Question

What is a euclidean isometry?

Answer 1

http://en.wikipedia.org/wiki/Euclidean_plane_isometry

Answer 2

in more depth actually i read that

Answer 3

are you familiar with euclidean isometries?

Answer 4

elaborate, what do you want?

Answer 5

well i need to show that a euclidean isometry has exactly one line of fixed points, or a single fixed point, or no fixed points, and a parallel family of invariant lines, or no fixed points and a single invariant line

Answer 6

but i dont know how to approach it just yet

Answer 7

euclidean isometries are distance preserving maps that maintain the usual euclidean distance. it's just a transformation..... all the points have a map.

Answer 8

yeah but i have to give a proof of it, its just part of my assigned homework problems i didnt get what it meant

Answer 9

I assume you an Euclidean isometry in two dimensional space? In which case, an isometry formally is a map
\[ T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \]
such that \( ||x - y || = ||T(x) - T(y)|| \).
( where \( || \cdot || \) is the usual Euclidean distance).

What you can prove--and is hopefully already in your class/lecture notes--is that we can generate all such transformations \( T \) from rotations about the origin, reflections in any line through the origin, and any displacement. In other words, an isometry \( T \) can be written as the finite product of an arrangement of those three sorts of transformations.

Now, with that result, it should now be clear what can and can't be fixed.
Under a rotation, only the origin is fixed
Under a reflection, the axis of reflection is fixed
Under a translation, no points are fixed

The composition of such elementary isometries only leaves fixed the 'intersection' of the fixed points in a way you'll have to nail down.

Hope that helps.

Answer 10

how many times can i tell you i love you

Answer 11

Aw ;-)

Answer 12

so wait in your answer, how does that show any of the 4 things listed?

Answer 13

like i have to show it in some way, does it have to be by a diagram or like what

Answer 14

say we wanted to show that a euclidean isometry has a single fixed points, i dont know i just picked it because it seems easy to show

Answer 15

so i know this has a rotation property behind it, because by rotation you fix a point and you rotate it relative to that point

Answer 16

but i still dont know how to show it