Group of translations by a fixed distance

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In summary, the conversation discusses the need for a group to solve a physics problem involving broken symmetry and the desire for a matrix representation and rule for getting the covariant derivative. The suggested solution is to use a "projective geometry" representation, where translation by a matrix multiplication is possible. This forms a group under multiplication with an identity when all translation values are equal to zero.
  • #1
Mentz114
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I think I need this group ( if it exists) to help solve a physics problem.
Not being a mathematician, what I'd really like is a matrix representation
and a rule for getting the covariant derivative in the event of a broken symmetry.

I'd be much obliged if anyone can give me any information.

M.
 
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  • #2
To represent translation by a matrix multiplication, you will have to use a "projective geometry" representation: each point (x,y,z) is represented by (x,y,z,1) with the understanding that (ax, ay, az, a) (a not 0) represents the same point. Then a translation by <u, v, w> can be written as
[tex]\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right)\left(\begin{array}{c}x \\ y \\ z \\ 1\end{array}\right)= \left(\begin{array}{c}x+ u \\ y+ v \\ z+ w \\ 1\end{array}\right)[/tex]
 
  • #3
HallsofIvy, thank you very much. This looks promising. Seems to form a group under multiplication, with an identity if u=v=w=0.

[tex] \left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right)\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right) = \left(\begin{array}{ccccccc}1 && 0 && 0 && 2u \\ 0 && 1 && 0 && 2v\\ 0 && 0 && 1 && 2w \\ 0 && 0 && 0&& 1\end{array}\right)

[/tex]
 
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1. What is a group of translations by a fixed distance?

A group of translations by a fixed distance is a mathematical concept that refers to a set of transformations in which every point in a figure is moved by the same distance and in the same direction. The distance and direction of the movement remain constant for every point in the figure.

2. How is a group of translations by a fixed distance different from a regular translation?

A regular translation involves moving the entire figure by a certain distance and direction. However, in a group of translations by a fixed distance, each point in the figure is moved individually by the same distance and direction. This results in a figure that is the same shape and size as the original, but in a different location.

3. What is the significance of a fixed distance in a group of translations?

The fixed distance in a group of translations is important because it ensures that each point in the figure is moved by the same amount, maintaining the overall shape and size of the figure. Without a fixed distance, the figure could become distorted or change in size.

4. Can a group of translations by a fixed distance be applied to any figure?

Yes, a group of translations by a fixed distance can be applied to any figure, as long as the distance and direction of the movement remain constant for every point in the figure. This concept is often used in geometry and other areas of mathematics to study the properties of shapes and figures.

5. How is a group of translations by a fixed distance used in real-world applications?

A group of translations by a fixed distance is used in real-world applications such as map-making and computer graphics. In map-making, it is used to accurately represent the location and size of geographical features. In computer graphics, it is used to create animations and manipulate images by moving individual points or pixels by a fixed distance and direction.

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