Mathematics for Computer Graphics

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Mathematics for Computer Graphics
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Spaces

• Computer graphics is concerned with the
  representation of geometric elements, e.g.,
  points and line segments.
• Types of abstract spaces found useful for their
  representation include
• Vector space
• Affine space
• Euclidean space

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Vector Space

• Two types of entities scalars and vectors
• Scalars ordinary real numbers and operations (,
  ) on them which are commutative, associative,
  and distributive
• Vectors set of elements with two operations
• Vector-vector addition (commutative and
  associative)
• Scalar-vector multiplication (distributive)

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Vector-vector Addition

• u v v u
• u ( v w ) (u v ) w
• there is 0 (identity) such as for any vector v, 0
  v v
• for every vector v, there is another vector w
  such as v w 0 . w is written " -v " (inverse)
• Example in R3

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Scalar-vector Multiplication

• (ab)v a (bv)
• 1v v
• (a b)v av bv
• a(v w) av aw

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One Particular Vector Space the Plane We chose
an origin, and every point of the plane is
matched with the vector that links the point to
the chosen origin

• Addition of two vector (the parallelogram rule,
  or the head-to-tail rule)

• Scalar-vector multiplication

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Linear Independence of Vectors

• A linear combination of n vectors
  is a vector
• The vectors are linearly independent if the only
  set of scalars such that
  is
• The greatest number of linearly independent
  vectors we can find in a vector space is the
  dimension of the space. Any set of n
  independently linear vectors in a space of
  dimension n forms a basis. Any vector can be
  expressed uniquely in terms of the basis vectors
  as
  . The scalar represents on this
  basis, which can be transformed to other basis
  using a transformation matrix.

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Affine Space

• Vectors in a vector space have no position or
  distance.
• Adding a third type of entity points to the
  vector space along with a point-to-point
  subtraction gives rise to an affine space that
  can accommodate the concept of positions.
• Given 2 point P and Q, we can form the difference
  of P and Q which lies in the vector Space
• Conversely, given a point P and a vector , we
  can add u to P and get another point in the
  affine space
• Properties to satisfy
• (P v) u P (v u)
• P u P if and only if u 0
• A vector can be written uniquely as
• A point can be written uniquely as

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Euclidean Space

• Affine space still lacks the concept of distance
  between points and length of a vector
• Euclidean space does. It is a space with
  scalars, vectors, and a new operation the inner
  (dot) product, with properties
• Commutative
• Distributive
• if
• 0
• If then and are orthogonal
• Magnitude of a vector is measured as
• Angle between 2 vectors

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2-D Cartesian Coordinate SystemSet of point
labeled with coordinate (x, y) is an affine
space.Two usual Cartesian reference frames

• Math frame

• Computer graphics frame

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Basic 2-D Geometric Transformation

• Translation
• Rotation
• Scaling