Chapter 4.1 Mathematical Concepts

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Chapter 4.1Mathematical Concepts
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Applied Trigonometry

• Trigonometric functions
• Defined using right triangle

h
y
a
x
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Applied Trigonometry

• Angles measured in radians
• Full circle contains 2p radians

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Applied Trigonometry

• Sine and cosine used to decompose a point into
  horizontal and vertical components

y
r
r sin a
a
x
r cos a
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Applied Trigonometry

• Trigonometric identities

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Applied Trigonometry

• Inverse trigonometric functions
• Return angle for which sin, cos, or tan function
  produces a particular value
• If sin a z, then a sin-1 z
• If cos a z, then a cos-1 z
• If tan a z, then a tan-1 z

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Applied Trigonometry

• Law of sines
• Law of cosines
• Reduces to Pythagorean theorem wheng 90 degrees

a
c
b
b
g
a
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Vectors and Matrices

• Scalars represent quantities that can be
  described fully using one value
• Mass
• Time
• Distance
• Vectors describe a magnitude and direction
  together using multiple values

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Vectors and Matrices

• Examples of vectors
• Difference between two points
• Magnitude is the distance between the points
• Direction points from one point to the other
• Velocity of a projectile
• Magnitude is the speed of the projectile
• Direction is the direction in which its
  traveling
• A force is applied along a direction

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Vectors and Matrices

• Vectors can be visualized by an arrow
• The length represents the magnitude
• The arrowhead indicates the direction
• Multiplying a vector by a scalar changes the
  arrows length

2V
V
V
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Vectors and Matrices

• Two vectors V and W are added by placing the
  beginning of W at the end of V
• Subtraction reverses the second vector

W
V
V W
W
W
V
V W
V
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Vectors and Matrices

• An n-dimensional vector V is represented by n
  components
• In three dimensions, the components are named x,
  y, and z
• Individual components are expressed using the
  name as a subscript

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Vectors and Matrices

• Vectors add and subtract componentwise

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Vectors and Matrices

• The magnitude of an n-dimensional vector V is
  given by
• In three dimensions, this is

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Vectors and Matrices

• A vector having a magnitude of 1 is called a unit
  vector
• Any vector V can be resized to unit length by
  dividing it by its magnitude
• This process is called normalization

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Vectors and Matrices

• A matrix is a rectangular array of numbers
  arranged as rows and columns
• A matrix having n rows and m columns is an n ? m
  matrix
• At the right, M is a2 ? 3 matrix
• If n m, the matrix is a square matrix

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Vectors and Matrices

• The entry of a matrix M in the i-th row and j-th
  column is denoted Mij
• For example,

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Vectors and Matrices

• The transpose of a matrix M is denoted MT and has
  its rows and columns exchanged

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Vectors and Matrices

• An n-dimensional vector V can be thought of as an
  n ? 1 column matrix
• Or a 1 ? n row matrix

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Vectors and Matrices

• Product of two matrices A and B
• Number of columns of A must equal number of rows
  of B
• Entries of the product are given by
• If A is a n ? m matrix, and B is an m ? p matrix,
  then AB is an n ? p matrix

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Vectors and Matrices

• Example matrix product

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Vectors and Matrices

• Matrices are used to transform vectors from one
  coordinate system to another
• In three dimensions, the product of a matrix and
  a column vector looks like

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Vectors and Matrices

• The n ? n identity matrix is denoted In
• For any n ? n matrix M, the product with the
  identity matrix is M itself
• InM M
• MIn M
• The identity matrix is the matrix analog of the
  number one
• In has entries of 1 along the main diagonal and 0
  everywhere else

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Vectors and Matrices

• An n ? n matrix M is invertible if there exists
  another matrix G such that
• The inverse of M is denoted M-1

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Vectors and Matrices

• Not every matrix has an inverse
• A noninvertible matrix is called singular
• Whether a matrix is invertible can be determined
  by calculating a scalar quantity called the
  determinant

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Vectors and Matrices

• The determinant of a square matrix M is denoted
  det M or M
• A matrix is invertible if its determinant is not
  zero
• For a 2 ? 2 matrix,

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Vectors and Matrices

• The determinant of a 3 ? 3 matrix is

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Vectors and Matrices

• Explicit formulas exist for matrix inverses
• These are good for small matrices, but other
  methods are generally used for larger matrices
• In computer graphics, we are usually dealing with
  2 ? 2, 3 ? 3, and a special form of 4 ? 4 matrices

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Vectors and Matrices

• The inverse of a 2 ? 2 matrix M is
• The inverse of a 3 ? 3 matrix M is

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Vectors and Matrices

• A special type of 4 ? 4 matrix used in computer
  graphics looks like
• R is a 3 ? 3 rotation matrix, and T is a
  translation vector

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Vectors and Matrices

• The inverse of this 4 ? 4 matrix is

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The Dot Product

• The dot product is a product between two vectors
  that produces a scalar
• The dot product between twon-dimensional vectors
  V and W is given by
• In three dimensions,

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The Dot Product

• The dot product satisfies the formula
• a is the angle between the two vectors
• Dot product is always 0 between perpendicular
  vectors
• If V and W are unit vectors, the dot product is 1
  for parallel vectors pointing in the same
  direction, -1 for opposite

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The Dot Product

• The dot product of a vector with itself produces
  the squared magnitude
• Often, the notation V 2 is used as shorthand for
  V ? V

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The Dot Product

• The dot product can be used to project one vector
  onto another

V
a
W
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The Cross Product

• The cross product is a product between two
  vectors the produces a vector
• The cross product only applies in three
  dimensions
• The cross product is perpendicular to both
  vectors being multiplied together
• The cross product between two parallel vectors is
  the zero vector (0, 0, 0)

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The Cross Product

• The cross product between V and W is
• A helpful tool for remembering this formula is
  the pseudodeterminant

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The Cross Product

• The cross product can also be expressed as the
  matrix-vector product
• The perpendicularity property means

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The Cross Product

• The cross product satisfies the trigonometric
  relationship
• This is the area ofthe parallelogramformed byV
  and W

V
V sin a
a
W
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The Cross Product

• The area A of a triangle with vertices P1, P2,
  and P3 is thus given by

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The Cross Product

• Cross products obey the right hand rule
• If first vector points along right thumb, and
  second vector points along right fingers,
• Then cross product points out of right palm
• Reversing order of vectors negates the cross
  product
• Cross product is anticommutative

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Transformations

• Calculations are often carried out in many
  different coordinate systems
• We must be able to transform information from one
  coordinate system to another easily
• Matrix multiplication allows us to do this

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Transformations

• Suppose that the coordinate axes in one
  coordinate system correspond to the directions R,
  S, and T in another
• Then we transform a vector V to the RST system as
  follows

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Transformations

• We transform back to the original system by
  inverting the matrix
• Often, the matrixs inverse is equal to its
  transposesuch a matrix is called orthogonal

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Transformations

• A 3 ? 3 matrix can reorient the coordinate axes
  in any way, but it leaves the origin fixed
• We must at a translation component D to move the
  origin

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Transformations

• Homogeneous coordinates
• Four-dimensional space
• Combines 3 ? 3 matrix and translation into one 4
  ? 4 matrix

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Transformations

• V is now a four-dimensional vector
• The w-coordinate of V determines whether V is a
  point or a direction vector
• If w 0, then V is a direction vector and the
  fourth column of the transformation matrix has no
  effect
• If w ? 0, then V is a point and the fourth column
  of the matrix translates the origin
• Normally, w 1 for points

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Transformations

• The three-dimensional counterpart of a
  four-dimensional homogeneous vector V is given by
• Scaling a homogeneous vector thus has no effect
  on its actual 3D value

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Transformations

• Transformation matrices are often the result of
  combining several simple transformations
• Translations
• Scales
• Rotations
• Transformations are combined by multiplying their
  matrices together

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Transformations

• Translation matrix
• Translates the origin by the vector T

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Transformations

• Scale matrix
• Scales coordinate axes by a, b, and c
• If a b c, the scale is uniform

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Transformations

• Rotation matrix
• Rotates points about the z-axis through the angle
  q

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Transformations

• Similar matrices for rotations about x, y

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Transformations

• Normal vectors transform differently than do
  ordinary points and directions
• A normal vector represents the direction pointing
  out of a surface
• A normal vector is perpendicular to the tangent
  plane
• If a matrix M transforms points from one
  coordinate system to another, then normal vectors
  must be transformed by (M-1)T

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Geometry

• A line in 3D space is represented by
• S is a point on the line, and V is the direction
  along which the line runs
• Any point P on the line corresponds to a value of
  the parameter t
• Two lines are parallel if their direction vectors
  are parallel

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Geometry

• A plane in 3D space can be defined by a normal
  direction N and a point P
• Other points in the plane satisfy

N
Q
P
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Geometry

• A plane equation is commonly written
• A, B, and C are the components of the normal
  direction N, and D is given by
• for any point P in the plane

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Geometry

• A plane is often represented by the 4D vector (A,
  B, C, D)
• If a 4D homogeneous point P lies in the plane,
  then (A, B, C, D) ? P 0
• If a point does not lie in the plane, then the
  dot product tells us which side of the plane the
  point lies on

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Geometry

• Distance d from a point P to a lineS t V

P
d
V
S
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Geometry

• Use Pythagorean theorem
• Taking square root,
• If V is unit length, then V 2 1

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Geometry

• Intersection of a line and a plane
• Let P(t) S t V be the line
• Let L (N, D) be the plane
• We want to find t such that L ? P(t) 0
• Careful, S has w-coordinate of 1, and V has
  w-coordinate of 0

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Geometry

• If L ? V 0, the line is parallel to the plane
  and no intersection occurs
• Otherwise, the point of intersection is