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Baris 27:
:<math>K = \pi (a+b) \left[1 + \sum_{n=1}^\infty \left(\frac{(2n-1)!!}{2^n n!}\right)^2 \frac{h^n}{(2n-1)^2}\right]</math>
== [[sistem Kordinat Kartesius]] ==
[[File:Ellipse-param.svg|thumb|Parameter bentuk:{{ubl
| ''a'': sumbu semi-mayor,
| ''b'': sumbu semi-minor,
| ''c'': eksentrisitas linear,
| ''p'': rektum semi-latus (biasanya <math>\ell</math>).
}}]]
=== persamaan standar ===
Bentuk standar elips dalam koordinat Cartesian mengasumsikan bahwa asalnya adalah pusat elips, ''x''- sumbu adalah sumbu utama, dan:
: fokus adalah poinnya <math>F_1 = (c,\, 0),\ F_2=(-c,\, 0)</math>,
: simpulnya adalah <math>V_1 = (a,\, 0),\ V_2 = (-a,\, 0)</math>.
For an arbitrary point <math>(x,y)</math> the distance to the focus <math>(c,0)</math> is
<math>\sqrt{(x - c)^2 + y^2 }</math> and to the other focus <math>\sqrt{(x + c)^2 + y^2}</math>. Hence the point <math>(x,\, y)</math> is on the ellipse whenever:
:<math>\sqrt{(x - c)^2 + y^2} + \sqrt{(x + c)^2 + y^2} = 2a\ .</math>
Removing the [[radical expression|radicals]] by suitable squarings and using <math>b^2 = a^2-c^2</math> produces the standard equation of the ellipse:
:<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,</math>
or, solved for ''y:''
:<math>y = \pm\frac{b}{a}\sqrt{a^2 - x^2} = \pm \sqrt{\left(a^2 - x^2\right)\left(1 - e^2\right)}.</math>
The width and height parameters <math>a,\; b</math> are called the [[semi-major and semi-minor axes]]. The top and bottom points <math>V_3 = (0,\, b),\; V_4 = (0,\, -b)</math> are the ''co-vertices''. The distances from a point <math>(x,\, y)</math> on the ellipse to the left and right foci are <math>a + ex</math> and <math>a - ex</math>.
It follows from the equation that the ellipse is ''symmetric'' with respect to the coordinate axes and hence with respect to the origin.
=== Parameters ===
==== Semi-major and semi-minor axes ====
Throughout this article <math>a</math> is the semi-major axis, i.e. <math>a \ge b > 0 \ .</math> In general the canonical ellipse equation <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1 </math> may have <math>a < b</math> (and hence the ellipse would be taller than it is wide); in this form the semi-major axis would be <math>b</math>. This form can be converted to the standard form by transposing the variable names <math>x</math> and <math> y</math> and the parameter names <math>a</math> and <math> b.</math>
==== Linear eccentricity ====
This is the distance from the center to a focus: <math>c = \sqrt{a^2 - b^2}</math>.
==== Eccentricity ====
The eccentricity can be expressed as:
: <math>e = \frac{c}{a} = \sqrt{1 - \left(\frac{b}{a}\right)^2}</math>,
assuming <math>a > b.</math> An ellipse with equal axes (<math>a = b</math>) has zero eccentricity, and is a circle.
==== Semi-latus rectum ====
The length of the chord through one focus, perpendicular to the major axis, is called the ''latus rectum''. One half of it is the ''semi-latus rectum'' <math>\ell</math>. A calculation shows:
: <math>\ell = \frac{b^2}a = a \left(1 - e^2\right).</math><ref>{{harvtxt|Protter|Morrey|1970|pp=304,APP-28}}</ref>
The semi-latus rectum <math>\ell</math> is equal to the [[radius of curvature]] at the vertices (see section [[#Curvature|curvature]]).
=== Tangent ===
An arbitrary line <math>g</math> intersects an ellipse at 0, 1, or 2 points, respectively called an ''exterior line'', ''tangent'' and ''secant''. Through any point of an ellipse there is a unique tangent. The tangent at a point <math>(x_1,\, y_1)</math> of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> has the coordinate equation:
:<math>\frac{x_1}{a^2}x + \frac{y_1}{b^2}y = 1.</math>
A vector [[parametric equation]] of the tangent is:
: <math>\vec x = \begin{pmatrix}x_1 \\ y_1\end{pmatrix} + s\begin{pmatrix}
\;\! -y_1 a^2 \\
\;\ \ \ x_1 b^2
\end{pmatrix}\ </math> with <math>\ s \in \mathbb{R}\ .</math>
'''Proof:'''
Let <math>(x_1,\, y_1)</math> be a point on an ellipse and <math display="inline">\vec{x} = \begin{pmatrix}x_1 \\ y_1\end{pmatrix} + s\begin{pmatrix}u \\ v\end{pmatrix}</math> be the equation of any line <math>g</math> containing <math>(x_1,\, y_1)</math>. Inserting the line's equation into the ellipse equation and respecting <math display-"inline">\frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1</math> yields:
: <math>
\frac{\left(x_1 + su\right)^2}{a^2} + \frac{\left(y_1 + sv\right)^2}{b^2} = 1\ \quad\Longrightarrow\quad
2s\left(\frac{x_1u}{a^2} + \frac{y_1v}{b^2}\right) + s^2\left(\frac{u^2}{a^2} + \frac{v^2}{b^2}\right) = 0\ .</math>
: There are then cases:
:# <math>\frac{x_1}{a^2}u + \frac{y_1}{b^2}v = 0.</math> Then line <math>g</math> and the ellipse have only point <math>(x_1,\, y_1)</math> in common, and <math>g</math> is a tangent. The tangent direction has [[normal (geometry)|perpendicular vector]] <math>\begin{pmatrix}\frac{x_1}{a^2} & \frac{y_1}{b^2}\end{pmatrix}</math>, so the tangent line has equation <math display="inline">\frac{x_1}{a^2}x + \tfrac{y_1}{b^2}y = k</math> for some <math>k</math>. Because <math>(x_1,\, y_1)</math> is on the tangent and the ellipse, one obtains <math>k = 1</math>.
:# <math>\frac{x_ 1}{a^2}u + \frac{y_1}{b^2}v \ne 0.</math> Then line <math>g</math> has a second point in common with the ellipse, and is a secant.
Using (1) one finds that <math>\begin{pmatrix} -y_1a^2 & x_1b^2 \end{pmatrix}</math> is a tangent vector at point <math>(x_1,\, y_1)</math>, which proves the vector equation.
If <math>(x_1, y_1)</math> and <math>(u, v)</math> are two points of the ellipse such that <math display="inline">\frac{x_1u}{a^2} + \tfrac{y_1v}{b^2} = 0</math>, then the points lie on two ''conjugate diameters'' (see [[#Conjugate diameters and the midpoints of parallel chords|below]]). (If <math>a = b</math>, the ellipse is a circle and "conjugate" means "orthogonal".)
=== Shifted ellipse ===
If the standard ellipse is shifted to have center <math>\left(x_\circ,\, y_\circ\right)</math>, its equation is
: <math>\frac{\left(x - x_\circ\right)^2}{a^2} + \frac{\left(y - y_\circ\right)^2}{b^2} = 1 \ .</math>
The axes are still parallel to the x- and y-axes.
=== General ellipse ===
{{Main|Matrix representation of conic sections}}
In [[analytic geometry]], the ellipse is defined as a quadric: the set of points <math>(X,\, Y)</math> of the [[Cartesian plane]] that, in non-degenerate cases, satisfy the [[Implicit and explicit functions|implicit]] equation<ref>{{cite book|url=https://books.google.com/books?id=yMdHnyerji8C|title=Precalculus with Limits|last1=Larson|first1=Ron|last2=Hostetler|first2=Robert P.|last3=Falvo|first3=David C.|publisher=Cengage Learning|year=2006|isbn=978-0-618-66089-6|page=767|chapter=Chapter 10|chapterurl=https://books.google.com/books?id=yMdHnyerji8C&pg=PA767}}
</ref><ref>{{cite book|url=https://books.google.com/books?id=9HRLAn326zEC|title=Precalculus|last1=Young|first1=Cynthia Y.|publisher=John Wiley and Sons|year=2010|isbn=978-0-471-75684-2|page=831|chapter=Chapter 9|chapterurl=https://books.google.com/books?id=9HRLAn326zEC&pg=PA831}}
</ref>
: <math>AX^2 + B X Y + C Y^2 + D X + E Y + F = 0</math>
provided <math>B^2 - 4AC < 0.</math>
To distinguish the [[degenerate conic|degenerate cases]] from the non-degenerate case, let ''∆'' be the [[determinant]]
:<math>\Delta = \begin{vmatrix}
A & \frac{1}{2}B & \frac{1}{2}D \\
\frac{1}{2}B & C & \frac{1}{2}E \\
\frac{1}{2}D & \frac{1}{2}E & F
\end{vmatrix} = \left(AC - \frac{B^2}{4}\right) F + \frac{BED}{4} - \frac{CD^2}{4} - \frac{AE^2}{4}.
</math>
Then the ellipse is a non-degenerate real ellipse if and only if ''C∆'' < 0. If ''C∆'' > 0, we have an imaginary ellipse, and if ''∆'' = 0, we have a point ellipse.<ref name="Lawrence">Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972.</ref>{{rp|p.63}}
The general equation's coefficients can be obtained from known semi-major axis <math>a</math>, semi-minor axis <math>b</math>, center coordinates <math>\left(x_\circ,\, y_\circ\right)</math>, and rotation angle <math>\Theta</math> (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:
:<math>\begin{align}
A &= a^2 (\sin\Theta)^2 + b^2 (\cos\Theta)^2 \\
B &= 2\left(b^2 - a^2\right) \sin\Theta \cos\Theta \\
C &= a^2 (\cos\Theta)^2 + b^2 (\sin\Theta)^2 \\
D &= -2A x_\circ - B y_\circ \\
E &= - B x_\circ - 2C y_\circ \\
F &= A x_\circ^2 + B x_\circ y_\circ + C y_\circ^2 - a^2 b^2.
\end{align}</math>
These expressions can be derived from the canonical equation <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> by an affine transformation of the coordinates <math>(x,\, y)</math>:
:<math>\begin{align}
x &= \left(X - x_\circ\right) \cos\Theta + \left(Y - y_\circ\right) \sin\Theta \\
y &= -\left(X - x_\circ\right) \sin\Theta + \left(Y - y_\circ\right) \cos\Theta.
\end{align}</math>
Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:
:<math>\begin{align}
a, b &= \frac{-\sqrt{2 \Big(A E^2 + C D^2 - B D E + (B^2 - 4 A C) F\Big)\left((A + C) \pm \sqrt{(A - C)^2 + B^2}\right)}}{B^2 - 4 A C} \\
x_\circ &= \frac{2CD - BE}{B^2 - 4AC} \\[3pt]
y_\circ &= \frac{2AE - BD}{B^2 - 4AC} \\[3pt]
\Theta &= \begin{cases}
\arctan\left(\frac{1}{B}\left(C - A - \sqrt{(A - C)^2 + B^2}\right)\right)
& \text{for } B \ne 0 \\
0 & \text{for } B = 0,\ A < C \\
90^\circ & \text{for } B = 0,\ A > C \\
\end{cases}
\end{align}</math>
{{Commons|Category:Parabolas}}
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