Vectors Tools for Graphics

To review vector arithmetic, and to relate vectors to objects of interest in graphics.

To relate geometric concepts to their algebraic representations.

To describe lines and planes parametrically.To distinguish points and vectors properly.To exploit the dot product in graphics topics.To develop tools for working with objects in 3D

space, including the cross product of two vectors.

Vector, Geometry and CG

Computer graphics objects

Objects to be drawn Shape position orientation

fundamental mathematical discipline to aid graphics is vector analysis transformation

Why vector analysis

2-D and 3-D coordinate systems

Vector Review

The difference between two points is a vector: v = Q - P;

Vector and Point

Turning this around, we also say that a point Q is formed by displacing point P by vector v; we say that v offsets P to form Q. Algebraically, Q is then the sum:

Q = P + v.

The sum of a point and a vector is a point: P + v = Q.

Vector representation

At this point we represent a vector through a list of its components: an n-dimensional vector is given by an n-tuple:

w = (w 1 , w 2 , . . . , w n )

Operation with Vectors

AddScale

Linear Combination of Vectors

a v + b w

Affine combination of vectors

A linear combination of vector is affine combination if

ex: 3 a + 2 b - 4 c

Convex combination of Vectors

Plus one more requirementai >= 0 I = 1…m.3a+.7b1.8a -.8bThe set of coefficients a 1 , a 2 , . . . , a m

is sometimes said to form a partition of unity, suggesting that a unit amount of material is partitioned into pieces.

The Magnitude of a vector

Note that if w is the vector from point A to point B, then |w| will be the distance from A to B

Unit vector

It is often useful to scale a vector so that the result has a length equal to one. This is called normalizing a vector, and the result is known as a unit vector. For example, we form the normalized version of a, denoted , by scaling it with the value 1/|a|:

Ex. a = (3, -4),

The dot product

The Angle Between Two Vectors.

The Sign of b.c and Perpendicularity.

The 2D Perp Vector.

The perp dot product

Orthogonal Projections

Calculate K and M

The distance from C to The Line

the distance from a point C to the line through A in the direction v is:

The Cross Product of Two Vectors

The cross product (also called the vector product) of two vectors is another vector. It has many useful properties, but the one we use most often is that it is perpendicular to both of the given vectors. The cross product is defined only for three-dimensional vectors.

Properties

Normal

Finding the Normal to a Plane