Equation of an Elipse

$\begin{tikzpicture} % grid \draw[help lines] (-2,-2) grid (2,2); % origin \draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1); \coordinate[label={[red]above left:$O$}] (O) at (0,0); \draw (0,0) ellipse (20mm and 10mm); \coordinate[label={[red]above left:$B$}] (B) at (0,1); \drawpoint{B}{.5mm}{black} \coordinate[label={[red]above right:$A$}] (A) at (2,0); \drawpoint{A}{.5mm}{black} \draw[black,dotted] (O) -- (A); \draw[black,dotted] (O) -- (B); \drawbrace{O}{A}{2mm}{blue}{$a$}{0}{-4mm}{mirror}; \drawbrace{O}{B}{2mm}{green}{$b$}{4mm}{0}{mirror}; \end{tikzpicture}$

In standard form:

$\begin{eqnarray*} (\frac{x}{{\color{blue}a}})^{2}+(\frac{y}{\color{green}b})^{2} &=& 1 \end{eqnarray*}$

When $O$ is not the centre of the ellipse, but rather an arbitrary point $C(j, k)$ then:

$\begin{eqnarray*} x &\mapsto& x-j \\ y &\mapsto& x-k \\ \end{eqnarray*}$

such that the equation becomes:

$\begin{eqnarray*} (\frac{x-j}{{\color{blue}a}})^{2}+(\frac{y-k}{\color{green}b})^{2} &=& 1 \end{eqnarray*}$

Equation of a Rotated Elipse

$\begin{tikzpicture} % grid \draw[help lines] (-2,-2) grid (2,2); % origin \draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1); \coordinate[label={[red]below left:$O$}] (O) at (0,0); \coordinate (X) at (5,0); \draw[rotate=60] (0,0) ellipse (20mm and 10mm); \coordinate[label={[red]above left:$B$}] (B) at (-.7,.8); \drawpoint{B}{.5mm}{black} \coordinate[label={[red]above right:$A$}] (A) at (1,1.7); \drawpoint{A}{.5mm}{black} \draw[black,dotted] (O) -- (A); \draw[black,dotted] (O) -- (B); \drawbrace{O}{A}{2mm}{blue}{$a$}{3mm}{-2mm}{mirror}; \drawbrace{O}{B}{2mm}{green}{$b$}{3mm}{3mm}{mirror}; % alpha \markangle{O}{X}{A}{3mm}{3mm}{$\alpha$}{cyan}{north west} \end{tikzpicture}$

$\begin{eqnarray*} (\frac{x\cos{{\color{cyan}\alpha}} + y\sin{{\color{cyan}\alpha}}}{{\color{blue}a}})^{2}+(\frac{x\sin{{\color{cyan}\alpha}} - y\cos{{\color{cyan}\alpha}}}{\color{green}b})^{2} &=& 1 \end{eqnarray*}$

Foci

$\begin{tikzpicture} % grid \draw[help lines] (-3,-3) grid (3,3); % origin \draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1); \coordinate[label={[red]above left:$O$}] (O) at (0,0); \draw (0,0) ellipse (30mm and 20mm); \coordinate[label={[red]above left:$B$}] (B) at (0,2); \drawpoint{B}{.5mm}{black} \coordinate[label={[red]above right:$A$}] (A) at (3,0); \drawpoint{A}{.5mm}{black} \coordinate[label={[red]above left:$B'$}] (B') at (0,-2); \drawpoint{B'}{.5mm}{black} \coordinate[label={[red]above left:$A'$}] (A') at (-3,0); \drawpoint{A'}{.5mm}{black} \draw[black,dotted] (A) -- (O) -- (A'); \draw[black,dotted] (B) -- (O) -- (B'); \drawbrace{O}{A}{2mm}{blue}{$a$}{0}{-4mm}{mirror}; \drawbrace{O}{B}{2mm}{green}{$b$}{4mm}{0}{mirror}; \coordinate[label={[cyan]above:$F_{1}$}] (F_{1}) at ({sqrt(5}, 0); \drawpoint{F_{1}}{.5mm}{cyan} \coordinate[label={[cyan]above:$F_{2}$}] (F_{2}) at (-{sqrt(5}, 0); \drawpoint{F_{2}}{.5mm}{cyan} \coordinate[label={[cyan]above:$P$}] (P) at (-1, {sqrt(32/9)}); \drawpoint{P}{.5mm}{cyan} \draw[cyan,dotted] (P) -- (F_{2}); \draw[cyan,dotted] (P) -- (F_{1}); \end{tikzpicture}$

$\begin{eqnarray*} {\overline{\color{red}{OF_{1}}} &=& \sqrt{{\color{blue}a}^{2} - {\color{green}b}^{2}} & \\ {\overline{\color{red}{OF_{2}}} &=& -\sqrt{{\color{blue}a}^{2} - {\color{green}b}^{2}} \end{eqnarray*}$

$\begin{eqnarray*} {\overline{\color{cyan}{PF_{2}}} + {\overline{\color{cyan}{PF_{1}}} &=& 2*{\color{blue}a} \end{eqnarray*}$

where ${\color{cyan}P}$ represents an arbitrary point on the ellipse perimeter.

Area and Perimeter

$\begin{tikzpicture} % grid \draw[help lines] (-2,-2) grid (2,2); % origin \draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1); \coordinate[label={[red]above left:$O$}] (O) at (0,0); \draw (0,0) ellipse (20mm and 10mm); \coordinate[label={[red]above left:$B$}] (B) at (0,1); \drawpoint{B}{.5mm}{black} \coordinate[label={[red]above right:$A$}] (A) at (2,0); \drawpoint{A}{.5mm}{black} \draw[black,dotted] (O) -- (A); \draw[black,dotted] (O) -- (B); \drawbrace{O}{A}{2mm}{blue}{$a$}{0}{-4mm}{mirror}; \drawbrace{O}{B}{2mm}{blue}{$b$}{4mm}{0}{mirror}; \coordinate[label={[red]above:$F$}] (F) at ({sqrt(2}, 0); \drawpoint{F}{.5mm}{red} \end{tikzpicture}$

$\begin{eqnarray*} A &=& \pi*a*b & \\ P &=& 4*a*E(e) & \\ e &=& \frac{\overline{OF}}{a} \end{eqnarray*}$

float wasElipseCircumference(float a, float b, integer precision) {
    float x = llListStatistics(LIST_STAT_MAX, [a, b]);
    float y = llListStatistics(LIST_STAT_MIN, [a, b]);
    float tol = llSqrt(llPow(.5, precision));
    if (precision * y < tol * x) return 4 * x;
    float s = 0;
    float m = 1;
    while (x - y > tol * y) {
        x = .5 * (x + y);
        y = llSqrt(x * y);
        m *= 2;
        s += m * llPow(x - y, 2);
    }
    return PI * (llPow(a + b, 2) - s) / (x + y);
}

Table of Contents

Equation of an Elipse

Equation of a Rotated Elipse

Foci

Area and Perimeter