Vectors Tools for Graphics

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Vectors Tools for Graphics
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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.

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Computer graphics objects

• Objects to be drawn
• Shape
• position
• orientation
• fundamental mathematical discipline to aid
  graphics is
• vector analysis
• transformation

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Why vector analysis
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2-D and 3-D coordinate systems
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Vector Review
The difference between two points is a vector v
Q - P
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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.

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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 )

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Operation with Vectors

• Add
• Scale

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Linear Combination of Vectors

• a v b w

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Affine combination of vectors

• A linear combination of vector is affine
  combination if
• ex 3 a 2 b - 4 c

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Convex combination of Vectors

• Plus one more requirement
• ai gt 0 I 1m
• .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.

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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
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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),
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The dot product
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The Angle Between Two Vectors.
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The Sign of b.c and Perpendicularity.
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The 2D Perp Vector.
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The perp dot product
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Orthogonal Projections
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Calculate K and M
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The distance from C to The Line
the distance from a point C to the line through
A in the direction v is
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Applications of Projection Reflections
r e - m. Because e a - m, this gives r a -
2m.
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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.
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Properties
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Normal
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Finding the Normal to a Plane