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);
}

fuss/mathematics/geometry/shapes/ellipses.txt ยท Last modified: 2017/02/22 18:30 (external edit)

Access website using Tor Access website using i2p


For the copyright, license, warranty and privacy terms for the usage of this website please see the license, privacy and plagiarism pages.