Here are presented some papers on the basics of ellipses.
Basics of Ellipses:
This paper introduces the ellipse from its simple pencil-and-paper construction and then proves
the standard ellipse equation
\begin{equation}
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.
\end{equation}
The Area of the Ellipses:
This paper finds the area of an ellipse using integral calculus.
The Ellipses from a Directrix, polar coordinates:
The object one can create from use of a directrix is proved to be an ellipse by transforming into the previous formulation we had of an ellipse. Polar coordiantes are introduced here.
Kepler's Laws from Newtonian Mechanics:
Now we're going to prove Kepler's three laws of planetary motion by assuming
Newtonian mechanics and making the approximation that the Sun is
'fixed' in space, and that the planets obey Newton's Law of gravitational
attraction, and that the gravitational interaction of the planets among
themselves is negligible. To follow this paper, the reader will need some basic vector calculus.
Kepler's First Law: The orbits of the planets around the Sun
are ellipses with the Sun at one focus of the ellipse.
Kepler's Second Law: A radial line connecting a planet to the Sun
sweeps out equal areas in equal times.
Kepler's Third Law: The square of a planet's period is proportional
to the cube of the length of its orbit's semimajor axis.