Dtyjlui

One thing I am finding challenging in my studies is in a situation involving two coordinate systems - the $x,y$ and $u,v$ of a real plane.

My understanding is that when relations such as the following are given:

$u = ax + by + e$ and $v = cx + dy + f$,

it is ambiguous if no more information is given, whether this information is providing the change of coordinates relating the coordinates of a point $P$ of the plane in one coordinate system to the other, OR if those relations specify a transformation mapping points in the $x,y$ plane to possibly distinct points of the $u,v$ plane.

Is this correct?

The book then proceeds to name relations such as the following:

$u = ax + by + e$ and $v = cx + dy + f$

affine changes of coordinates. And so if a point $(x,y)$ is substituted, am I to assume that $(u,v) = (ax + by + e, cx + dy + f)$ is a distinct point that is the image of $(x,y)$?

Amidst all these confusions, the book supplies a quadratic equation like $P(x,y) = 0$, and then they substitute for $x$ and $y$ using the relations $u = ax + by + e$ and $v = cx + dy + f$ , to obtain a polynomial in $u$ and $v$.

I am unsure how this new polynomial relates to the old one. Is it the old one expressed in a new coordinate system, or a new and different polynomial expressed in the new? I'm leaning towards the latter because sometimes a circle would go to an ellipse, and so it cannot be the description of the old curve in the new coordinate system.

If this is the case I fail to see the point of this exercise.

You're mostly correct. The given equations could indeed be interpreted in two different ways.
Suppose we're given a point $P$ with coordinates $(x,y)$ relative to some coordinate system. Then the two possible interpretations are:

1. The equations give the coordinates $(u,v)$ of some new/different/transformed point with respect to the same coordinate system.




2. The equations give the coordinates $(u,v)$ of the same point $P$ with respect to some different coordinate system.

The use of different symbols $(u,v)$ somewhat suggests interpretation #2. If I wanted the reader to interpret as in #1, I would probably denote the coordinates of the new point as something like $(x',y')$ to indicate that these coordinates till refer to the original coordinate system.

sometimes a circle would go to an ellipse, and so it cannot be the description of the old curve in the new coordinate system.

A given equation can represent either a circle or an ellipse depending on the scales of measurement along the coordinate axes. If the scales are the same on the two axes, then the equation $x^2+y^2 = 1$ obviously represents a circle. But if you use different scales on the two axes, it represents an ellipse. Using different scales is a little odd in a geometric study, but it makes sense if you're concerned only with algebraic relationships, and not with geometry.

It's quite common to fabricate a special coordinate system that makes the equation of a given curve especially simple. Suppose we have an ellipse with semi-axis lengths $2$ and $3$. We can set up a coordinate system as follows:

1. Origin at the center of the ellipse


2. The point on the ellipse that's 3 units away from the center has coordinates $(1,0)$
   


3. The point on the ellipse that's 2 units away from the center has coordinates $(0,1)$
   

Then, in this coordinate system, the ellipse has equation $x^2+y^2=1$, which makes it very simple algebraically. We haven't changed the shape of the curve, but we've made the algebra easier by our clever choice of coordinate system.

You can do even fancier tricks by using coordinate systems whose axes are not at right angles.

If you're studying linear algebra, then, in a few weeks, you'll probably learn about eigenvectors and diagonalization of matrices. These concepts were originally developed as ways of choosing coordinate systems that would make the equations of quadric surfaces especially simple, as in our ellipse example above.