Relationship between translation and rotation

Click For Summary

Homework Help Overview

The discussion revolves around the relationship between translations and rotations in the context of linear transformations in geometry. The original poster attempts to prove or disprove the statement that every translation can be expressed as a product of two non-involutory rotations.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the specific case where a translation is represented as a product of two reflections with parallel lines and question the generalization of this proof. There are inquiries about the nature of rotations, particularly whether they are considered around the origin or any axis, and the implications of fixed points in translations.

Discussion Status

The discussion is ongoing, with participants raising questions about the assumptions underlying the original statement and exploring the connections between translations, rotations, and reflections. Some participants suggest visualizing the mappings to better understand the relationships involved.

Contextual Notes

There are uncertainties regarding the definitions of rotations and translations, particularly concerning fixed points and the centers of rotation. The original poster expresses difficulty in generalizing their proof beyond a specific case.

kolua
Messages
69
Reaction score
3

Homework Statement


Prove or disprove: Every translation is a product of two non-involutory rotations.

Homework Equations

The Attempt at a Solution

:[/B]
I am not sure if I got the right proof for the special situation: A translation is the product of two reflections with parallel reflections lines. And the reflections lines can be written as the product of two non-involutory rotations of 90 degrees, which means that the translation is a product of two non-involutory rotations under this specific condition.

I don't know how to deal with the general conclusion. What happens when the two rotations have different centers? How should I prove the statement for a general case?
 
Physics news on Phys.org
Do you consider rotations only with the origin as fixed point or around any axis? Do translations have fixed points?
 
fresh_42 said:
Do you consider rotations only with the origin as fixed point or around any axis? Do translations have fixed points?
no, i don't think there fixed pionts should be taken into consideration. should they?
 
A rotation is usually a linear map with ##0## as center. A translation maps this point to another one, so how can they be related?
 
fresh_42 said:
A rotation is usually a linear map with ##0## as center. A translation maps this point to another one, so how can they be related?
by reflection?
 
There is a connection between reflections and rotations, yes, but a translation moves everything, a rotation has a fixed center point and a reflection even a fixed axis. Have you tried to draw the different mappings? You should do it.
 
fresh_42 said:
There is a connection between reflections and rotations, yes, but a translation moves everything, a rotation has a fixed center point and a reflection even a fixed axis. Have you tried to draw the different mappings? You should do it.
so there is no such relationship and I should disprove statement?
 
Yes. Take any point and see what a translation does and where you can get by rotations.
 

Similar threads

What should be the geometries of two contacting solids that may have a relative rotation and translation along the same axis?
  • · Replies 8 ·
Replies
8
Views
2K
Find the formula for X=tP+sQ under a translation and a rotation
  • · Replies 3 ·
Replies
3
Views
2K
A Set in the plane is never isometric to a proper subset
  • · Replies 11 ·
Replies
11
Views
2K
How Do You Calculate Rotations and Translations in Geometry?
  • · Replies 4 ·
Replies
4
Views
2K
MCNP6.2 - Combination of transformations
  • · Replies 1 ·
Replies
1
Views
3K
Euclidean Group Maps: Proving Injectivity, Surjectivity, and Inverses
  • · Replies 3 ·
Replies
3
Views
2K
MHB The product γ is a rotation or a translation
  • · Replies 11 ·
Replies
11
Views
1K
Understanding work in translation and rotation of a magnetic dipole
  • · Replies 1 ·
Replies
1
Views
2K
Simple Transformation of a Function: translation, reflection, sketch
  • · Replies 3 ·
Replies
3
Views
2K
[Linear Algebra] rotational matrices
  • · Replies 7 ·
Replies
7
Views
2K