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GEOMETRIC MODELINGAND 2D

TRANSFORMATIONLecture 2- Prof. Ehsan T Esfahani

Spring 2016

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Mathematical background

Goal:

Study of geometrical properties such as shape, size,

properties of space, and positioning of CAD model

By the end of this lecture(s) you should be able to perform

the following 2D operations:

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Definitions

Scalar (): Numbers representing magnitude and

quantities such as length, area, volume, speed

Vectors (): Set of scalars [ ] representing bothdirection and magnitude

Points (P): Representation of location in space

[ ]Scalar: Latin alphabets

Points: () Vector: [ ]

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Operations on points and vectors:

Vector operations

Addition:

,

+ + + +

Scalar Multiplication:

Vector multiplications (dot and cross)

Operations on Points

Point-point subtraction results in a vector, This is the sameas point-vector addition

Point-point addition is not defined

, > 1

Q-P = + =Q

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Vector Space

Vector Space is a set of vector on which two operations

are defined:

Vector Addition

Scalar Multiplication

Vector space lack position specification

We can not precisely define 3D geometry in vector space

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Affine Space

Space elements:

Vector space

Points

There is no special point

Defined operations:

Vector-Vector addition

Scalar-vector multiplication

Point-vector addition (equivalent to point-point subtraction)

Scalar-scalar operations

We defined all the 2D transformation in the affine space

Parallelism and ratio of lengths

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Parametric Line (Affine space)

Set of all points ( ) passing through

in the direction of vector

All the points on the line (P, S, R,) can

be defined using different values of a

Generalization (Affine Sum)

+

+ (1 ) + and + 1

= , & = 1

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ConvexityAn object is convexifffor any two points in the object all

points on the line segments between these points are also

in the object

Convex Not Convex

Convex Hull Convex hall of a set X of points is thesmallest convex set that contains X

= , & = 1 i

1 23 456

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Affine Space

The universe of parallel lines

No angle (there is no dot product)

No measurement of length (there is no dot product) No Special point (origin)

Look at affine space as a vector space with no

commitment to any origin point

= , & = 1

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Euclidian Space

Affine Space + inner product

Using inner product we can find:

Distance Angle

Orthogonal projection

Length of a vector (distance between two points)

Angle between two vectors (or lines)

,

. > 0

.

cos.

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Orthogonal Projection

Direction of : Same as Amplitude of : cos cos and cos . .

.

.

Cross Product , sin 2

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Linear Combination

if we have several vectors , , . . . , , and scalars ,, . . . , , we can form the linear combination

=

Set of vectors , , . . . , are linear independent iff

=

0 , 0

This means that one vector can not be represented in

terms of other vectors

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Dimension of Vector Space

Minimum number of linearly independent vectors needed

to span the space

Spanning the space: representing any possible vectorsin the space

Any set of linearly independent vectors form a basis

For a given basis, any vector in that space can be

uniquely represented by linear combination of basis

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Frame

Basis (, , . . . , ) + origin ()

Every vector can be written as

=

Every point can be written as

+ =

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Homogenous Coordinates

n+1 dimension to represent n dimensional space

Examples in (Can be generalized to )

Points wx,wy,wz,w , , , 0

Vectors , , , 0 [, , ] Easy to distinguish points and vecotrs

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Using Frames in HC (2D Example)

Points

1

Vectors

0

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Points and Vector Relationship

Vector + Vector = Vector

Point + Vector = Point

Point Point = Vector

0+0 + 0

1+0 + 1

1

1

0