KIPA Game Engine Seminars
Day 14
Jonathan Blow
Ajou University
December 12, 2002
1
Matrix Tricks
• Review of the dot product trick
– a dot b = aTb
2
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
3
Dot Product as a 3x3 Matrix
• (plus a step at the end to extract the value)
4
Quaternion-based rotation
• x’ = qxq*
• Discussion of how to expand this using
Hamilton’s multiplication rules
5
Cross Product
• As determinant of pseudo-matrix containing
(i, j, k)
– Formulation from Lounesto’s book “Clifford
Algebras and Spinors”
6
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)
7
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?
8
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
9
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
10
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)
11
Clifford Product
• of a vector times a bivector
12
Clifford Inverse
• of a vector
13
Mirror transform
• Matrix with determinant –1
• Example with X, Y, Z axes
14
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 lefthanded space
– If the view transform reverses handedness, and is
pointed back into the same room, you have a mirror.
15
Least-Squares Fitting
of Sample Data
• Was asked as a question last week
• Will do derivation now on whiteboard
16
Static Mesh LOD
• (block-based)
• Using alpha blending to blend between
pieces
• (demonstration of software in progress)
17