Equation of an Ellipse
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Cartesian Equation of an Ellipse
The general equation for an ellipse where its major, or longer, axis is horizontal is : 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 \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1}
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Where the major axis is vertical, it is: 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 \frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1}
- The center is located at 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 (h,k)} .
- The foci are found at a distance of 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 c} from the centre along the major axis, where 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 c = \sqrt{a^2 - b^2}} .
- The eccentricity of the ellipse can be found from the formula: 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 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 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 x = \pm \frac{a}{e}}
- The major axis has a length of 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 2a} and the minor one 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 2b} .
- The sum of the distance from each point to each of the foci is 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 2a} .
Parametric representation
Standard parametric representation
Using trigonometric functions, a parametric representation of the standard 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} is:
- 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 (x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\ .}
The parameter t (called the eccentric anomaly in astronomy) is not the angle of 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 (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
