comp trans lec

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Composite transformations:

The heart of computer graphics

Lecturer: Dr Dan Cornford

[email protected]

http://wiki.aston.ac.uk/DanCornford

CS2150, Computer Graphics, Aston University, Birmingham, UKhttp://wiki.aston.ac.uk/C2150

October 12, 2009

Dan Cornford Composite Transformations 1/18


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Outline: Composite transformations

Combining transformations.

Matrix times a matrix.

How to transform things in OpenGL.

Efficiency.

Inverse transformations.



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Composition of transformations

Big advantage of homogeneous coordinates is that

transformations can be very easily combined.

All that is required is multiplication of the transformationmatrices.

This makes otherwise complex transformations very easy

to compute.



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For instance if we wanted to rotate an object about some

point, p.

To simply write down the transformation would be quite a

challenge!However, the transformation matrix can be computed by:

1 translate object by p,2 rotate object by angle ,3 translate object by p.


M i l i li i


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Matrices multiplication

Given two matrices, A and B if we want to multiply Bby A

(that is form AB) then if A is (nm), Bmust be (m p).

This produces a result, C= AB, which is (n p), withelements:

cij =m

k=1

aikbkj

Basically we multiply the first row of A with the first column

of Band put this in the c1,1 element of C. And so on ....

The identity matrix (for multiplication) is denoted I:

I=

1 0 00 1 0

0 0 1

.

This means that AI= A = IA.


M t i b i


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Matrices basics

From GameDev.net

Unlike scalar multiplication, AB= BA this means that

applying a scaling after a translation will not have the sameeffect as a translation after a scaling.

Multiplying composite transformation matrices is at the core of allmodern graphics libraries.


C iti f t f ti


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Recall our previous example of rotation about a point p,

which can be written as:

T(p)R()T(p) =

1 0 px0 1 py0 0 1

cos sin 0sin cos 0

0 0 1

1 0 px0 1 py0 0 1

=cos sin px(1 cos ) + py sin sin cos py(1 cos ) pxsin

0 0 1

.

Note the ordering of the transformation matrices.

For those interested, MATLAB provides an excellent

platform for investigating these sort of transformations,

since its natural matrix format makes things very easy to

code.


T f ti d O GL


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Transformations and OpenGL

Basic commands are:

glTranslate#(dx, dy, dz)

glScale#(sx, sy, sz)

glRotate#(angle, x, y, z)

The matrices are applied to the vertices in the oppositeorder they are specified (pre-multiplied by existing

transformation matrix).

Can define our own matrices: glLoadMatrix and

glMultMatrix.


Transformations and OpenGL stacks


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From GameDev.net

There are two important matrices in OpenGL

GL PROJECTION and GL MODELVIEW.

OpenGL stores these as composite transformation

matrices.

OpenGL maintains matrix stacks which are used to store

the composite transformation matrices (of all

transformations so far specified).




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Consider the following code fragment:

g l L o a d I d e n t i t y ( ) ;

g lR ot at e ( 45 .0 , 0 .0 , 0 .0 , 0 .0 ); /* a pp ly t ra ns fo rm at io n R */g l T ra n sl a t e ( 2. 0 , 0 .0 , 0 . 0) ; /* a pp ly t r an s fo r ma t io n T */

g l B e g i n ( G L _ P O I N T S ) ;

g l V e r t e x 3 f ( v ) ; /* d ra w t ra ns fo rm ed v er te x v */

glEnd();

The effect of this is:the GL MODELVIEW matrix successively contains I, IR, and

finally I R T;

the transformed vertex is I R T v = (I (R (T v)));

the transformations to vertex v effectively occur in theopposite order to which they were specified.

In reality only a single multiplication of the vertex by the GL MODELVIEW

matrix occurs; the Rand T matrices are already multiplied into a

single composite matrix before being applied to v.




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From GameDev.net

We use glPushMatrix() and glPopMatrix() to save and

remove the current matrix from its stack.

This is normally the GL MODELVIEW matrix.

Stacks play a large role in defining scene graphs and

animating models.


Efficiency your job


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Efficiency your job

The composition of transformations involving translation,

scaling and rotation leads to transformation matrices M of

the general form:

M=r1,1 r1,2 txr2,1 r2,2 ty

0 0 1

.

How many elementary operations (+,, , /) are required

to multiply a vector [x, y,w]

by this matrix?


Efficiency


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Efficiency

The composition of transformations involving translation,

scaling and rotation leads to transformation matrices M of

the general form:

M=r1,1 r1,2 txr2,1 r2,2 ty

0 0 1

.

Multiplying a point [x, y,w] by this matrix would take nine

multiplies and six adds.


Efficiency


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Efficiency

Using the fixed structure in the final row of the matrix, the

actual transformation can be written:

x = r1,1x+ r1,2y+ tx ,

y = r2,2y+ r2,1x+ ty ,

This takes four multiplies and four adds.

Parallel hardware adders and multipliers effectively remove

this concern in modern graphics.


Inverse transformations


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If a general transformation (which may be composite) is

given by the 3 3 matrix M, then the inverse

transformation, which maps the new image back to the

original, untransformed one is given by M1.

The matrix inverse is defined so that MM1 = Iwhere I is

the 3 3 identity matrix,

I=

1 0 0

0 1 0

0 0 1

.

Inverses only exist for one to one transformations, for many

operations they are not defined.




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The translation matrix has inverse:

T1 =

1 0 tx0 1 ty0 0 1

,

The scaling matrix has inverse:

S1 =

1sx

0 0

0 1sy 0

0 0 1

.

You should be able to work out the inverse of the rotation

matrix ...


Inverse of composite transformations


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Inverse of composite transformations

For a composite transformation matrix C= AB the inverseis less obvious.

We know (AB)1 = B1A1 from linear algebra thus:

C1 = B1A1 .

This is logical the inverse transformations are applied in

the opposite order.


Summary


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Summary

Having finished this lecture you should:

be able to compute composite transformation matrices;

understand the way OpenGL implements transformations;

be able to improve the efficiency of applying

transformations;

understand the role and concept of an inverse

transformation.

If you have trouble with the maths use the extra example sheets

(and solutions) on the web site.


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