A translation α (also known as "slide") is a bijective mapping from R2 to R2 that sends point (x,y) to (x′,y′) such that the α has equations {x′=x+ay′=y+b for some a,b∈R. In other words, α((x,y))=(x′,y′). Most of the time α (or any lower case greek letter) is used to represent any transformations (i.e. half turns, reflection, rotation, dilation, etc.). We shall use the greek letter τ for translations.
For example, let A=(c,d) and B=(e,f) be points in R2. Then there are unique numbers a and b such that e=c+a and f=d+b. Thus, the unique translation that takes A to B has equations
x′=x+(e−c) and y′=y+(f−d). We shall denote this unique translation by τA,B.
A simple illustration would be this:
Given A=(5,6) and B=(7,10), find τA,B((1,2)).
τA,B has equations {x′=x+(7−5)y′=y+(10−6). Hence, τA,B((1,2))=(3,6).□
Given A=(1,2),B=(3,4), and C=(5,6), find τA,BτB,CτA,C((1,2)).
The answer is (9,10).
Find the equations for τA,BτB,A.
They are x′=x and y′=y.
(4,3)(4,−3)(−4,3)(−4,−3)
If a transformation of X=x+4 and Y=y−3 is applied, what is the new position of the origin in the xy-coordinate system?
Let τA,B represent a linear translation in the xy-plane from point A to point B.