2IV60 Computer Graphics Basic Math for CG

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2IV60 Computer GraphicsBasic Math for CG

• Jack van Wijk
• TU/e

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Overview

• Coordinates, points, vectors
• Matrices

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nD Space

• nD Space ?n (typically)
• n number of dimensions
• Examples
• 1D space time, along a line or curve
• 2D space plane, sphere
• 3D space the world we live in
• 4D space 3D time

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Coordinates

• 2D Cartesian coordinates

x
y
(x,y)
(x,y)
y
x
Standard Screen (output,
input)
HB A-1
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Polar coordinates
y
(x,y)
r
?
x
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3D coordinates 1

• 3D Cartesian coordinates

(x,y,z)
z
(x,y,z)
z
x
y
x
y
Right-handed Left-handed
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3D coordinates 2

• 3D Cartesian coordinates

(x,y,z)
(x,y,z)
z (middle)
z (middle)
y (index)
x (thumb)
x (thumb)
y (index)
Right-handed Left-handed
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3D coordinates 3

• Cylinder coordinates

(x,y,z)
z
y
x
?
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3D coordinates 4

• Spherical coordinates

(x,y,z)
z
f
r
y
x
?
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Points

• Point position in nD space
• Notation P (HB), also P, p, p en p
• (x,y,z) (HB), also (x1, x2, x3), (Px, Py, Pz),
• (r, ?, z), ( r, ?, ?),

HB A-2
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Vectors 1

• Vector
• arrow
• multiple interpretations (displacement, velocity,
  force, )
• has a magnitude and direction
• has no position
• Notation V (HB), also V, v, v en v
• (Vx, Vy, Vz) (HB), also (x, y, z), (x1, x2, x3)

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Vectors 2
y
P2
y2
P1
y1
V directed line segment, or difference between
two points
x
x1
x2
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Vectors 3

• Length of a vector

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Vectors 4

• Direction of a vector Direction angles.
• Unit vector V
• Magnitude info is removed, direction is kept.

z
g
b
y
x
a
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Vector addition

• Add components, put vector head to tail

y
y
W
VW
W
V
V
x
x
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Scalar multiplication of vector

• Multiplication components with scalar s

y
y
2V
V
x
x
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Combining vector operations

• Infinite line through P with direction V
• L(t) P Vt, t ??

V
y
P
t
x
t parameter along line t ?a, b line segment
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Vector multiplication 1

• Scalar product or dot product
• Product of parallel components, gives 1 real
  value

W
V
q
W cos q
V
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Vector multiplication 2

• Scalar product

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Vector multiplication 3

• Vector product or cross product gives a vector
  (in 3D)
• n perpendicular to V and W

V?W
W
q
V
n
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Vector multiplication 5

• Scalar product

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Vector multiplication 6

• Scalar product
• scalar
• Test if vectors are perpendicular
• cos
• project,

• Vector product
• vector
• Get a vector perpendicular to two given vectors
• sin
• surface area,

HB A-2
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Exercise 1

• Given a point P.
• Requested Reflect a point Q with respect to P.

Q
W P Q Q Q 2W 2P Q or P (P
Q )
P
Q
We dont need coordinates!
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Exercise 1

• Given a point P.
• Requested Reflect a point Q with respect to P.

Q
Alternative P is halfway Q and Q P (Q
Q)/2 2P Q Q Q 2P Q
P
Q
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Exercise 2

• Given a line L L(t) P Vt .
• Requested Reflect a point Q with respect to L.

We know Q Q 2 W W L(t) Q W. V 0
L(t)
V
y
P
t
x
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Exercise 2
We know L(t) P Vt Q Q 2 W W L(t)
Q W. V 0
Substitute to get t (L(t) Q).V 0 (P Vt
Q).V 0 V .V t (P Q).V 0 t ((Q P).V) /
(V .V)
Then Q Q 2 (P Vt Q) 2P Q
V((Q P).V / V .V)
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Steps to be made

• Write down what you know
• Eliminate, substitute, etc. to get he result
• Check the result
• Does it make sense?
• Is there a simpler derivation?

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Exercise 3

• Given a triangle with vertices P, Q en R, where
  the angle PQR is perpendicular.
• Requested Rotate the triangle around the line PQ
  over an angle a . What is the new position R of
  R?

P
Q
R
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Circle in space
y
C(?)
r
?
x
A1, B1, A.B 0

• C(?) (r cos ?, r sin ?, 0)
• (0,0,0) r cos ? (1,0,0) r sin ?
  (0,1,0)
• P r cos ? A r sin ? B

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Exercise 3
A (R Q) / R Q B (R Q)?(P Q) / (R
Q)?(P Q) R Q R Q cos ? A R
Q sin ? B
P
Q
R
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Use

• Scalar product, cross product
• coordinate independent definitions
• vector algebra
• Dont use
• arccos, arcsin
• y f(x)

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Very short intro to Linear Algebra

• System of linear equations
• Such systems occur in many, many applications.
• They are studied in Linear Algebra.

HB A-5
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Very short intro to Linear Algebra

• System of linear equations
• Typical questions
• Given u, v, w, what are x, y, z?
• Can we find a unique solution?

HB A-5
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Very short intro to Linear Algebra

• System of linear equations
• Crucial in computer graphics
• Transforming geometric objects
• Change of coordinates

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Example transformation
y
P (x,y)
x
y
P
V
U
y
x
x
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What is a matrix?

• Matrix
• Mathematical objects with operations
• Matrix in computer graphics
• Defines a coordinate frame
• Defines a transformation
• Handy tool for manipulating transformations

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Matrix

• Matrix rectangular array of elements
• Element quantity (value, expression, function,
  )
• Examples

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Matrix

• r ? c matrix r rows, c columns
• mij element at row i and column j.

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Matrix

• r ? c matrix r rows, c columns
• mij element at row i and column j.

r ? c elements
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Matrix

• r ? c matrix r rows, c columns
• mij element at row i and column j.

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Matrix

• r ? c matrix r rows, c columns
• mij element at row i and column j.

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Matrix
Column vector matrix with c 1 Used for
vectors (and points)
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Matrix

• Matrix as collection of column vectors

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Operations on matrices

• Multiplication with scalar
• simple
• Addition
• simple
• Matrix-matrix multiplication
• More difficult, but the most important

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Scalar Matrix multiplication

• Matrix M multiplied with scalar s

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Scalar Matrix addition
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Scalar Matrix addition

• Just add elements pairwise
• A and B must have the same number of rows and
  columns
• Generalization of vector and scalar addition

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Matrix Matrix multiplication
j 1 k?

• cij dot product of row vector ai and column
  vector bj
• columns A must be the same as rows B n p
• C m ? q matrix

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Matrix Matrix multiplication
Example
i 3
j 2
i3, j2
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Matrix Matrix multiplication
Example
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Matrix Matrix multiplication
Example
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Matrix Matrix multiplication
Example
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Matrix Matrix multiplication

• AB ? BA
• Matrix matrix multiplication is not commutative!
• Order matters!
• A(BC) ABAC
• Matrix matrix multiplication is distributive

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Example transformation sequence (coordinate
version)
Unclear! Error-prone!
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Example transformation sequence (matrix vector
version)
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Example transformation sequence (compact matrix
vector version)
Matrix vector notation allows for compactness
and genericity!
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Matrix Transpose
Transpose matrix Interchange rows and columns. r
? c matrix M ? c ? r matrix MT
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Matrix Inverse
Simple algebra puzzle. Let ax b. What is x?
Linear algebra puzzle. Let MU V. What is U?
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Matrix Inverse
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Matrix Inverse examples
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Matrix Inverse examples
Does not exist, cannot be inverted.
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Matrix Inverse

• Does not always exist
• In general if determinant M 0, matrix
  cannot be inverted
• Inverse for n 1 and n 2 easy
• Inverse for higher n use library function
• Important special case orthonormal matrix.

HB A-5
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Overview

• Coordinates, points, vectors
• definitions, operations, examples
• Matrices
• definitions, operations, examples