Equation of an Ellipse - Revision history

Khanh at 18:31, 14 November 2021

2021-11-14T18:31:36Z

Lila at 20:40, 18 October 2021

2021-10-18T20:40:30Z

Lila at 20:39, 18 October 2021

2021-10-18T20:39:36Z

Lila at 20:34, 18 October 2021

2021-10-18T20:34:57Z

Khanh: Created page with "* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/ Equations of Ellipses], Lumen Learning"

2021-09-17T20:52:49Z

Created page with "* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/ Equations of Ellipses], Lumen Learning"

New page

* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/ Equations of Ellipses], Lumen Learning

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 For <math>u \in [0,\, 1],</math> this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing <math>u.</math> The left vertex is the limit <math display="inline">\lim_{u \to \pm \infty} (x(u),\, y(u)) = (-a,\, 0)\;.</math>
 ==Resources==
 * [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/ Equations of Ellipses], Lumen Learning

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 [[Image:ElipseAnimada.gif|thumb]]
-[[Image:Ellipse axis.png|right|250px]]
+[[Image:Ellipse axis.png|thumb]]
 ==Cartesian Equation of an Ellipse==

← Older revision		Revision as of 20:39, 18 October 2021
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		+	[[Image:ElipseAnimada.gif\|thumb]]
		+	[[Image:Ellipse axis.png\|right\|250px]]
		+
		+	==Cartesian Equation of an Ellipse==
		+	The general equation for an ellipse where its major, or longer, axis is horizontal is : <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>.<br />
		+	Where the major axis is vertical, it is: <math>\frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1</math>
		+	* The center is located at <math>(h,k)</math>.
		+	* The foci are found at a distance of <math>c</math> from the centre along the major axis, where <math>c = \sqrt{a^2 - b^2}</math>.
		+	* The eccentricity of the ellipse can be found from the formula: <math>b^2 = a^2(1-e^2)</math> 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 <math>x = \pm \frac{a}{e}</math>
		+	* The major axis has a length of <math>2a</math> and the minor one <math>2b</math>.
		+	* The sum of the distance from each point to each of the foci is <math>2a</math>.
		+
	== Parametric representation ==		== Parametric representation ==
	[[File:Elliko-sk.svg\|thumb\|The construction of points based on the parametric equation and the interpretation of parameter ''t'', which is due to de la Hire]]		[[File:Elliko-sk.svg\|thumb\|The construction of points based on the parametric equation and the interpretation of parameter ''t'', which is due to de la Hire]]

← Older revision		Revision as of 20:34, 18 October 2021
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		+	== Parametric representation ==
		+	[[File:Elliko-sk.svg\|thumb\|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 <math>\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2} = 1</math> is:
		+	: <math>(x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\ .</math>
		+
		+	The parameter ''t'' (called the ''eccentric anomaly'' in astronomy) is not the angle of <math>(x(t),y(t))</math> with the ''x''-axis, but has a geometric meaning due to Philippe de La Hire.
		+
		+	===Rational representation===
		+	With the substitution <math display="inline">u = \tan\left(\frac{t}{2}\right)</math> and trigonometric formulae one obtains
		+	:<math>\cos t = \frac{1 - u^2}{u^2 + 1}\ ,\quad \sin t = \frac{2u}{u^2 + 1}</math>
		+
		+	and the ''rational'' parametric equation of an ellipse
		+	: <math>\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\;,</math>
		+
		+	which covers any point of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> except the left vertex <math>(-a,\, 0)</math>.
		+
		+	For <math>u \in [0,\, 1],</math> this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing <math>u.</math> The left vertex is the limit <math display="inline">\lim_{u \to \pm \infty} (x(u),\, y(u)) = (-a,\, 0)\;.</math>
		+
		+
		+	==Resources==
	* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/ Equations of Ellipses], Lumen Learning		* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/ Equations of Ellipses], Lumen Learning