Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 2004 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["ParaPlot`"]
(*MathSource Item:0207-986*)

Needs["PlaneCurveFormulas`"]


Needs["PlaneCurveGenerator`"]

Needs["Graphics`ImplicitPlot`"]
(*MathSource Item:0200-529*)

some equations derivation for ellipse

the equation for ellipse with focuses at {+-c,0} and sum of distance 2*a is
Sqrt[(x-c)^2+(y-0)^2]+Sqrt[(x-(-c))^2+(y-0)^2]==2*a
we show the getting rid of the square roots.

ellipse = {Sqrt[(x - c)^2 + (y - 0)^2] + 
          Sqrt[(x - (-c))^2 + (y - 0)^2], 2*a}

ellipse - ellipse[[1, 2]]

now square both sides. We know the expression on both sides are positive.

%^2

Expand@%

% - (4 (a^2) + (c^2) + 2 c x + (x^2) + (y^2))

%^2

Expand@%

% - Last@%

Collect[%, {x, y}]

Now we have it in the normal polynomial of degree 2 form.