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2D Transforms: Vector Spaces, Transforms in 2D, and Composing Transforms, Slides of Computer Graphics

The basics of 2d transforms, including vector spaces, transforms in 2d, composing transforms, and homogeneous coordinates. Topics include translation, scaling, rotation, and shearing, as well as their representation with matrices and homogeneous coordinates.

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2012/2013

Uploaded on 04/30/2013

aradhana 🇮🇳

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2D Transforms

•Reviewed the math we’re going to

use in this course

•Points

•Vectors

•Matrices

•Linear interpolation

•Rays, planes, etc.

Last Time

Today

•Vector spaces and coordinate frames

•Transforms in 2D

•Composing Transforms

Vector Spaces

•Let’s think for a minute about what x

and y coordinates really mean

(x,y)

(u,v)

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Partial preview of the text

Download 2D Transforms: Vector Spaces, Transforms in 2D, and Composing Transforms and more Slides Computer Graphics in PDF only on Docsity!

2D Transforms

• Reviewed the math we’re going to

use in this course

• Points

• Vectors

• Matrices

• Linear interpolation

• Rays, planes, etc.

Last Time

Today

• Vector spaces and coordinate frames

• Transforms in 2D

• Composing Transforms

Vector Spaces

• Let’s think for a minute about what x

and y coordinates really mean

(x,y)

(u,v)

y (^) j i

Vector Spaces

• For illustration, let’s flip things around

(u,v)

y (^) j

= (-u,v)(x,y) y

Vector Spaces

• Any pair of non-parallel, non-anti-

parallel vectors can define a vector space in 2D

• We’re used to thinking about spaces

defined by orthogonal, normalized, axis-aligned vectors

• But there’s no reason this is the only

way to do things

Vector Space Terms

• Linear Vector Space: Any space made

up of vectors and scalars

• Euclidean Space: Vector space with a

distance metric

• Affine Space: Vector space with an

origin

• The Cartesian plane is both affine and

Euclidean

• We call this type of space a frame

2D Transforms

• What am I talking about when I say

“transforms”?

Translation (x,y)

(x’,y’)

Scaling
Rotation

General Form

• The transformations we consider are of

the following form:

Transformation Matrix

• Remember that the transformation matrix

describes the change in vector space:

x’

y’

Object vs. World

Space

• Let’s stop and think about why we’re

doing this...

• We can define the points that make up

an object in “object space”

• Whatever is most convenient, often

centered around the origin

• Then, at run time, we can put the

objects where we want them in “world space”

Homogeneous

Coordinates

• We refer to this as a homogeneous

coordinate:

• This mathematical construct allows us to

• Represent affine transforms with a single

matrix

• Do calculations in projective space

(vectors are unique only up to scaling)

Homogeneous

Coordinates

• For points, w must be non-zero

• If w=1, the point is “normalized”

• If w!=1, can normalize by

Vectors in

Homogeneous

Coordinates

• Remember last time, I mentioned that

it would be useful that we could represent points and vectors the same way?

• Here’s the payoff

• Can use homogeneous coordinates to

represent vectors, too

• What is w?

• Remember, vectors don’t have a

“position”

Vectors in

Homogeneous

• Since vectors don’t have a position,Coordinates

they should not be affected by translation

• What about rotation/scaling?

• Set w=0 for vectors:

These have no effect

Summing up

Translation

• We will represent translation with a

matrix of the following form:

u is the x-offset v is the y-offset

Scaling

• Want to stretch or shrink the entire

space in one or more dimensions

• α = stretch in x, β = stretch in y

Original α=1, β=

α=2, β=

α=1, β=

Scaling

• Scaling is centered around the origin

• Points either get pulled toward the

origin or pushed away from it

Scaling

• So, given what we know, how would we

construct a scaling matrix?

• Assume we have an object centered

around the origin, and want to scale it by α in x, and β in y

• x’ = αx

• y’ = βy

Summing up Scaling

• We will represent scaling with a matrix

of the following form:

α is the scale factor in the x-direction β is the scale factor in the y-direction

Rotation

• Like scaling, rotations are centered

about the origin

Rotation

Composing

Transforms

• So why did we go through the trouble to

use homogeneous coordinates for our points, and do our transforms using square transformation matrices?

• To make composing transforms easy!

• Composing 2 transforms is just

multiplying the 2 transform matrices WARNING: The order in which matrix multiplicationstogether are performed may (and usually does) change the result! (i.e. they are not commutative)

2D Transforms: Vector Spaces, Transforms in 2D, and Composing Transforms, Slides of Computer Graphics

Related documents

Partial preview of the text

Download 2D Transforms: Vector Spaces, Transforms in 2D, and Composing Transforms and more Slides Computer Graphics in PDF only on Docsity!

2D Transforms

• Reviewed the math we’re going to

• Points

• Vectors

• Matrices

• Linear interpolation

• Rays, planes, etc.

Last Time

Today

• Vector spaces and coordinate frames

• Transforms in 2D

• Composing Transforms

Vector Spaces

• Let’s think for a minute about what x

Vector Spaces

• For illustration, let’s flip things around

Vector Spaces

• Any pair of non-parallel, non-anti-

• We’re used to thinking about spaces

• But there’s no reason this is the only

Vector Space Terms

• Linear Vector Space: Any space made

• Euclidean Space: Vector space with a

• Affine Space: Vector space with an

• The Cartesian plane is both affine and

• We call this type of space a frame

2D Transforms

• What am I talking about when I say

General Form

• The transformations we consider are of

• Remember that the transformation matrix

Object vs. World

Space

• Let’s stop and think about why we’re

• We can define the points that make up

• Whatever is most convenient, often

• Then, at run time, we can put the

Homogeneous

Coordinates

• We refer to this as a homogeneous

• This mathematical construct allows us to

• Represent affine transforms with a single

• Do calculations in projective space

Homogeneous

Coordinates

• For points, w must be non-zero

• If w=1, the point is “normalized”

• If w!=1, can normalize by

Vectors in

Homogeneous

Coordinates

• Remember last time, I mentioned that

• Here’s the payoff

• Can use homogeneous coordinates to

• What is w?

• Remember, vectors don’t have a

Vectors in

Homogeneous

• Since vectors don’t have a position,Coordinates

• What about rotation/scaling?

• Set w=0 for vectors:

Summing up

Translation

• We will represent translation with a

Scaling

• Want to stretch or shrink the entire

• α = stretch in x, β = stretch in y

Scaling

• Scaling is centered around the origin

• Points either get pulled toward the

Scaling

• So, given what we know, how would we

• Assume we have an object centered

• x’ = αx

• y’ = βy

Summing up Scaling