Background%20Mathematics

1
Background Mathematics

• Aaron Bloomfield
• CS 445 Introduction to Graphics
• Fall 2006

2
Mathematical Foundations

• A very brief review of some mathematical tools
  well employ
• Geometry (2D, 3D)
• Trigonometry
• Vector and affine spaces
• Points, vectors, and coordinates
• Dot and cross products
• Linear transforms and matrices

3
3D Geometry

• To model, animate, and render 3D scenes, we must
  specify
• Location
• Displacement from arbitrary locations
• Orientation
• Well look at two types of spaces
• Vector spaces
• Affine spaces
• We will often be sloppy about the distinction

4
Vector Spaces

• Given a basis for a vector space
• Each vector in the space is a unique linear
  combination of the basis vectors
• The coordinates of a vector are the scalars from
  this linear combination
• Best-known example Cartesian coordinates
• Note that a given vector will have different
  coordinates for different bases

5
Vectors And Points

• We commonly use vectors to represent
• Direction (i.e., orientation)
• Points in space (i.e., location)
• Displacements from point to point
• But we want points and directions to behave
  differently
• Ex To translate something means to move it
  without changing its orientation
• Translation of a point different point
• Translation of a direction same direction

6
Affine spaces vs. Vector spaces

• Vector spaces
• All vectors originate at the origin
• Thus, can be denoted with (x,y,z) coordinates
• The head of the vector
• The origin is absolute
• Affine spaces
• Vectors can start at an arbitrary point
• Thus, can be denoted via (x,y,z) and starting
  point P
• The origin is relative to the current context
• Only when the distinction matters will we specify

7
Affine Spaces

• To be more rigorous, we need an explicit notion
  of position
• Affine spaces add a third element to vector
  spaces points (P, Q, R, )
• Points support these operations
• Point-point subtraction Q - P v
• Result is a vector pointing from P to Q
• Vector-point addition P v Q
• Result is a new point
• P 0 P
• Note that the addition of two points is not
  defined

Q
v
P
8
Affine Spaces

• Points, like vectors, can be expressed in
  coordinates
• The definition uses an affine combination
• Net effect is same expressing a point in terms
  of a basis
• Thus the common practice of representing points
  as vectors with coordinates
• Be careful to avoid nonsensical operations
• Point point
• Scalar point

9
Affine Lines An Aside

• Parametric representation of a line with a
  direction vector d and a point P1 on the line
• P(a) Porigin ad
• Restricting 0 ? a produces a ray
• Setting d to P - Q and restricting 0 ? a ? 1
  produces a line segment between P and Q

10
Dot Product

• The dot product or, more generally, inner product
  of two vectors is a scalar
• v1 v2 x1x2 y1y2 z1z2 (in 3D)
• Useful for many purposes
• Computing the length of a vector
• length(v) v sqrt(v v)
• Normalizing a vector, making it unit-length
• Divide each coordinate by the length
• Computing the angle between two vectors
• u v u v cos(?)
• Checking two vectors for orthogonality
• Does the angle equal 90º?
• Projecting one vector onto another

11
Cross Product

• The cross product or vector product of two
  vectors is a vector
• Cross product of two vectors is orthogonal to
  both
• Right-hand rule dictates direction of cross
  product
• Cross product is handy for finding surface
  orientation
• Lighting
• Visibility

12
Linear Transformations

• A linear transformation
• Maps one vector to another
• Preserves linear combinations
• Thus behavior of linear transformation is
  completely determined by what it does to a basis
• Turns out any and all linear transformations can
  be represented by a matrix

13
Matrices

• By convention, matrix element Mrc is located at
  row r and column c
• This is called row-major order
• By (OpenGL) convention, vectors are columns
• And 2-D matrices are in column-major order!

14
Matrices

• Matrix-vector multiplication applies a linear
  transformation to a vector
• Recall how to do matrix multiplication

15
Matrix Transformations

• A sequence or composition of linear
  trans-formations corresponds to the product of
  the corresponding matrices
• Note the matrices to the right affect vector
  first
• Note order of matrices matters!
• The identity matrix I has no effect in
  multiplication
• Some (not all) matrices have an inverse

16
3D Scene Representation

• Scene is usually approximated by 3D primitives
• Point
• Line segment
• Polygon
• Polyhedron
• Curved surface
• Solid object
• etc.

17
3D Point

• Specifies a location
• Represented by three coordinates
• Infinitely small

typedef struct Coordinate x Coordinate
y Coordinate z Point
(x,y,z)
18
3D Vector

• Specifies a direction and a magnitude
• Represented by three coordinates
• Magnitude V sqrt(dx dx dy dy dz dz)
• Which is the square root of the dot product
• Has no location (in affine spaces!)
• Dot product of two 3D vectors
• V1V2 dx1dx2 dy1dy2 dz1dz2
• V1V2 V1 V2 cos(Q)

(dx,dy,dz)
typedef struct Coordinate dx Coordinate
dy Coordinate dz Vector
(dx2,dy2 ,dz2)
Q
19
3D Line Segment

• Use a linear combination of two points
• Parametric representation
• P P1 t (P2 - P1), (0 ? t ? 1)

typedef struct Point P1 Point P2 Segment
P2
P1
20
3D Ray

• Line segment with one endpoint at infinity
• Parametric representation
• P P1 t V, (0 lt t lt ?)

typedef struct Point P1 Vector V Ray
V
P1
21
3D Line

• Line segment with both endpoints at infinity
• Parametric representation
• P P1 t V, (-? lt t lt ?)

typedef struct Point P1 Vector V Line
V
P1
22
3D Plane

• A combination of three non-linear points
• Implicit representation
• N d 0, or
• ax by cz d 0
• N is the plane normal
• Unit-length vector
• Perpendicular to plane

typedef struct Vector N Distance d Plane
P3
P2
P1
23
3D Polygon

• Area inside a sequence of coplanar points
• Triangle
• Quadrilateral
• Convex
• Star-shaped
• Concave
• Self-intersecting
• Points are in counter-clockwise order
• Holes (use gt 1 polygon struct)

typedef struct Point points int npoints
Polygon
24
3D Sphere

• All points at distance r from point (cx, cy,
  cz)
• Implicit representation
• (x - cx)2 (y - cy)2 (z - cz)2 r 2
• Parametric representation
• x r cos(?) cos(?) cx
• y r cos(?) sin(?) cy
• z r sin(?) cz

typedef struct Point center Distance
radius Sphere