CM0304 Graphics I Graphics Hardware I.1 Graphics Systems
CMT107 Visual Computing
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I.3 Vectors and Matrices
Xianfang Sun
School of Computer Science & Informatics
Cardiff University
• Vector Operations
• Vector Geometry
• Vector Projection
➢ 3D Vectors
• Cross Product
• 3D Vector Geometry
➢ Matrices
• Special Matrices
• Matrix Operations
• Determinant
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➢ A vector is a directed line segment, characterised by:
• Direction
• But NOT Position
➢ A vector with length 0 is a zero vector, denoted by 0
• Zero vectors doesn’t have direction.
➢ A vector with length 1 is a unit vector.
➢ A vector u with the same length but opposite direction of
vector v is the negative vector of v, denoted by u = -v.
➢ Two vectors are equal iff they have the same length
and the same direction.
• Two zero vectors are always equal, though their directions are
undefined.
Vector Operations
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➢ A vector u multiplied by a scalar denoted by u has the
same direction of u if >0 and the opposite direction if
<0. The length of u is || times of the length of u.
➢ The sum w of two vectors u and v:
follows the head-to-tail rule. That is,
if the head of u is connected to the tail of v, then w is the
directed line segment from the tail of u to the head of v.
➢ The subtraction of vector v from vector w is the addition of vector
w and vector -v
u = w – v = w + (-v)
Vector Operations
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➢ A nD vector is represented by:
➢ The sum and subtraction of two vectors are:
➢ The multiplication of a vector v by a scalar is defined by
➢ The inner product (dot product, scalar product) of
two vectors is:
vuvuvu +++==
Vector Geometry
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➢ A vector has direction and length.
• The length is defined by
• The direction is parallel to the direction from the origin to the
point in nD Euclidean space.
• The angle between two vectors u and v is calculated
➢ Normalisation of a vector v gives a unit vector v’, which
has length 1:
➢Vectors u and v are orthogonal if , which means that they
are perpendicular to each other, and the angle between these two
vectors are 90.
vvv +++== vvv
Vector Projection
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➢The vector projection wp of a vector w on a nonzero vector v is a
vector parallel to v, defined by
where is a scalar calculated by
➢The vector w then can be represented by the sum of wp and vector
u, which is perpendicular to v (and wp ).
Cross Product
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➢ Denote two 3D vectors and by , the vector
product (also called cross product, outer product) of and is
defined by
➢ If is the angle between and , then the length is:
➢ The direction satisfies right-hand rule
• is perpendicular to both
zyx ],,[=v
http://en.wikipedia.org/wiki/File:Right_hand_rule_cross_product.svg
http://en.wikipedia.org/wiki/File:Right_hand_rule_cross_product.svg
3D Vector Geometry
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➢ A point in 3D space can be represented by a 3D vector:
➢ A directed line segment from to can be represented
by vector:
➢ Let and are two directed line segments on a plane,
then the normal direction is determined by the cross
product of and :
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➢ A matrix is a rectangular array of scalars, arranged in rows and
columns. The individual items in a matrix are called its elements or
entries. The number of rows and columns are referred to as the row
and column dimensions.
➢ The following matrix A has row dimension m and column dimension
n, or simply, m x n dimension. aij is an element of the matrix
➢ The transpose of an m x n matrix A, denoted by AT, is the n x m
matrix obtained by interchanging the rows and columns of A.
➢ To save the space, the matrix is often written as A = [aij]mxn, or
simply, A = [aij], if the dimension of the matrix is implicitly known.
Special Matrices
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➢ A square matrix is a matrix which has the same row and column
dimension.
➢ A symmetric matrix is a square matrix that is equal to its transpose.
Let A=[aij] be a symmetric matrix, then A = A
T. Its elements satisfy
➢ A diagonal matrix is a matrix (usually square matrix) in which the
elements outside the main diagonal are all zero, i.e., A=[aij],
➢ An identity matrix, denoted by I, is a square diagonal matrix with
1’s on the diagonal and 0’s elsewhere
= if ,0
otherwise.0
Special Matrices
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➢ A row matrix is a matrix of dimension 1 x n. It is also called a row
➢ A column matrix is a matrix of dimension m x 1, also called a
column vector.
Matrix Operations
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➢ Scalar-Matrix multiplication is defined by multiplying each element
by the scalar
➢ Matrix-Matrix Addition of two matrices of the same dimension is
defined by adding corresponding elements of the two matrices
➢ Matrix-Matrix Multiplication of an m x l dimensional matrix A and
an l x n dimensional matrix B is defined by
➢ Inverse of a Square Matrix A is a square matrix B, such that
• Denote by B = A-1
ba +=+= BAC
Orthogonal Matrix
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➢ An orthogonal matrix is a square matrix with real entries whose
columns and rows are orthogonal unit vectors (i.e., orthonormal
➢ Equivalently, a matrix Q is orthogonal if its transpose is equal to
its inverse:
which entails
Determinant
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➢ The determinant is a value associated with a square matrix,
denoted by det(A), det A, or |A|. It is defined as
• where Aij is the (i, j) minor matrix of A, which is obtained by
deleting the ith row and the jth column of A.
➢ The determinant of a 2 x 2 matrix is calculated by
➢ The determinant of a 3 x 3 matrix is calculated by
312213332112322311322113312312332211
11131312121111
aaaaaaaaaaaaaaaaaa
+−=+−== AAAA
Cross Product Using Determinant
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➢ The cross product of two 3D vectors can be calculated using
determinant as follows:
122112211221
yxyxxzxzzyzy
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➢ What are the characteristics of a vector?
➢ What operations are defined for vectors.
➢ How to calculate the vector projection onto
another vector?
➢ How to calculate cross product? What is the
geometric meaning of cross product?
➢ How to do matrix operations?
➢ What is a orthogonal matrix?
➢ How to calculate determinant?
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