Group theory for scaling , translation and rotation

model · 2011-12-19 23:00:52

Are groups used for translation , scaling , mirrored and rotation. if so then how? can any one please explain me via an example .
Thanks

bobbym · 2011-12-20 00:11:38

Hi model;

Usually matrices are used for that.

Bob · 2011-12-20 01:33:13

hi model,

Does a set of transformations form a group under combination of transforms?

Answer: Sometimes. (maybe always?)

eg. The set of translations form a group.

(i) There is an identity, the 'stay where you are' translation.

(ii) Each translation has an inverse translation.

(iii) A combination of two translations is also a translation So the set is closed.

(iv) To combine three translations you would just use the rules of arithmetic so combining is associative.

Similarly the set of rotations about a single point.

But the set of reflections is not a group because two reflections do not make another reflection.

You may be wondering: "Is the set of all possible transformations, a group?"

I'm not sure.

Identity, inverse and closure are all ok. But I'm unsure about associativity.

I'll think about it.

EDIT: I think associativity does hold. I'm now working on a rigorous proof.

FURTHER EDIT:

Transformations that are not bijective will not have inverses.

Rotations, reflections, translations, glide reflections, shears, stretches and enlargements all appear to be ok.

But it is possible to define a transformation that maps the whole plane onto, for example, a single line. Such a transformation would have no inverse.

Bob

Last edited by Bob (2011-12-20 10:36:45)

Bob · 2011-12-24 00:23:22

hi

I think what follows is a proof that all bijective transformations of the plane may be combined associatively.

The proof came to me fairly quickly but seemed too easy, so I've been trying to find a loophole ever since.

As I haven't found one, I thought I'd post it. Comments welcome, but, if I'm wrong, break it to me gently please.

See first diagram.

Each circle represents the points of a cartesian plane.

P, Q and R are transformations such that all points are transformed and no two points are transformed to the same point.
(they are bijective mappings)

A, B, C and D are points in the plane such that

P(A) = B Q(B) = C R(C) = D

I shall adopt the 'reverse order' convention for the mappings namely P (on A) followed by Q will be written QP(A).

When a point is transformed by one transformation and the result transformed again by another, the result will, of course, be another point; and since each transformation is bijective, the combination will be too. Thus, for example, QP must also represent a transformation of points in the plane.

To Prove:

R(QP) = (RQ)P

Proof:

QP(A)= Q(B) = C

Therefore, R(QP)(A) = R(C) = D

RQ(B) = R(C) = D

Therefore (RQ)P(A) = RQ(B) = D

Therefore, R(QP) = (RQ)P

Examples.

second diagram

R1 and R2 are rotations of 180 about the highlit (highlighted?) points. T is a translation of 3 across and 4 up.

If F is translated by T and then rotated by R2 the resulting shape must be congruent to F but rotated so R2T is another rotation of 180 around the point shown.

Thus we can see that (R2T)R1 = R2(TR1)

third diagram

R is a reflection in the line x = 5

T is the translation 3 across and 4 up.

T(RR) = T since the reflections 'cancel out'.

TR is shown. It is a glide reflection; specifically, reflect in the line x = 6.5, and 'glide' up parallel to this line 4 units.

R(T) is shown. If you apply the glide reflection to this you have to reflect in x = 6.5 and go up 4. This brings the shape to the same position as T(RR). So T(RR) = (TR)R.

Bob

Last edited by Bob (2011-12-24 02:00:42)

anonimnystefy · 2011-12-24 00:34:33

hi bob

haven't got any chance to talk to you.why do you go online-offline all the time?

Bob · 2011-12-24 00:50:23

hi Stefy,

Sorry if you've missed me. I've been busy with lots of jobs so it's difficult to say when I'll be on-line.

Bob

anonimnystefy · 2011-12-24 01:41:08

yes,i understand that.

but as a true Englishman you should be able to see the tense i've used in post #5.if the last sentnce were like this: 'why are you going online-offline all the time?' then i would take the answer you'd given me.but since i used present simple which expresses a true fact and a continuous and lasting habit,i was actually asking you why i never 'see' you and have never 'saw' online for a longer time since i joined this forum.

Last edited by anonimnystefy (2011-12-24 01:41:58)

Bob · 2011-12-24 02:13:38

hi Stefy,

When I'm having breakfast (around GMT 8.00) I log on and see what's happening. But if I'm leaving my computer to do something else then I log off, otherwise you'd be posting away, wondering why I'm ignoring you when, actually, I'm not even in the room.

At lunch (around GMT 1300) I log back in again for a while. Again I log out once I'm off elsewhere.

I may log in again at some time between GMT 1600 and GMT 1900 and again maybe late at GMT 2300.

But none of this is guaranteed as I'm often out doing other things. eg. This week I was trying to get some manhole covers up as the architect for our building project needs to know what is underneath. Then I went back to school to set up the lighting for a dance show. Then I had to drive to a nearby town to 'rescue' my son as his car battery was dead. Today, I've been cleaning up chairs (been in a shed for years) to use for my daughter's Christmas party.

As you can tell, I try to keep busy. I often wonder how I ever managed to do a job as well!

Don't know if you celebrate it, but here's a wish anyway:

Have a happy Christmas! :)

Bob

Last edited by Bob (2011-12-24 02:17:53)

anonimnystefy · 2011-12-24 02:25:03

bob bundy wrote:

I often wonder how I ever managed to do a job as well!

You're very modest!

We don't celebrate Christmas now,we celebrate it on 7th of January.

anyway,i must tell you that you're a very busy man.how many dependent children do you have?

Bob · 2011-12-24 02:33:30

hi Stefy,

I have three children. My eldest, David, works in Germany. There are some old posts where he made a contribution. You could probably find them if you're interested. He has not joined the forum though.

My second son still lives at home. He has only had this car for a month and it looks like the battery needs replacing.

My daughter lives in Chester. She is expecting a baby in February.

All three went to University and have 'firsts' in mathematics.

Post #4 is now complete.

Bob

Last edited by Bob (2011-12-24 02:40:14)

anonimnystefy · 2011-12-24 03:23:13

hi bob

so no dependent children.

i think we should stop chatting here.my fault though.

Math Is Fun Forum

#1 2011-12-19 23:00:52

Group theory for scaling , translation and rotation

#2 2011-12-20 00:11:38

Re: Group theory for scaling , translation and rotation

#3 2011-12-20 01:33:13

Re: Group theory for scaling , translation and rotation

#4 2011-12-24 00:23:22

Re: Group theory for scaling , translation and rotation

#5 2011-12-24 00:34:33

Re: Group theory for scaling , translation and rotation

#6 2011-12-24 00:50:23

Re: Group theory for scaling , translation and rotation

#7 2011-12-24 01:41:08

Re: Group theory for scaling , translation and rotation

#8 2011-12-24 02:13:38

Re: Group theory for scaling , translation and rotation

#9 2011-12-24 02:25:03

Re: Group theory for scaling , translation and rotation

#10 2011-12-24 02:33:30

Re: Group theory for scaling , translation and rotation

#11 2011-12-24 03:23:13

Re: Group theory for scaling , translation and rotation

Board footer