Relationship between translation and rotation

In summary, the given statement that every translation is a product of two non-involutory rotations can be disproven by considering the general case. While a translation can be seen as the product of two reflections, the reflections must have parallel reflection lines and can be written as the product of two non-involutory rotations of 90 degrees. However, in the general case where the two rotations have different centers, this relationship does not hold. Additionally, translations do not have fixed points, making it difficult to establish a connection between translations and rotations. Therefore, the statement is not true and should be disproven.
  • #1
kolua
69
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?
 
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  • #2
Do you consider rotations only with the origin as fixed point or around any axis? Do translations have fixed points?
 
  • #3
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?
 
  • #4
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?
 
  • #5
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?
 
  • #6
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.
 
  • #7
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?
 
  • #8
Yes. Take any point and see what a translation does and where you can get by rotations.
 

1. What is the difference between translation and rotation?

Translation refers to the movement of an object from one point to another in a straight line, while rotation refers to the movement of an object around a fixed point in a circular motion.

2. How are translation and rotation related in physics?

In physics, translation and rotation are both considered types of motion and are governed by Newton's laws of motion. Translation involves the displacement of an object, while rotation involves the object's angular displacement and velocity.

3. Can an object have both translation and rotation at the same time?

Yes, an object can have both translation and rotation simultaneously. This is known as rotational motion, where an object moves in a circular path while also translating along a straight line.

4. How is translation and rotation used in robotics?

In robotics, translation and rotation are essential for the movement and control of robotic arms and other mechanical parts. Translation allows the robot to move in a straight line, while rotation allows it to manipulate objects in different directions.

5. What are some real-life examples of the relationship between translation and rotation?

Some real-life examples of the relationship between translation and rotation include the movement of a rolling ball, the motion of a spinning top, and the movement of a car on a curved road. In all these cases, there is a combination of translation and rotation occurring.

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