Mapping functions and bijections

[Svensken]

Homework Statement

Hello! I am stuck, having wondered about this question for quite some time now and I am not too sure how to solve it


Denote the xy-plane by P. Let C be some general curve in P defined by the equation
f(x,y) = 0, where f(x,y) is some algebraic expression involving x and y.

Let x0, y0 and θ be real numbers and define bijections T, R : P → P by the rules
T(x,y) = (x−x0,y−y0) and
R(x,y) = (xcosθ−ysinθ,xsinθ+ycosθ).
Thus T is the parallel translation of P that takes (x0,y0) to the origin, and R is the rotation θ radians anticlockwise about the origin.

(i) Verify carefully that if B : P → P is any bijection then B(C) is defined by the equation
f(B−1(x,y)) = 0. (note that B-1 means the inverse)

(ii) Deduce that T(C) is the curve defined by the equation
f(x+x0,y+y0) = 0
and R(C) by the equation f(xcosθ+ysinθ,−xsinθ+ycosθ) = 0.

Homework Equations

I think that the way to go about this is to note that C is a subset of P, and since b:p is any bijection it can be proved...i really have no intuition and would greatly appreciate some help in the right direction :)

 

[HallsofIvy]

Let (x, y) be any point on B(C). Then B^{-1}(C) is a point on C and therefore satisfies that equation.