside • one overlapping hemisphere, lines outside

25. Culling • Inside a closed sphere see one side of triangles only • Outside, see the other side, only. • culling - don't draw those we can't see • Need to tell OpenGL

26. glCullFace(face) • face GL_FRONT, GL_BACK, GL_FRONT_AND_BACK • face says which are NOT to be rendered. • Also need glEnable(GL_CULL_FACE);

27. sphereInBox1.cpp • Note what happens when box is opened. • Show where the culling code is. • cull some other faces – have them try – with ball, box, lid, front or back • do experiment

28. Discuss behavior under transformation.Order preserving, order reversing,glFrontFace(GL_CW)

29. curves in 2-space plane curve: x = f(t) y = g(t) t∈[a,b] or (-∞, b], [a, ∞), (-∞, ∞)

30. curveDrawerMJB.cpp • run • endpoints • number of intervals and interval size • y = x*x

31. curveDrawerMJB.cpp modifications y=sin(x) • x=t y=sin(t) • pick good a and b • pick good MAX and MIN

32. curveDrawerMJB.cpp modifications circle • x=cos(t) y=sin(t) • pick good a and b • pick good MAX and MIN