Содержание
- 2. First, a Word About Hammers requirements for this to be a good idea a way of
- 3. Hammers (cont.) by definition, not the optimal way to solve problems, BUT computers are very fast
- 4. What are “advanced game math problems”? problems that are ammenable to mathematical modeling state the problem
- 5. Prerequisites linear algebra vector, matrix symbol manipulation at least calculus concepts what derivatives mean comfortable with
- 6. Overview of Lecture random assortment of example problems breifly mentioned 5 specific example problems in some
- 7. A Look Forward linear equations Ax = b linear inequalities Ax >= b linear programming min
- 8. Applications to Games graphics, physics, ai, even ui computational geometry visibility contact curve fitting constraints integration
- 9. Applications to Games (cont.) don’t forget... The Elastohydrodynamic Lubrication Problem Solving Optimal Ownership Structures “The two
- 10. Specific Examples #1a: Ease Cubic Fitting warm up with an ease curve cubic x(t)=at3+bt2+ct+d x’(t)=3at2+2bt+c 4
- 11. Specific Examples #1a: Ease Cubic Fitting (cont.) x(t)=at3+bt2+ct+d, x’(t)=3at2+2bt+c x(0) = a03+b02+c0+d = d = 0
- 12. Specific Examples #1a: Ease Cubic Fitting (cont.) d = 0, a+b+c+d = 1, c = 0,
- 13. Specific Examples #1a: Ease Cubic Fitting (cont.) or, x(0) = d = 0 x(1) = a
- 14. Specific Examples #1b: Cubic Spline Fitting same technique to fit higher order polynomials, but they “wiggle”
- 15. Specific Examples #1b: Cubic Spline Fitting (cont.) a0 b0 c0 d0 a1 b1 c1 d1 .
- 16. Specific Examples #2: Minimum Cost Network Flow what is the cheapest flow route(s) from sources to
- 17. Specific Examples #2: Minimum Cost Network Flow (cont.) min cost: minimize cTx the sum of the
- 18. Specific Examples #3: Points in Polys point in convex poly defined by planes n1 · x
- 19. Specific Examples #3: Points in Polys (cont.) closest point in two polys min (x2-x1)2 s.t. A1x1
- 20. Specific Examples #3: Points in Polys (cont.) how do we stack x1,x2 into single x given
- 21. Specific Examples #3: Points in Polys (cont.) more points, more polys! min (x2-x1)2 + (x3-x2)2 +
- 22. Specific Examples #4: Contact model like IK constraints a = Af + b a >= 0,
- 23. Specific Examples #5: Joint Limits in CCD IK how to do child-child constraints in CCD? parent-child
- 24. What Unifies These Examples? linear equations Ax = b linear inequalities Ax >= b linear programming
- 25. QP is a Superset of Most quadratic programming min ½xTQx + cTx s.t. Ax >= b
- 26. Karush-Kuhn-Tucker Optimality Conditions get us to MLCP for QP form “Lagrangian” L(x,u,v) = ½ xTQx +
- 27. Karush-Kuhn-Tucker Optimality Conditions (cont.) L(x,u,v) = ½ xTQx + cTx - uT(Ax - b) - vT(Dx
- 28. This is an MLCP x v u Q -DT -AT D 0 0 A 0 0
- 29. Modeling Summary a lot of interesting problems can be formulated as MLCPs model the problem mathematically
- 30. Solving MLCPs (where I hope I made you hungry enough for homework) Lemke’s Algorithm is only
- 31. Playing Around With MLCPs PATH, a MCP solver (superset of MLCP!) really stoked professional solver free
- 32. References for Lemke, etc. free pdf book by Katta Murty on LCPs, etc. free pdf book
- 34. Specific Examples #5: Constraints for IK compute “forces” to keep bones together a1 = A11 f1
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