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Lecture 1

This document discusses different types of geometric transformations. It defines geometric transformations as one-to-one mappings that usually map a set to itself, with an inverse mapping. Geometric transformations are classified based on whether they preserve angles and distances (isometric), angles and ratios of distances (similarity), or parallelism and collinearity (affine and projective). Homogeneous coordinates are introduced to represent points in a way that allows geometric transformations to be written as matrix multiplications. Isometric transformations preserve lengths, angles and areas through rotations, translations and reflections, while similarity transformations additionally include uniform scaling.

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60 views10 pages

Lecture 1

This document discusses different types of geometric transformations. It defines geometric transformations as one-to-one mappings that usually map a set to itself, with an inverse mapping. Geometric transformations are classified based on whether they preserve angles and distances (isometric), angles and ratios of distances (similarity), or parallelism and collinearity (affine and projective). Homogeneous coordinates are introduced to represent points in a way that allows geometric transformations to be written as matrix multiplications. Isometric transformations preserve lengths, angles and areas through rotations, translations and reflections, while similarity transformations additionally include uniform scaling.

Uploaded by

Hola Gola
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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ME F318 Computer Aided Design

Lecture 1: Geometric Transformations

BITS Pilani Dr. Amit R. Singh


Pilani Campus

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What is a Geometric Transformation?

• A one-to-one mapping of elements from a set.


• Usually, it maps a set to itself.
• The inverse mapping must exist.

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Classification of Geometric
Transformations

• Based on dimensions of the set.


• Based on some “special property” of the Transformation.

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Classification based on “special property”

• Isometric Transformations: preserve angles and distances.


• Similarity Transformations: preserve angles and ratios of
distances.
• Affine Transformations: preserve parallelism.
• Projective Transformations: preserve collinearity.

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Homogeneous Coordinates

Definition
If (x, y ) are the Euclidean coordinates of a point P, in R2 , and if
(x1 , x2 , x3 ) are any three real numbers such that x1 /x3 = x and
x2 /x3 = y , then the triple (x1 , x2 , x3 ) is said to be a set of
homogeneous coordinates for P.

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Why Homogeneous Coordinates?

• One motivation: to define intersection of parallel lines.


• It “homogenizes” equations:
→ l1 x + l2 y + l3 = 0 becomes l1 x1 + l2 x2 + l3 x3 = 0.
→ a11 x 2 + 2a12 xy + a22 y 2 + 2a13 x + 2a23 y + a33 = 0.
becomes
a11 x12 + 2a12 x1 x2 + a22 x22 + 2a13 x1 x3 + 2a23 x2 x3 + a33 x32 = 0.
• It helps us write translations as a matrix multiplication.

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Isometric Transformations

Definition
The general form of an 2D isometric transformation is
 0   
x  cos θ − sin θ tx x
y 0  =   sin θ cos θ ty  y 
1 0 0 1 1

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Properties of Isometric Transformations

• When  = 1, orientation is preserved and the transformation is


a rigid body motion. Also called as Euclidean Transformations.
• When  = −1, orientation is reversed and the transformation
is a reflection composed with a rigid body motion.
• This transformation preserves lengths, angles and areas. “Iso”
= same and “Metry” = distance.

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Similarity Transformations

Definition
The general form of a similarity transformation is
 0   
x s cos θ −s sin θ tx x
y 0  =  s sin θ s cos θ ty  y 
1 0 0 1 1

The upper left 2 × 2 matrix is a rotation matrix R multiplied by a


scalar s.

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Properties of Similarity Transformations

• It preserves shape but not necessarily the size.


Mathematically, it preserves ratio of lengths and areas.
• The parameter s provides isotropic scaling.

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