The Group of Translations of The Plane
A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. If you have two such translations a and b, they can be composed to form a new translation a ∘ b as follows: first follow the prescription of b, then that of a. For instance, if
- a = "move North-East for 3 miles"
and
- b = "move South-East for 4 miles"
then
- a ∘ b = "move East for 5 miles"
(see Pythagorean theorem for why this is so, geometrically).
The set of all translations of the plane with composition as operation forms a group:
- If a and b are translations, then a ∘ b is also a translation.
- Composition of translations is associative: (a ∘ b) ∘ c = a ∘ (b ∘ c).
- The identity element for this group is the translation with prescription "move zero miles in whatever direction you like".
- The inverse of a translation is given by walking in the opposite direction for the same distance.
This is an Abelian group and our first (nondiscrete) example of a Lie group: a group which is also a manifold.
Read more about this topic: Examples Of Groups
Famous quotes containing the words group, translations and/or plane:
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—Gore Vidal (b. 1925)
“Woe to the world because of stumbling blocks! Occasions for stumbling are bound to come, but woe to the one by whom the stumbling block comes!”
—Bible: New Testament, Matthew 18:7.
Other translations use “temptations.”
“We’ve got to figure these things a little bit different than most people. Y’know, there’s something about going out in a plane that beats any other way.... A guy that washes out at the controls of his own ship, well, he goes down doing the thing that he loved the best. It seems to me that that’s a very special way to die.”
—Dalton Trumbo (1905–1976)