Vectors tools for graphics
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Vectors Tools for Graphics. Vector, Geometry and CG. 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.

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Vectors Tools for Graphics

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Vectors tools for graphics

Vectors Tools for Graphics


Vector geometry and cg

Vector, Geometry and CG

  • 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.


Computer graphics objects

Computer graphics objects

  • Objects to be drawn

    • Shape

    • position

    • orientation

  • fundamental mathematical discipline to aid graphics is

    • vector analysis

    • transformation


Why vector analysis

Why vector analysis


2 d and 3 d coordinate systems

2-D and 3-D coordinate systems


Vector review

Vector Review

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


Vector and point

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

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

Operation with Vectors

  • Add

  • Scale


Linear combination of vectors

Linear Combination of Vectors

  • a v + b w


Affine combination of vectors

Affine combination of vectors

  • A linear combination of vector is affine combination if

  • ex: 3 a + 2 b - 4 c


Convex combination of vectors

Convex combination of Vectors

  • Plus one more requirement

  • ai >= 0 I = 1…m

  • .3a+.7b

  • 1.8a -.8b

  • The 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

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

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 dot product


The angle between two vectors

The Angle Between Two Vectors.


The sign of b c and perpendicularity

The Sign of b.cand Perpendicularity.


The 2d perp vector

The 2D Perp Vector.


The perp dot product

The perp dot product


Orthogonal projections

Orthogonal Projections


Calculate k and m

Calculate K and M


The distance from c to the line

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 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

Properties


Normal

Normal


Finding the normal to a plane

Finding the Normal to a Plane


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