Matrix Operations

Vectors
Example Vectors
	Geometric Vectors
		Directed Line Segments
		Scalar-Vector Multiplication
			Changes Length
			Changes Length
		Vector-Vector Addition
			Head-to-Tail Axiom
Euclidean Spaces
	Inner (dot) product
		u·v=v·u
		(au+bv)·w=au·w+bv·w
		v·v>0(v!=0)
		0·0=0
		if u·v=0, then u and v orthogonal
	\|v\|=sqrt(v·v)
	Add concepts of affine spaces, such as points
	P-Q is a vector
	\|P-Q\|=sqrt((P-Q)·(P-Q))
	Angle between two vectors
		u·v=\|u\|\|v\|cos(theta)
	Equation for a circle
		\|Q-P\|=r
		sqrt((Q-P)·(Q-P))=r
		(Q-P)·(Q-P)=r2
Dot Product
Projections
	Finding the shortest distance between a point to a line or plane
	Start with two vectors
	Divide one into two parts
		One parallel , one orthogonal to the second original vector
	v is first vector, w is second vector
	w=av+u
	Parallel part, v
	Orthogonal part, u
	u·v=0
	w·v=av·v+u·v=av·v
	a=-(w·v)/(v·v)
	av is projection of w onto v
	u=w-[(w·v)/(v·v)]v
Normalized Vectors
	Length of One
	Unit vector
	v'=v/\|\|v\|\|
Normalized Vectors
OpenGL and Affine Spaces
	Positions always defined relative to the origin of the current coordinate system
	Transform the coordinate system to take positions relative to some other point
	Necessary for computing distances, angles, etc.
Matrices
	nxm array of scalars
		n Rows
		m Columns
	If m=n then square matrix
	The elements of a matrix A are {aij}, i=1,...,n, j=1,...,n
	A=[aij]
	Transpose of A of n is mxn matrix with rows and columns interchanged
	AT=[aji]
	Single row or single column
		Row Matrix
		Column Matrix
	b=[bi]
	Transpose of a row matrix is a column matrix
	Transpose of a column matrix is a row matrix
Matrices
Identity Matrix
	1's along diagonal
	I=[aij] aij={1 if i=j, 0 otherwise
	AI=A
	IB=B
Matrix Operations
	Scalar-matrix multiplication
		aA=[aaij]
		a(bA)=(ab)A
		abA=baA
	Matrix-matrix addition
		C=A+B=[aij+bij]
		Commutative
		Associative
	Matrix-matrix multiplication
		Product of nxl and lxm is nxm
		Middle dimmension MUST match
		C=AB=[cij]
		cij=Sum[k=1,l]aikbkj
		Associative
		NOT Commutative
			AB!=BA
			May in fact not even be defined
			Order matters
			Application to Transformations
Row and Column Matrices
	Re presentation of vectors or points
	p=[x y z]T
	pT=[x y z]
	Transformation - Square matrix
	Point - Column matrix of 2, 3, or 4 pts
	p'=Ap
		Transforms p by A
	p'=ABCp
		Concatenations of transformations
	(AB)T=BTAT
Cross Product
	Provides a vector orthogonal to two others
		w=uxv=[a2b3-a3b2 a3b1-a1b3 a1b2-a2b1]T
	Magnitude provides sine of angle between u and v
		\|sin(theta)\|=\|uxv\|/\|u\|\|v\|
Cross Product
Applications to 3 D Graphics
	Vertex
	Matrix
		Identity
		Rotation
		Translation
		Scale
		Projection
	Mathematics
		Multiplication
		Dot Product
		Cross Product
Homogeneous Coordinates
	In affine spaces points and vectors can be confused
		P=[x y z]T
		V=[x y z]T
	Instead use
		P=[x y z P0]T
			P=a1v1+a2v2+a3v3+P0
		V=[x y z 0]T
			V=b1v1+b2v2+b3v3
	Affine transformations preserve lines
Points
Line Equations
	y-y1=m(x-x1), three equations in 3D
		Point- Slope Form
		m=slope, (x,y)=point 0, (x1, y1)=point 1
	L=Q+tw
		Point-Normal Form
		L is the line
		Q is a point on the line
		t is the unknown
		w is a directional vector for the line
		w must be from one point to another
		w=P2-Q where P2 is the second point
Plane Equations
	p=ax+by+cz+d
		Three point form
		p is the plane
		N=[p2-p1]x[p3-p1]=[a,b,c]T (cross product)
		N*P1=-D (dot product)
	p=(X-P)·v=0
		Point-Normal form
		X is unknown
		P is point on plane
		v is normal to plane
Line-Plane Intersections
	Substitute for x ,y, or z in 3 point form
		Works well for slope intercept form
	Substitute line for X in point normal form
		Works well when line in point normal form
		Carried out: t=(((P-Q)·v)/(w·v))w
		Intersection point at: Q+(((P-Q)·v)/(w·v))w
	t is time (unknown)
	v,w are vectors
	Q,P,X are points
Translation Matrix
Rotation Matrix
Scaling Matrix
Simple Perspective Projection Matrix
Mutiple Transformations
	Rotate about fixed point
		S=(T)R(-T)p
	Instance Transformation
		Multiple uses of Same object
		M=TRS
	Order of operations
		Right to left - fewer component operations
		Precalculate matrix ops
		Not unique effect but resulting matrix is
	Rotations relative to another angle
		Translate to origin
		Rotate to align with current orientation
		Rotate by specified amount
		Rotate to unalign with orientation
		R=T(p0)Rx(-Ox)Ry(-Oy)Rz(O)Ry(Oy)Rx(Ox)T(-p0)
	Rotation about an arbitrary axis
		Rotate to the axis first
		Apply a rotation to the rotation
		Brings it into the correct frame of reference or orientation first
OpenGL
	Provides 2 Frames
		Camera frame, fixed
		World frame
	Current Transformation Matrix
		Applied to everything
		Changing CTM changes state of environment
		4x4 Matrix
OpenGL Matrix Operations
	Replacement Matrices
		glLoadIdentity()
		glLoadMatrixf(pointer to matrix);
	Matrix Transformations
		glRotatef(angle, vx, vy, vz);
		glTranslatef(dx, dy, dz);
		glScalef(sx, sy, sz);
		glMultMatrixf(myarray);
	Rotation about a fixed point
		glMatrixMode(GL_MODELVIEW);
		glLoadIdentity();
		glTanslatef(4.0, 5.0, 6.0);
		glRotatef(45.0, 1.0, 2.0, 3.0);
		glTranslatef(-4.0, -5.0, -6.0);
	Matrices are generated left to right
	Matrix farthest to the right is applied first
Matrix Stacks
	Saving of states
	Transform only part of a scene
		Scale a single object
		Rotate a heirarchical object
		Apply transformation then restore original state
		glPushMatrix();
		glScalef(3.0, 4.0, 5.0);
		/* Draw Scaled Object */
		glPopMatrix();

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