# Vectors

Cross Product

Provides a vector orthogonal to two others

Magnitude provides sine of angle between u
and v
|sin(theta)|=|uxv|/|u||v|

Applications to 3D 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

Affine transformations preserve lines

Points

 Point, same notation as vector. Keep track of units Point Origin Specify Standard Space Homogenized Point. Intersection with standard plane/space Fails when h=0

Line Equations

, three equations in 3D
Point- Slope Form
m=slope, (x,y)=point 0,
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
where is the second point

Plane Equations

p=ax+by+cz+d
Three point form
p is the plane
(cross
product)
(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

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

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
Matrix Transformations
glRotatef(angle, vx, vy, vz);
glTranslatef(dx, dy, dz);
glScalef(sx, sy, sz);
glMultMatrixf(myarray);
glMatrixMode(GL_MODELVIEW);
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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