2d and 3d transformations, homogeneous coordinates ... · pdf file2d and 3d transformations,...

2D and 3D Transformations,Homogeneous Coordinates

Lecture 03

Patrick [email protected]

Centre for Image AnalysisUppsala University

Computer GraphicsNovember 6 2006

Patrick Karlsson (Uppsala University) Transformations and Homogeneous Coords. Computer Graphics 1 / 23

Reading InstructionsChapters 4.1–4.9.

Edward Angel.“Interactive Computer Graphics: A Top-downApproach with OpenGL”,Fourth Edition, Addison-Wesley, 2004.


Todays lecture ...in the pipeline


Scalars, points, and vectors

Scalars α, β

Real (or complex) numbers.

Points P, QLocations in space (but no size or shape).

Vectors u, vDirections in space (magnitude but no position).


Mathematical spaces

Scalar fieldA set of scalars obeying certain properties. New scalars can be formedthrough addition and multiplication.

(Linear) Vector spaceMade up of scalars and vectors. New vectors can be created throughscalar-vector multiplication, and vector-vector addition.

Affine spaceAn extended vector space that include points. This gives us additionaloperators, such as vector-point addition, and point-point subtraction.


Data types

Polygon based objectsObjects are described using polygons.

A polygon is defined by its vertices (i.e., points).

Transformations manipulate the vertices, thus manipulates theobjects.

Some examples in 2DScalar α 1 float.

Point P(x , y) 2 floats.

Vector v(x , y) 2 floats.

Matrix M 4 floats.


Transformations

To create and move objects we need to be able to transformobjects in different ways.

There are many classes of transformations.

TransformationsTranslate (Move around.)

Rotate

Scale

Shear (Scaling and rotation.)


TransformationsTranslation

Simply add a translation vector

x ′ = x + dx

y ′ = y + dy

P(x,y)

P’(x’,y’)


TransformationsRotation

P(x , y) in polar coordinates

x = r cos(φ) y = r sin(φ)

P ′(x ′, y ′) in polar coordinates

x ′ = r cos(θ + φ) = r cos(φ) cos(θ) − r sin(φ) sin(θ)

y ′ = r sin(θ + φ) = r cos(φ) sin(θ) − r sin(φ) cos(θ)

Substitute x and y

x ′ = x cos(θ) − y sin(θ)

y ′ = x sin(θ) + y cos(θ)

θφ

r

r

P(x,y)

P’(x’,y’)

x

y


TransformationsRotation

Arbitrary rotationTranslate the rotation axis to the origin.

Rotate.

Translate back.

y

x


TransformationsScaling around the origin

Multiply by a scale factor

x ′ = sxx

y ′ = syy

x

y

P(x,y)

P’(x’,y’)

x x’


TransformationsShear

Shear in the x direction

x ′ = x + y cot(θ)

y ′ = y

x

y

θ

P(x,y) P’(x’,y’)


Affine transforms are linear!

αf (x1 + x2) = αf (x1) + αf (x2)

The linearity implies that in order to move an object we only needto transform the individual vertices making up the object, since theinner points are defined by linear combinations of the vertices.

P0

P1

P(α) = (1− α)P0 + αP1 α = [0, 1]


Vector mathshort revision

The inner (or dot) product is written as u · v . If u · v = 0, thenvectors u and v are orthogonal.

The squared magnitude of a vector v is given by the inner productv · v = ||v ||2.

The angle between two vectors u and v is given byu·v

||u||||v || = cos(θ).

Two nonparallel vectors can generate a third vector that isorthogonal to them both by using the cross product n = u × v .The new vector n can be then be used to create a vector that isorthogonal to both u and n by performing w = u × n. The threevectors u, n, and w are mutually orthogonal.

The magnitude of a cross product gives the sine of the anglebetween the vectors, | sin(θ)| = ||u×v ||

||u||||v || .


Vectors and matrices

Translation „x ′

y ′

«=

„xy

«+

„dxdy

«

Rotation „x ′

y ′

«=

„cos(θ) − sin(θ)sin(θ) cos(θ)

« „xy

«

Scaling „x ′

y ′

«=

„sx 00 sy

« „xy

«

Shearx „x ′

y ′

«=

„1 cot(θ)0 1

« „xy

«


Translation is different!

Translation in 2D „x ′

y ′

«=

„xy

«+

„dxdy

«

“Stepping up one dimension”0@ x ′

y ′

W

1A =

0@ 1 0 dx0 1 dy0 0 1

1A 0@ xyW

1A

If W = 1, then this called a Homogeneous coordinate0@ x ′

y ′

1

1A =

0@ 1 0 dx0 1 dy0 0 1

1A 0@ xy1

1A


Using homogeneous coordinatesTranslation

P′ = T (dx , dy)P

0@ x ′

y ′

1

1A =

0@ 1 0 dx0 1 dy0 0 1

1A 0@ xy1

1A

Rotation

P′ = R(θ)P

0@ x ′

y ′

1

1A =

0@ cos(θ) − sin(θ) 0sin(θ) cos(θ) 0

0 0 1

1A 0@ xy1

1A

Scaling

P′ = S(sx , sy )P

0@ x ′

y ′

1

1A =

0@ sx 0 00 sy 00 0 1

1A 0@ xy1

1A

Shearx

P′ = Hx (θ)P

0@ x ′

y ′

1

1A =

0@ 1 cot(θ) 00 1 00 0 1

1A 0@ xy1

1APatrick Karlsson (Uppsala University) Transformations and Homogeneous Coords. Computer Graphics 17 / 23

Concatenation of transformations

Observe!The order of the matrices (right to left) is important!

P′ = T−1(S(R(T (P)))) = (T−1SRT )P

M = T−1SRT

P′ = MP

y

x


Homogeneous coordinatesWhat about the last element W? x

yW

⇒

αxαyαW

⇒

x/Wy/W

W/W

⇒

x/Wy/W

1

︸︷︷︸

This is called to homogenize.

Cartesian coordinates:(xc

yc

)=

(x/Wy/W

)h

If W = 0: A point at infinity = a vector xy0


Stepping up to three dimensions

Translation and scaling is the same.

Rotation is a little more tricky, and becomes Rx , Ry , and Rz .

Observe that RxRy 6= RyRx .


3D transformations

Translation

P′ = T (dx , dy , dz)P

0BB@x ′

y ′

z′

1

1CCA =

0BB@1 0 0 dx0 1 0 dy0 0 1 dz0 0 0 1

1CCA0BB@

xyz1

1CCA

Scaling

P′ = S(sx , sy , sz)P

0@ x ′

y ′

z′

1A =

0BB@sx 0 0 00 sy 0 00 0 sz 00 0 0 1

1CCA0BB@

xyz1

1CCA


3D transformations

Rotation around the x-axis

P′ = Rx (θ)P

0BB@x ′

y ′

z′

1

1CCA =

0BB@1 0 0 00 cos(θ) − sin(θ) 00 sin(θ) cos(θ) 00 0 0 1

1CCA0BB@

xyz1

1CCA

Rotation around the y-axis

P′ = Ry (θ)P

0BB@x ′

y ′

z′

1

1CCA =

0BB@cos(θ) 0 sin(θ) 0

0 1 0 0− sin(θ) 0 cos(θ) 0

0 0 0 1

1CCA0BB@

xyz1

1CCA

Rotation around the z-axis

P′ = Rz(θ)P

0BB@x ′

y ′

z′

1

1CCA =

0BB@cos(θ) − sin(θ) 0 0sin(θ) cos(θ) 0 0

0 0 1 00 0 0 1

1CCA0BB@

xyz1

1CCA


Lecture 04

Do not miss tomorrows exiting episode:3D viewing and projections .Room P2344 at 13:15 - 15:00.


2d and 3d transformations, homogeneous coordinates ... · pdf file2d and 3d transformations,...

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