1 KIPA Game Engine Seminars Jonathan Blow Ajou University December 12, 2002 Day 14

2 Matrix Tricks Review of the dot product trick –a dot b = a T b

3 Cross Product as Matrix The skew-symmetric matrix is a trick for turning a cross product into a matrix multiply Useful when you are doing math and need to transform cross products, invert them, etc “skew-symmetric” because it is almost equal to its transpose, but for some negated terms

4 Dot Product as a 3x3 Matrix (plus a step at the end to extract the value)

5 Quaternion-based rotation x’ = qxq* Discussion of how to expand this using Hamilton’s multiplication rules

6 Cross Product As determinant of pseudo-matrix containing (i, j, k) –Formulation from Lounesto’s book “Clifford Algebras and Spinors”

7 v cross (v cross a) A common expression to run into We see r x (n x r) a lot in physics / rotation kinds of applications Show what this means in terms of the skew symmetric matrix (n x r) = -(r x n)

8 Cross Product and Dot Product Cross product as area of parallelogram As lengths of vectors times sin(theta) Dot product as lengths of vectors times cos(theta) What happens when we square, add, square root these terms?

9 Cross Product and Dot Product It’s almost as though the cross and dot product are projections of the result of some “bigger product” onto orthogonal spaces –Review of unit vector and cos^2 + sin^2 = 1 Re-discussion of the idea that any linear item can be broken into parallel and orthogonal components

10 Clifford Basis Vectors View the products of two vectors as creating bivectors, not vectors in the same space Algebra a bit different from Hamilton’s

11 Clifford Product of a vector times a vector How does this compare to the dot and cross products? Clifford product decomposed as (a dot b) and (a wedge b)

12 Clifford Product of a vector times a bivector

13 Clifford Inverse of a vector

14 Mirror transform Matrix with determinant –1 Example with X, Y, Z axes

15 Portals, Teleports and Mirror Transforms You can attach arbitrary transforms to portals (since you are basically “restarting” the viewing process at each portal) A portal can look into somewhere spatially disjoint A portal can look from a right-handed into a left- handed space –If the view transform reverses handedness, and is pointed back into the same room, you have a mirror.

16 Least-Squares Fitting of Sample Data Was asked as a question last week Will do derivation now on whiteboard

17 Static Mesh LOD (block-based) Using alpha blending to blend between pieces (demonstration of software in progress)