CS 428: Fall 2010 Introduction to ... - Rutgers University

CS 428: Fall 2010

Introduction to

Computer GraphicsComputer Graphics

Geometric Transformations

9/15/2010 1Andrew Nealen, Rutgers, 2010

Topic overview

� Image formation and OpenGL (last week)

� Modeling the image formation process

� OpenGL primitives, OpenGL state machine

� Transformations and viewing� Transformations and viewing

� Polygons and polygon meshes

� Programmable pipelines

� Modeling and animation

� Rendering

Andrew Nealen, Rutgers, 2010 9/15/2010 2

Topic overview

� Image formation and OpenGL

� Transformations and viewing (next weeks)

� Linear algebra review, Homogeneous coordinates

� Geometric + projective transformations� Geometric + projective transformations

� Viewing, Viewports, Clipping

� Polygons and polygon meshes

� Programmable pipelines

� Modeling and animation

� RenderingAndrew Nealen, Rutgers, 2010 9/15/2010 3

Transformations in CG

� Specify placement of objects in the world

� relative to the configuration in which they are

defined

� Allow for reuse of objects � Allow for reuse of objects

in different places, sizes

� Specify the camera

position

� Specify the camera model

(projection)


Transformations in CG

� The “where” is specified by translations and

rotations (= rigid body motions)

� Shape changes include

� For now we will only use linear deformations

� Linear algebra!Andrew Nealen, Rutgers, 2010 9/15/2010 5

Representations in CG

� Computations should not depend on

coordinate system (such as midpoint/origin)

� Need careful accounting of points and vectors

� Both (3 tuples of floating point values)3ℜ∈� Both (3 tuples of floating point values)

� Vectors

� Displacements, velocities, directions, trajectories,

surface normals, etc.

� Points

� Locations!


3ℜ∈

Vector/point operations

� Vector + vector = vector

� Point + vector = point

� Point + point = invalid!

� Street address analogy

� Point – point = vector

� Works!Andrew Nealen, Rutgers, 2010 9/15/2010 7

Vector review

� [p + q]i = pi + qi addition

� [s p]i = s ∙ pi scalar multiplication

� || p || = sqrt[ (pi)2 ] length


Vector review

� p ∙ q = Σ pi ∙ qi dot product

� Normalization


θ

p

q

p

pp =ˆ

θcos⋅⋅ qp

Perpendicular vectors

, 0

( , ) ( , )v v v vx y y x⊥

< >=

= ⇒ = ± −

v w

v v

v

v⊥In 2D only!


Linear combination + Basis

� Linear combination

� λ1 ∙ v1 + λ2 ∙ v2 + … + λn ∙ vn with λi∈ R

� Linear independence of vectors v1, …, vn

� λ ∙ v + … + λ ∙ v = 0 only when λ = … = λ = 0� λ1 ∙ v1 + … + λn ∙ vn = 0 only when λi = … = λ n = 0

� Basis of n-dimensions is a set of n linearly

independent vectors

� Every vector in Rn has a unique set of λ’s to

represent it → Cartesian coordinates


Inner (dot) product

� Defined for vectors:

, || || || || cosθ< >= ⋅ ⋅v w v w

L

θ

L v

wL

cosθ || ||

,

|| ||L

=

< >=

w

v w

v

Projection of w onto v


v

q

q’

2 2 2

:

(1) dist( , ) || ||L ′+ = −0

Pythagoras

q q q p

l

Distance between point and line

p0

q’

2 2 2

22

2

,(2)

|| ||

dist( , ) || ||

,|| || .

|| ||

L

L

< − >=

′⇒ = − − =

< − >= − −

0

0

00

q p v

v

q q q p

q p vq p

v

L


Representation of a plane

in 3D space

� A plane π is defined by a normal n and one

point in the plane p0.

� A point q ∈ plane ⇔ < q – p0 , n > = 0

� The normal n is perpendicular to all vectors in � The normal n is perpendicular to all vectors in

the plane


n

p0

q

π

Distance between point and plane

� Geometric way:

� Project (q - p0) onto n!

0| , |dist

< − >=

q p n


n

p0

q

π

dist =n

Coordinates

� Connect drawing plane/space with R2 or R3

� Coordinate origin and axes are problem

specific

� Example: orthogonal coordinates in the lower � Example: orthogonal coordinates in the lower

corner of this room

� Affine spaces have

� No fixed origin

� No fixed axes

� (which is not the case in linear spaces)


CoordinatesAffine space

� “An affine space is a vector space that's

forgotten its origin” – John Baez

� In R3, the origin, lines and planes through the

origin and the whole space are linear origin and the whole space are linear

� points, lines and planes in general as well as the

whole space are the affine subspaces.

Linear subspace Affine subspace9/15/2010 17Andrew Nealen, Rutgers, 2010

PrimitivesPoints


PrimitivesLines


PrimitivesTriangles


PrimitivesShapes


PrimitivesShapes … are tessellated


PrimitivesPositioning

� Absolute coordinates?


PrimitivesPositioning

� Transformation +

relative coordinates

� Translation

� Rotation

� Scaling

� Shearing

� Affine maps /

Transformations!


Affine mapsAffine combinations

� The set

is an affine combination of vectors v (or of

=⋅=∈ ∑ ∑= =

n

i

n

i

iii mitvvVv0 0

1,| λλ

is an affine combination of vectors vi (or of

points pi).

v0

v1

vp0

p

p1


Affine maps Barycentric coordinates

� Given and affine space A with coordinate

system B={p0,…pn}

� For a point with

the λ are known as barycentric coordinates

∑=

⋅=n

i

i pp0

1λ ∑=

=n

i

i

0

1λ

the λi are known as barycentric coordinates

� Physical interpretation:

� Points pi have mass λi → p ist the centroid (= center of

mass)

p0

p

p1110 =+ λλ

01

0

1

01

1

0

pp

pp

pp

pp

−

−=

−

−=

λ

λ


p0 p1

p2

p

( )( )( )( )( )( )( )( )210

201

210

210

,,

,,

,,

,,

pppA

pppA

pppA

pppA

∆∆

=

∆∆

=

λ

λ

Affine maps Barycentric coordinates

( )( )( )( )( )( )210

102

210

,,

,,

,,

pppA

pppA

pppA

∆∆

=

∆

λ221100 pppp ⋅+⋅+⋅= λλλ

( )( ) ( ) ( )02012102

1,, pppppppA −×−=∆


Affine mapsConvex hull

� The set

is the convex hull co{p0,...,pn} of points p0,...,pn

{ }

=≥=⋅== ∑ ∑= =

n

i

n

i

iiiin niundpppppco0 0

0 ,...,0,0,1,|,..., λλλ

� The convex hull contains all convex

combinations of the points

� Convex combinations =

affine combinations /w

barycentric coordinates

greater/equal to zero p0

p1

p2

p3

p4

p5


Affine maps…as linear maps

� A map Φ: Rn→ Rm is affine

� when Φ can be represented as Φ(v)=A(v)+b

where A is a linear map and b ∈ Rm

� Affine maps have a linear part (multiplication) � Affine maps have a linear part (multiplication)

and a translation (additive)

Linear

transformation Translation

+

=

0

0

0

222120

121110

020100

'

'

'

z

y

x

z

y

x

aaa

aaa

aaa

z

y

x


Affine transformations

� Preserve parallel lines

� lines → lines, planes → planes

� Might not preserve length and angles

� But do preserve relative length along lines� But do preserve relative length along lines

� If they do preserve length and angles then the

transformation is an isometry

� Affine = linear + translation


Affine maps…as linear maps

� Leads to the use of projective geometry

� 2D points and vectors represented as

(x, y, w) → homogeneous coordinates

w = 1 pointw = 1 point

w = 0 vector

� Point (0, 0, 0) not allowed, so domain

R3 – {(0, 0, 0)}

� If w ∈ (0,1] then (x, y, w) → (x/w, y/w, 1)

A point


What is w ?2D case!

� A kind of a type

� Points + “points at

infinity”

� Points at infinity are not� Points at infinity are not

affected by translation

� Infinite # of points

correspond to (x, y, 1)

→ {(tx, ty, t) | t ≠ 0}

� Line through origin

– {origin}Andrew Nealen, Rutgers, 2010 9/15/2010 32

Homogeneous coordinates

� Works nicely for points and vectors

� Adding and scaling works too

� More in “projections”, where w ∈ [0,1]


Linear transformation

� Purely linear transformation

� Origin does not move

� New coordinate axes are lin. comb. of old ones


Linear transformation

� Purely linear transformation


Affine transformationas a linear transformation + translation in n dimensions

� Origin moves → translation


Affine transformationas a linear transformation in n+1 dimensions

� Origin moves → translation


What is so great about this?

� Easy to implement

� Checks for errors in the implementation

� Can always check the w coordinate to make sure

that points and vectors remain unchangedthat points and vectors remain unchanged

� Unified representation for linear + translation

� Can compose many transformations into a single

matrix through concatenation


M = Mrot ∙ Mscale ∙ Mtranslate ∙ …

CS 428: Fall 2010 Introduction to ... - Rutgers University

Documents

Transcript of CS 428: Fall 2010 Introduction to ... - Rutgers University