Equation of an Ellipse

Cartesian Equation of an Ellipse

The general equation for an ellipse where its major, or longer, axis is horizontal is : ${\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1$.
Where the major axis is vertical, it is: ${\frac {(y-k)^{2}}{a^{2}}}+{\frac {(x-h)^{2}}{b^{2}}}=1$

• The center is located at $(h,k)$.
• The foci are found at a distance of $c$ from the centre along the major axis, where $c={\sqrt {a^{2}-b^{2}}}$.
• The eccentricity of the ellipse can be found from the formula: $b^{2}=a^{2}(1-e^{2})$ where e is eccentricity. A higher eccentricity makes the curve appear more 'squashed', whereas an eccentricity of 0 makes the ellipse a circle.
• The directrices are the lines $x=\pm {\frac {a}{e}}$
• The major axis has a length of $2a$ and the minor one $2b$.
• The sum of the distance from each point to each of the foci is $2a$.

Parametric representation

The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire

Standard parametric representation

Using trigonometric functions, a parametric representation of the standard ellipse ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ is:

$(x,\,y)=(a\cos t,\,b\sin t),\ 0\leq t<2\pi \ .$

The parameter t (called the eccentric anomaly in astronomy) is not the angle of $(x(t),y(t))$ with the x-axis, but has a geometric meaning due to Philippe de La Hire.

Rational representation

With the substitution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u = \tan\left(\frac{t}{2}\right)} and trigonometric formulae one obtains

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos t = \frac{1 - u^2}{u^2 + 1}\ ,\quad \sin t = \frac{2u}{u^2 + 1}}

and the rational parametric equation of an ellipse

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x(u) &= a\frac{1 - u^2}{u^2 + 1} \\ y(u) &= \frac{2bu}{u^2 + 1} \end{align}\;,\quad -\infty < u < \infty\;,}

which covers any point of the ellipse Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1} except the left vertex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-a,\, 0)} .

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \in [0,\, 1],} this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u.} The left vertex is the limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{u \to \pm \infty} (x(u),\, y(u)) = (-a,\, 0)\;.}

Resources

• Equations of Ellipses, Lumen Learning

Licensing

Content obtained and/or adapted from:

• Ellipse, Wikibooks: Conic Sections under a CC BY-SA license
• Ellipse, Wikipedia under a CC BY-SA license