Relationship between translation and rotation

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SUMMARY

The discussion centers on the mathematical relationship between translations and rotations, specifically addressing the statement that every translation is a product of two non-involutory rotations. The participants conclude that while a translation can be represented as a product of two reflections with parallel lines, this does not hold true in general, particularly when considering rotations with different centers. The consensus is that translations do not have fixed points, unlike rotations, which solidifies the argument against the original statement.

PREREQUISITES
  • Understanding of linear transformations in geometry
  • Familiarity with the concepts of rotations and reflections
  • Knowledge of fixed points in geometric transformations
  • Basic principles of vector spaces and mappings
NEXT STEPS
  • Study the properties of linear transformations in geometry
  • Explore the relationship between reflections and rotations in detail
  • Investigate fixed points in various geometric transformations
  • Learn about the implications of non-involutory rotations in transformations
USEFUL FOR

Mathematicians, geometry students, and educators interested in the relationships between different geometric transformations, particularly those focusing on translations, rotations, and reflections.

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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?
 
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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.
 

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