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Points, Vectors, Lines, Spheres and Matrices
©Anthony Steed 2001-2003
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• Points• Vectors• Lines• Spheres• Matrices
• 3D transformations as matrices
• Homogenous co-ordinates
Overview
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Basic Maths
In computer graphics we need mathematics both for describing our scenes and also for performing operations on them, such as projection and various transformations.
Coordinate systems (right- and left-handed), serve as a reference point.
3 axes labelled x, y, z at right angles.
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Co-ordinate Systems
X
Y
Z
Right-Handed System(Z comes out of the screen)
X
Y
Z
Left-Handed System(Z goes in to the screen)
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Points, P (x, y, z)
Gives us a position in relation to the origin of our coordinate system
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Vectors, V (x, y, z)
Represent a direction (and magnitude) in 3D space
Points != VectorsVector +Vector = VectorPoint – Point = VectorPoint + Vector = PointPoint + Point = ?
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Vectors, V (x, y, z)
v
w
v + w
Vector additionsum v + w
v2v
(-1)v (1/2)V
Scalar multiplication of vectors (they remain parallel)
v
w
Vector differencev - w = v + (-w)
v
-w
v - w
x
y
Vector OP
P
O
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Vectors V
Length (modulus) of a vector V (x, y, z)|V| =
A unit vector: a vector can be normalised such that it retains its direction, but is scaled to have unit length: ||V of modulus
Vvector ^
V
VV
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Dot Product
u · v = xu ·xv + yu ·yv + zu ·zv
u · v = |u| |v| coscos = u · v/ |u| |v|
This is purely a scalar number not a vector. What happens when the vectors are unit What does it mean if dot product == 0 or == 1?
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Cross Product
The result is not a scalar but a vector which is normal to the plane of the other 2
Can be computed using the determinant of:
u x v = i(yvzu -zvyu), -j(xvzu - zvxu), k(xvyu - yvxu)
Size is u x v = |u||v|sin Cross product of vector with it self is null
uuu
vvv
zyx
zyx
kji
uuu
vvv
zyx
zyx
kji
uuu
vvv
zyx
zyx
kji
uuu
vvv
zyx
zyx
kji
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Parametric equation of a line (ray)
Given two points P0 = (x0, y0, z0) and P1 = (x1, y1, z1) the line passing through them can be expressed as:
P(t) = P0 + t(P1 - P0)x(t) = x0 + t(x1 - x0)
y(t) = y0 + t(y1 - y0)
z(t) = z0 + t(z1 - z0)
=
With - < t <
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Equation of a sphere
a
bchypotenuse
a2 + b2 = c2
P
xp
yp
(0, 0)
r
x2 + y2 = r2
Pythagoras Theorem:
Given a circle through the origin with radius r, then for any point P on it we have:
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Equation of a sphere
If the circle is not centred on the origin:
(0, 0)
P
xp
yp
r
xc
yc ab
a
ba2 + b2 = r2
We still have
buta = xp- xc
b = yp- yc
So for the general case (x- xc)2 + (y- yc)
2 = r2
(xp,yp)
(xc,yc)
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Equation of a sphere
Pythagoras theorem generalises to 3D giving
a2 + b2 + c2 = d2 Based on that we can easily
prove that the general equation of a sphere is:
(x- xc)2 + (y- yc)
2 + (z- zc)2 = r2
and at origin: x2 + y2 + z2 = r2
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Matrix Math
©Anthony Steed 2001
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Vectors and Matrices
Matrix is an array of numbers with dimensions M (rows) by N (columns)• 3 by 6 matrix• element 2,3
is (3)
Vector can be considered a 1 x M matrix•
zyxv
100025114311212003
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Types of Matrix
Identity matrices - I
Diagonal
1001
1000010000100001
Symmetric
• Diagonal matrices are (of course) symmetric
• Identity matrices are (of course) diagonal
4000010000200001
fecedbcba
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Operation on Matrices
Addition• Done elementwise
Transpose• “Flip” (M by N becomes N by M)
s d r c
q b p a
s r
q p
d c
b a
389
724
651
376
825
941T
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Operations on Matrices
Multiplication• Only possible to multiply of dimensions
– x1 by y1 and x2 by y2 iff y1 = x2
• resulting matrix is x1 by y2
– e.g. Matrix A is 2 by 3 and Matrix by 3 by 4• resulting matrix is 2 by 4
– Just because A x B is possible doesn’t mean B x A is possible!
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Matrix Multiplication Order
A is n by k , B is k by m C = A x B defined by
BxA not necessarily equal to AxB
k
l
ljilij bac1
*
*****
*****
*.
*.*.*.*.*.
*****
*...
*.*.*.*.*.
*****.....
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Example Multiplications
____
____
1011
12
101132
__________________
010001100
111013322
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Inverse
If A x B = I and B x A = I then
A = B-1 and B = A-1
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3D Transforms
In 3-space vectors are transformed by 3 by 3 matrices
ziyfxczhyexbzgydxa
ihg
fed
cba
zyx
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Scale
Scale uses a diagonal matrix
Scale by 2 along x and -2 along z
zcybxac
ba
zyx
000000
1046200
010002
543
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Rotation
Rotation about z axis
Note z values remain the same whilst x and y change
1000)θcos()θsin(-0)θsin()θcos(
Y
X
θ yx
yθ cosxθ sin yθ sin-xθ cos
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Rotation X, Y and Scale
About X
About Y
Scale (should look familiar)
)θcos()θsin(-0
)θsin()θcos(0
001
)θcos(0)θsin(010
)θsin(-0)θcos(
cb
a
000000
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Homogenous Points
Add 1D, but constrain that to be equal to 1 (x,y,z,1)
Homogeneity means that any point in 3-space can be represented by an infinite variety of homogenous 4D points• (2 3 4 1) = (4 6 8 2) = (3 4.5 6 1.5)
Why?• 4D allows as to include 3D translation in matrix
form
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Homogenous Vectors
Vectors != Points Remember points can not be added If A and B are points A-B is a vector Vectors have form (x y z 0) Addition makes sense
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Translation in Homogenous Form
Note that the homogenous component is preserved (* * * 1), and aside from the translation the matrix is I
1
1010000100001
1 czbyax
cba
zyx
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Putting it Together
R is rotation and scale components T is translation component
TR
TTT
RRR
RRR
RRR
.
1
0
0
0
321
987
654
321
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1432
1432010000100001
1
1432010000100001
1000001001000001
1
YZXYZXZYX
1442
1000011001000001
1442
1000001001000001
1432010000100001
1
YZXZYXZYX
Order Matters
Composition order of transforms matters• Remember that basic vectors change so
“direction” of translations changed
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Exercises
Calculate the following matrix: /2 about X then /2 about Y then /2 about Z (remember “then” means multiply on the right). What is a simpler form of this matrix?
Compose the following matrix: translate 2 along X, rotate /2 about Y, translate -2 along X. Draw a figure with a few points (you will only need 2D) and then its translation under this transformation.
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Matrix Summary
Rotation, Scale, Translation Composition of transforms The homogenous form
