What's the definition of Ellipse?

Question

I needed to prove that a parametric curve is an Ellipses and only then I realized that the Ellipses were not defined in the textbook.

However, I could not use the definition for the "standard" Ellipses where $$\frac{x(t)^2}{a^2}+\frac{y(t)^2}{a^2}=1 $$ because it's a generalized ellipses with rotation. I could not really use the General ellipse $$Ax^2+Bxy+Cy^2 +Dx+Ey+F=0, B^2-4AC<0$$ because then there was no point of the proof.

The only sensible definition I found from the Wikipedia was

Any ellipse is an affine image of the unit circle with equation $x^2+y^2=1$

But that had gone a bit outside the scope of the course.

What's the definition of Ellipse? How am I suppose to say mathematically that an Ellipses is an Ellipses?

Answer 1

Any ellipse has the form of a deformed circle along any axis $$(x'(s),y'(s))= s (x \cos \phi + y \sin\phi, -x \cos \phi+y \cos \phi)$$

$$ x^2+y^2=1 \longrightarrow \left( \cos \phi\ (a x) + \sin \phi\ (b y)\right)^2 + \left(-\sin \phi\ (a x) + \cos \phi \ (b y) \right) ^2, $$ its center translated by any vector $(x_0,y_0)$ $$\longrightarrow \left( \cos \phi\ (a\ (x-x_0) + \sin \phi\ (b \ (y-y_0)\right)^2 + \left(-\sin \phi\ (a \ (x-x_0)) + \cos \phi \ (b \ (y-y_0)) \right) ^2 $$

Now, using Mathematica for short,

   1=(Cos[\[Phi]] (a (x - x0)) + 
    Sin[\[Phi]] (b (y - y0)))^2 + (-Sin[\[Phi]] (a (x - x0)) + 
      Cos[\[Phi]] (b (y - y0)))^2 // Expand // FullSimplify

$$1=a^2 (x-\text{x0})^2+b^2 (y-\text{y0})^2$$

On paper, complete the squares of x,y and collect all additive constants

$$A x^2 + B x = A \ \left(x^2 + \frac{B x}{A} + \frac{B^2}{4 A^2}\right) - \frac{B^2}{4 A} = A \left(x +\frac{B}{2 A} \right)^2 - \frac{B^2}{4 A}$$

The coefficients of the complete squares have to be positive, the constant negative. Different signs yield a hyperbola, one sign zero with a linear term a parabola.

The constant zero a pair of lines, two positive coefficients and a zero constant a point.

Easier classification of quadratic forms via projective geometry.

Answer 2

There are probably a few different ways to define an ellipse but one definition is: the set of all points where the sum of the distances between a given point and two fixed points (the foci) is constant (the constant is the length of the major axis). Watch the bottom bar at the bottom of this animation - the two lengths change but the sum doesn't. https://www.youtube.com/watch?v=XLjnTgXXgXk

Answer 3

You are given the parametric equation $$ P=\vec a\cos t+ \vec b\sin t $$ If $\vec a\perp\vec b$ then you can choose orthogonal cartesian axes directed as $\vec a$ and $\vec b$ so that: $$ x(t)=|\vec a|\cos t,\quad y(t)=|\vec b|\sin t $$ and from these you get $$ {x^2\over |\vec a|^2}+{y^2\over |\vec b|^2}=1. $$ It can be proved that this equation also holds for an ellipse in oblique coordinates, hence the above proof works even if $\vec a$ and $\vec b$ are not orthogonal. In that case $\vec a$ and $\vec b$ give direction and magnitude of a pair of conjugate semidiameters of the ellipse.