Linear Transformations and Matrices Jim Van Verth ([email protected]) Lars M. Bishop ([email protected])

Linear Transformationsand Matrices

Jim Van Verth ([email protected])

Lars M. Bishop ([email protected])

Essential Math for Games

Transformation

• Have some geometric data

• How to apply functions to it?

• Also desired: combine multiple steps into single operation

• For vectors: linear transformations

Essential Math for Games

Transformations

• A transformation T:VW is a function that maps elements from vector space V to W

• The function

f(x, y) = x2 + 2y

is a transformation because it maps R2 into R

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Linear Transformation

• Two basic properties: T(x + y) = T(x) + T(y) T(ax) = aT(x)

• Follows that T(0) = 0 T(ax+y) = aT(x) + T(y)

Essential Math for Games

Linear Transformations

• Basis vectors span vector space

• Know where basis goes, know where rest goes

• So we can do the following: Transform basis Store as columns in a matrix Use matrix to perform linear transforms

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Linear Transforms

• Example:

• (1,0) maps to (1,2)

• (0,1) maps to (2,1)

• Matrix is

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What is a Matrix?

• Rectangular m x n array of numbers

• M rows by n columns

• If n=m, matrix is square

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Matrix Concepts

• Number at row i and column j of matrix A is element Aij

• Elements in row i make row vector

• Elems in column j make column vector

• If at least one Aii (diagonal from upper left to lower right) are non-zero and all others are zero, is diagonal matrix

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Transpose

• Represented by AT

• Swap rows and columns along diagonal

• ATij = Aji

• Diagonal is invariant

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Transpose

• Transpose swaps transformed basis vectors from columns to rows

• Useful identity

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Transforming Vectors

• Represent vector as matrix with one column

• # of components = columns in matrix

• Take dot product of vector w/each row

• Store results in new vector

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Transforming Vectors

• Example: 2D vector

• Example: 3D vector to 2D vector

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Row Vectors

• Can also use row vectors• Transformed basis stored as rows• Dot product with columns• Pre-multiply instead of post-multiply

• If column default, represent row vector by vT

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Row vs. Column

• Using column vectors, others use row vectors Keep your order straight!

• Transpose to convert from row to column (and vice versa)

Row vector order (DirectX)

Column vector order (us, OpenGL)

Essential Math for Games

Matrix Product

• Want to combine transforms

• What matrix represents ?

• Idea: Columns of matrix for S are xformed basis Transform again by T

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Matrix Product

• In general, element ABij is dot product of row i from A and column j from B

or

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Matrix product (cont’d)

• Number of rows in A must equal number of columns in B

• Generally not commutative

• Is associative

Essential Math for Games

Block Matrices

• Can represent matrix with submatrices

• Product of block matrix contains sums of products of submatrices

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Identity

• Identity matrix I is square matrix with main diagonal of all 1s

• Multiplying by I has no effect A I = A

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Inverse

• A-1 is inverse of matrix A such that

• A-1 reverses what A does• A is orthogonal if AT = A-1

Component vectors are at right angles and unit length

I.e. orthonormal basis

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Computing Inverse

• Only square matrices have inverse

• Inverse doesn’t always exist

• Zero row, column means no inverse

• Use Gaussian elimination or Cramer’s rule (see references)

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Computing Inverses

• Most interactive apps avoid ever computing a general inverse

• Properties of the matrices used in most apps can simplify inverse

• If you know the underlying structure of the matrix, you can use the following:

Essential Math for Games

Computing Inverse

• If orthogonal, A-1 =AT

• Inverse of diagonal matrix is diagonal matrix with A-1

ii = 1/Aii

• If know underlying structure can use

• We’ll use this to avoid explicit inverses

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Storage Format

• Row major Stored in order of rows

Used by DirectX

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Storage Format (cont’d)

• Column Major Order Stored in order of columns

Used by OpenGL, and us

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Storage Format (cont’d)

• Note: storage format not the same as multiplying by row vector

• Same memory footprint: Matrix for multiplying column vectors in

column major format Matrix for multiplying row vectors in row

major format

• I.e. two transposes return same matrix

Essential Math for Games

System of Linear Equations

• Define system of m linear equations with n unknowns

b1 = a11x1 + a12 x2 + … + a1n xn

b2 = a21x1 + a22 x2 + … + a2n xn

…

bm = am1x1 + am2 x2+ … + amn xn

Essential Math for Games

System of Linear Equations

• Matrix multiplication encapsulates linear system

Essential Math for Games

• Matrix multiplication encapsulates linear system

• Solve for x

• Multiplying by inverse imprecise

• Use Gaussian elimination, or other methods (see refs)

System of Linear Equations

Essential Math for Games

References

• Anton, Howard and Chris Rorres, Elementary Linear Algebra, 7th Ed, Wiley & Sons, 1994.

• Axler, Sheldon, Linear Algebra Done Right, Springer Verlag, 1997.

• Blinn, Jim, Notation, Notation, Notation, Morgan Kaufmann, 2002.