# Mapping Circle to Ellipse with Dilation?

• Stefk
In summary, the problem at hand involves the mapping ##F_{a,b}##, which associates points in the plane with coordinates ##(x, y)## to points with coordinates ##(ax, by)##. The task is to show that the set of points satisfying the equation $$\left( \frac u a \right)^2 + \left( \frac v b \right)^2 = 1$$ is the image of the circle of radius 1 centered at the origin under the mapping ##F_{a,b}##. This can be done by parametrizing the circle with ##u = a\sin(t)## and ##v = b\cos(t)## and applying the mapping.
Stefk

## Homework Statement

The problem comes from S. Lang's "Basic mathematics", chapter 7, §1:

"Consider the following generalization of a dilation. Let ##a > 0, b > 0##. To each point ##(x, y)## of the plane, associate the point ##(ax, by)##. Thus we stretch the x-coordinate by ##a## and the y-coordinate by ##b##. This association is a mapping which we may denote by ##F_{a,b}##.

Show that the set of points ##(u, v)## satisfying the equation $$\left( \frac u a \right)^2 + \left( \frac v b \right)^2 = 1$$ is the image of the circle of radius 1 centered at ##O## under the map ##F_{a,b}##."

## Homework Equations

Equation of a circle of radius ##r## centered at ##(a, b)##: $$(x - a)^2 + (y - b)^2 = r^2$$
Equation of a circle of radius ##r## centered at the origin: $$x^2 + y^2 = r^2$$
Equation of a circle of radius 1 centered at the origin: $$x^2 + y^2 = 1$$

## The Attempt at a Solution

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I've spent quite some time on this problem and I still can't see how to solve it. I understand by subsequent indications and further reading on wikipedia that the equation is the cartesian equation of an ellipse and that the exercise is a way of viewing the ellipse as an irregularly dilated or stretched circle. However, I don't see any rigorous, algebraic way of coming to that equation from the circle equation using the ##F_{a,b}## mapping, nor the other way around (e.g. using an inverse mapping ##F^{-1}_{a,b}##). Applying the mapping to ##u## and ##v## on the left-hand side doesn't even seem to make sense. I might be missing something obvious...

PS: there isn't any development on ellipses in Lang's book (at least not in the part where the exercise comes from or before) so a solution can't rely on anything else that what was stated in the problem, except basic notions of coordinates, distance between points and Pythagora's theorem.

Use parametrisation.

Let u = a sin t and v = b cos t.

Stefk said:
To each point ##(x, y)## of the plane, associate the point ##(ax, by)##.

Equation of a circle of radius 1 centered at the origin: $$x^2 + y^2 = 1$$

These are the relevant bits.

The first bit means, whenever you see ##x##, write ##ax##, whenever you see ##y## write ##by## (you can write it ##x \to ax## and ##y \to by## or something like that).

Now apply that to the second bit quoted above and take it from there.

## 1. What is a circle?

A circle is a shape that is formed by a set of points that are equidistant from a central point. It is a closed curve and has no corners or edges.

## 2. How is a circle different from an ellipse?

A circle is a special type of ellipse where the distance from the center to any point on the curve is the same. An ellipse, on the other hand, has two focal points and the sum of the distances from these points to any point on the curve is constant.

## 3. What is a dilation?

A dilation is a transformation that changes the size of an object while keeping its shape and orientation the same. It involves multiplying the coordinates of the points of the object by a scale factor.

## 4. How do you find the equation of an ellipse?

The standard form of an ellipse's equation is (x-h)²/a² + (y-k)²/b² = 1, where (h,k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.

## 5. What is the relationship between a circle and a dilation?

A circle can be created by dilating a point with a scale factor of 1 from a single point. Similarly, a circle can be dilated to create an ellipse by using a scale factor greater than 1 or less than 1.

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