# Background Mathematics Aaron Bloomfield CS 445: Introduction to Graphics Fall 2006

advertisement ```Background Mathematics
Aaron Bloomfield
CS 445: Introduction to Graphics
Fall 2006
Mathematical Foundations
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A very brief review of some mathematical tools
we’ll employ
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Geometry (2D, 3D)
Trigonometry
Vector and affine spaces
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Points, vectors, and coordinates
Dot and cross products
Linear transforms and matrices
2
3D Geometry
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To model, animate, and render 3D scenes, we
must specify:
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We’ll look at two types of spaces:
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Location
Displacement from arbitrary locations
Orientation
Vector spaces
Affine spaces
We will often be sloppy about the distinction
3
Vector Spaces
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Given a basis for a vector space:
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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
4
Vectors And Points
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We commonly use vectors to represent:
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Direction (i.e., orientation)
Points in space (i.e., location)
Displacements from point to point
But we want points and directions to behave
differently
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Ex: To translate something means to move it without
changing its orientation
Translation of a point = different point
Translation of a direction = same direction
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Affine spaces vs. Vector spaces
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Vector spaces
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All vectors originate at the origin
Thus, can be denoted with (x,y,z) coordinates
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The origin is absolute
Affine spaces
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The head of the vector
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
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Affine Spaces
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To be more rigorous, we need an explicit notion of
position
Affine spaces add a third element to vector
spaces: points (P, Q, R, …)
Q
Points support these operations
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Point-point subtraction:
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v
Result is a vector pointing from P to Q
Vector-point addition:
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Q-P=v
Result is a new point
P+0=P
P+v=Q
P
Note that the addition of two points is not defined
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Affine Spaces
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Points, like vectors, can be expressed in
coordinates
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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
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Point + point
Scalar * point
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Affine Lines: An Aside
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Parametric representation of a line with a direction
vector d and a point P1 on the line:
P(a) = Porigin + ad
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Restricting 0  a produces a ray
Setting d to P - Q and restricting 0  a  1
produces a line segment between P and Q
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Dot Product
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The dot product or, more generally, inner product
of two vectors is a scalar:
v1 • v2 = x1x2 + y1y2 + z1z2 (in 3D)
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Useful for many purposes
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Computing the length of a vector
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Normalizing a vector, making it unit-length
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u • v = |u| |v| cos(θ)
Checking two vectors for orthogonality
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Divide each coordinate by the length
Computing the angle between two vectors:
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length(v) = |v| = sqrt(v • v)
v
Does the angle equal 90&ordm;?
Projecting one vector onto another
θ
u
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Cross Product
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The cross product or vector product of two vectors is a
vector:
 y1 z 2  y 2 z1 
v1  v 2   ( x1 z 2  x 2 z1)
 x1 y 2  x 2 y1 
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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
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Lighting
Visibility
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Linear Transformations
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A linear transformation:
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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
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Matrices
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By convention, matrix element Mrc is located at row
r and column c:
 M11 M12
 M21 M22
M
 
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Mm1 Mm2
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This is called row-major order
By (OpenGL) convention,
vectors are columns:
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And 2-D matrices are in
column-major order!
 M1n 
 M2n 
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 
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 Mmn 
 v1 

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v   v 2
 v 3 
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Matrices
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Matrix-vector multiplication
transformation to a vector:
applies
a
linear
 M11 M12 M13  vx 
M  v  M 21 M 22 M 23  vy 
M31 M32 M33  vz 
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Recall how to do matrix multiplication
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Matrix Transformations
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A sequence or composition of linear transformations corresponds to the product of the
corresponding matrices
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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:
M 1 Mv   v
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3D Scene Representation
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Scene is usually approximated by 3D primitives
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Point
Line segment
Polygon
Polyhedron
Curved surface
Solid object
etc.
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3D Point
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Specifies a location
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Represented by three coordinates
Infinitely small
typedef struct {
Coordinate x;
Coordinate y;
Coordinate z;
} Point;
(x,y,z)
Origin
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3D Vector
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Specifies a direction and a magnitude
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Represented by three coordinates
Magnitude ||V|| = sqrt(dx dx + dy dy + dz dz)
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Which is the square root of the dot product
Has no location (in affine spaces!)
typedef struct {
Coordinate dx;
Coordinate dy;
Coordinate dz;
} Vector;
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(dx,dy,dz)
Q
(dx2,dy2 ,dz2)
Dot product of two 3D vectors
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V1&middot;V2 = dx1dx2 + dy1dy2 + dz1dz2
V1&middot;V2 = ||V1 || || V2 || cos(Q)
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3D Line Segment
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Use a linear combination of two points
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Parametric representation:
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P = P1 + t (P2 - P1),
typedef struct {
Point P1;
Point P2;
} Segment;
(0  t  1)
P2
P1
Origin
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3D Ray
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Line segment with one endpoint at infinity
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Parametric representation:
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P = P1 + t V,
(0 &lt;= t &lt; )
typedef struct {
Point P1;
Vector V;
} Ray;
V
P1
Origin
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3D Line
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Line segment with both endpoints at infinity
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Parametric representation:
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P = P1 + t V,
(- &lt; t &lt; )
typedef struct {
Point P1;
Vector V;
} Line;
P1
V
Origin
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3D Plane
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A combination of three non-linear points
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Implicit representation:
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typedef struct {
Vector N;
Distance d;
} Plane;
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N is the plane “normal”
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N = (a,b,c)
N + d = 0, or
ax + by + cz + d = 0
P2
P3
P1
d
Unit-length vector
Perpendicular to plane
Origin 22
3D Polygon
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Area “inside” a sequence of coplanar points
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Triangle
Quadrilateral
Convex
Star-shaped
Concave
Self-intersecting
typedef struct {
Point *points;
int npoints;
} Polygon;
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Points are in counter-clockwise order
Holes (use &gt; 1 polygon struct)
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3D Sphere
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All points at distance “r” from point “(cx, cy, cz)”
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Implicit representation:
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(x - cx)2 + (y - cy)2 + (z - cz)2 = r 2
Parametric representation:
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x = r cos() cos(Q) + cx
y = r cos() sin(Q) + cy
z = r sin() + cz
r
typedef struct {
Point center;
Distance radius;
} Sphere;
Origin
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