1 Points, Vectors, Lines, Spheres and Matrices ©Anthony Steed 2001-2003.

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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