Mathematics for Computer Graphics

1
Mathematics for Computer Graphics

• Computer Graphics Algorithms make use of many
  mathematical concepts and techniques.
• Coordinate Reference Frames
• Graphics packages typically require that
  coordinate parameters be specified with respect
  to Cartesian reference frames. But in many
  applications, non-Cartesian coordiante systems
  are useful Spherical, Cylindriacl and others.

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• We must convert any non-Cartesian description to
  Cartesian coordinates.
• The most common two orientations for a Cartesian
  screen are

(0,0)
x
y
y
x
(0,0)
The origin in the lower-left corner of the screen.
The origin at the upper-left corner of the screen.
It is possible in some graphics packages to
select a position, such as the center of the
screen, for the coordinate origin.
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• (2) Polar Coordinates in the xy Plane
• The coordinate position is specified with a
  radial distance r from the coordinate origin, and
  an angular displacement ? from the horizontal.
• Positive angular displacements are
  counterclockwise, and negative angular
  displacements are clockwise.

r
?
4

• Angle ? can be measured in degrees, with one
  complete counterclockwise revolution about the
  origin as 360o.
• The relation between Cartesian and polar
  coordinates is shown here

5

• Using the definition of the trigonometric
  functions, we transform from polar coordinates to
  Cartesian coordinates with the expressions
• x r cos ? y r sin ?
• The inverse transformation from Cartesian to
  polar coordinates is

P
y
r
?
O
x
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• Example
• ? 60, r 32
• --gt x r cos ? 32 0.5 16
• y r sin ? 32 0.866 27.712
• Example
• x 60, y 24
• --gt r sqr(602 242) sqr(4176) 64.621
• ? tan-1(24/60) tan-1(0.4) 21.8o

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• Angular values can be specified in degree or they
  can be given in dimensionless units (radians), ?
  s / r, where s is the length of the circular
  arc subtending ?, and r is the radius of the
  circle.
• Total angular distance around point P is the
  length of the circle perimeter 2?r/r 2?.

s
?
r
P
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• (3) Three Dimensional Cartesian Reference Frames
• Most computer graphics packages use the right
  handed Cartesian Coordinates, in which the right
  hand thumb points in the positive z direction.

y axis
P
x axis
z axis
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(4) Points and Vectors

• A point is a position specified with coordinate
  values in some reference frame, so that the
  distance from the origin depends on the choice of
  reference frame.
• Here we have coordinate specification in two
  reference frames. In frame A, point coordinates
  are given by the values of the ordered pair
  (x,y). If frame B, the same point has coordinate
  (0,0) and the distance to the origin of frame B
  is 0.

P
Frame B
y
OB
x
OA
Frame A
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• A vector is defined as the difference between two
  points positions.

P2
y2
V
y1
P1
x1
x2
V P2 - P1 (x2 - x1, y2 - y1) (Vx,
Vy) Where Vx and Vy are the projections of the x
and y axes.
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• The vector magnitude can be calculated using
  Pythagorean theorem
• The direction of the vector can be given in terms
  of the angular displacement from the x axis as
• A vector has the same properties (magnitude and
  direction) no matter where we position the
  vector.
• The vector magnitude is independent of the
  coordinate system.

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Example P1 (5,2) , P2 (8,4) --gt V
(3,2) V sqr(32 22) 3.605 ? tan-1(2/3)
33.69o
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• If we change the coordinate representation, the
  values for the vector components change.
• For a three-dimensional Cartesian space, we
  calculate the vector magnitude as
• Vector direction is given with the direction
  angles, ?, ?, ?, that the vector makes with each
  of the coordinate axes.

x
V
?
?
y
?
z
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• Direction angles are the positive angles that the
  vector makes with each of the positive coordinate
  axes. We calculate these angles as
• These three values are called the direction
  cosines of the vector.
• One property of them is
• cos2? cos2? cos2? 1
• By definition, the sum of two vectors is obtained
  by adding corresponding components
• V1V2 (V1xV2x, V1yV2y, V1zV2z)

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• Example
• A vector in the three dimensional space, P1
  (3,4,2) , P2 (9,12,16)
• Vx(9-3)6, Vy(12-4)8, Vz (16-2) 14
• --gt V sqr(6282142) sqr(3664196)sqr(296)
  17.2
• cos ? Vx/V 6/17.2 0.348 cos2?
  cos2? cos2? ?
• --gt ? cos-1(0.348) 69.583
  (.348)2(.465)2(.813)2.998
• cos ? Vy/V 8/17.2 0.465
• --gt ? cos-1(0.465) 62.282
• cos ? Vx/V 14/17.2 0.813
• --gt ? cos-1(0.813) 35.515

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• Two vectors can be added geometrically by
  positioning the two vectors end to end and
  drawing the resultant vector from the start of
  the first one to the tip of the second one.
• Scalar multiplication of a three-dimensional
  vector is defined as
• aV (aVx , aVy, aVz)
• There is no addition of vectors and scalars.

y
V2
y
V1 V2
V2
V1
V1
x
x
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• Examples
• V1 (4,7,-2), V2 (3,-3,8)
• Addition
• V1 V2 (7,4,6)
• Scalar Product
• 4V1 (16,28,-8)
• Dot Product

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• There are two types of vector multiplication
• (Dot product) Vector multiplication for producing
  a scalar is defined as
• V1 . V2 V1 V2 cos ?, 0 ? ? ? ?
• where ? is the angle between the two vectors.
• In addition to the coordinate-independent
• form of the scalar product, we can
• express this product in specific
• coordinate representation
• V1 . V2 V1xV2x V1yV2y V1zV2z

V1
V2
?
V2 cos ?
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• The dot product of a vector with itself is simply
  another statement of the Pythagorean theorem.
• The scalar product of two vectors is zero if and
  only if they are perpendicular.
• Dot products are commutative
• V1.V2 V2.V1
• Dot products are distributive with respect to
  vector addition
• V1 . (V2 V3 ) V1 . V2 V1 . V3

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• (Cross Product) Vector multiplication for
  producing a non scalar is defined as
• V1 ? V2 u V1 V2 sin ?, 0 ? ? ? ?
• where u is a unit vector (its magnitude is 1)
  that is perpendicular to both V1 and V2. The
  direction of u is determined by the right hand
  rule.
• The cross product of two
• vectors is a vector that is
• perpendicular to the plane
• of the two vectors and with
• magnitude equal to the area
• of the parallelogram formed
• by the two vectors.

V1 ? V2
V2
?
V1
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• We can also express the cross product in terms of
  vector components in a specific reference frame,
  in the Cartesian coordinate system we calculate
  the components of the cross product as
• V1 ? V2 (V1yV2z - V1zV2y , V1zV2x - V1xV2z ,
  V1xV2y - V1yV2x)
• The cross product of any two parallel vectors is
  zero.
• The cross product of a vector with itself is
  zero.
• The cross product is not commutative but
  anticommutative
• V1 ? V2 - (V2 ? V1 )

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• The cross product is not associative
• V1 ? (V2 ? V3 ) ? (V1 ? V2 ) ? V3
• The cross product is distributive with respect to
  vector addition
• V1 ? (V2 V3 ) (V1 ? V2 ) (V1 ? V3 )

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(5) Matrices

• A matrix is a rectangular array of quantities
  (numbers, functions or numerical expressions).
• We identify matrices according to the number of
  rows and number of columns.
• To multiply a matrix A by a scalar value s, we
  multiply each element ajk by the scalar.
• Matrix addition is defined only for matrices that
  have the same size, by adding corresponding
  elements.

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• We can multiply an mn A matrix by an np B
  matrix to form the matrix product np AB. We then
  obtain the product matrix by forming sums of the
  products of the elements in the row vectors of A
  with the corresponding elements in the column
  vectors of B.
• AB ? BA because

while