قطع ناقص

هذا المقال هو عن the geometric figure. إذا كنت تريد الاستخدامات الأخرى، انظر قطع ناقص (توضيح).

An ellipse obtained as the intersection of a cone with an inclined plane.

The rings of Saturn are circular, but when seen partially edge on, as in this image, they appear to be ellipses. In addition, the planet itself is an ellipsoid, flatter at the poles than the equator. Picture by ESO

القطع الناقص أو الإهليلج (ellipse) (الكلمة آتية من اللاتينية بمعنى نقص absence) هو المنحني الجبري المستوي الذي يحقق أن مجموع بعد أي نقطة من هذا المنحنى عن نقطتين ثابتين داخله ( تدعيان البؤرتين foci واحده بؤرة focus) يبقى ثابتا.

القطع الناقص وبعض خصائصه

القطع الناقص هو أيضا أحد أنواع القطوع المخروطية, فعند قطع مخروط بمستوى لا يمر بقاعدته يصبح التقاطع بين المخروط والمستوي قطعا ناقصا.

 عناصر القطع الناقص

انظر أيضاً: features of conic sections

The ellipse and some of its mathematical properties.

 رسم القطوع الناقصة

 طريقة الدبابيس والحبل

Drawing an ellipse with two pins, a loop, and a pen.

 طريقة ترامل

Trammel of Archimedes (ellipsograph) animation

 الهندسة الإقليدية

 المعادلات

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is ${\displaystyle{\displaystyle\left({\frac{x}{a}}\right)^{2}+\left({\frac{y}{b}}\right)^{2}=1.}}$

This means any noncircular ellipse is a squashed circle. If we draw an ellipse twice as long as it is wide, and draw the circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse. So the area enclosed by an ellipse is easy to calculate—it's the lengths of elliptic arcs that are hard.

 البؤرة

The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

${\displaystyle{\displaystyle f={\sqrt{a^{2}-b^{2}}}.}}$

 Eccentricity

The eccentricity of the ellipse (commonly denoted as either e or ${\displaystyle{\displaystyle\varepsilon}}$) is

${\displaystyle{\displaystyle e=\varepsilon={\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}}={\sqrt{1-\left({\frac{b}{a}}\right)^{2}}}=f/a}}$

(where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening factor ${\displaystyle{\displaystyle g=1-{\frac{b}{a}}=1-{\sqrt{1-e^{2}}},}}$

${\displaystyle{\displaystyle e={\sqrt{g(2-g)}}.}}$

 Directrix

 Ellipse as hypotrochoid

An ellipse (in red) as a special case of the hypotrochoid with R = 2r.

The ellipse is a special case of the hypotrochoid when R = 2r.

 المساحة

The area ${\displaystyle{\displaystyle A_{\text{ellipse}}}}$ enclosed by an ellipse is:

${\displaystyle{\displaystyle A_{\text{ellipse}}=\pi ab}}$

where a and b are one-half of the ellipse's major and minor axes respectively.

An ellipse defined implicitly by ${\displaystyle{\displaystyle Ax^{2}+Bxy+Cy^{2}=1}}$ has area ${\displaystyle{\displaystyle{\frac{2\pi}{\sqrt{4AC-B^{2}}}}}}$.

The area formula πab is easy to understand: start with a circle of radius b (so its area is πb²) and stretch it by a factor a/b to make an ellipse. This increases the area by the same factor: πb²(a/b) = πab.

For the ellipse in standard form, ${\displaystyle{\displaystyle{\frac{x^{2}}{a^{2}}}+{\frac{y^{2}}{b^{2}}}=1}}$, and hence ${\displaystyle{\displaystyle y=\pm{\sqrt{\frac{a^{2}b^{2}-b^{2}x^{2}}{a^{2}}}}}}$, with horizontal intercepts at ± a, the area ${\displaystyle{\displaystyle A_{\text{ellipse}}}}$ can be computed as twice the integral of the positive square root:

${\displaystyle{\displaystyle{\begin{aligned} \displaystyle A_{\text{ellipse}}&\displaystyle=\int_{-a}^{a}2b{\sqrt{1-x^{2}/a^{2}}}\,dx\\ &\displaystyle={\frac{b}{a}}\int_{-a}^{a}2{\sqrt{a^{2}-x^{2}}}\,dx.\end{aligned}}}}$

The second integral is the area of a circle of radius ${\displaystyle{\displaystyle a}}$, i.e., ${\displaystyle{\displaystyle\pi a^{2}}}$; thus we have ${\displaystyle{\displaystyle A_{\text{ellipse}}={\frac{b}{a}}A_{\text{circle}}=\pi ab}}$.

The area formula can also be proven in terms of polar coordinates using the coordinate transformation ${\displaystyle{\displaystyle{\mathbf{T}}(r,\theta)=(ra\cos\theta,rb\sin\theta).}}$

Any point inside the ellipse with x-intercept a and y-intercept b can be defined in terms of r and ${\displaystyle{\displaystyle\theta}}$, where ${\displaystyle{\displaystyle 0\leqslant r\leqslant 1}}$ and ${\displaystyle{\displaystyle 0\leqslant\theta\leqslant 2\pi}}$.

To define the area differential in such coordinates we use the Jacobian matrix of the coordinate transformation times ${\displaystyle{\displaystyle dr\,d\theta}}$: ${\displaystyle{\displaystyle dA_{\text{ellipse}}=\det{\begin{pmatrix}{\frac{\partial{\mathbf{T}}}{\partial r}}&{\frac{\partial{\mathbf{T}}}{\partial\theta}}\\ \end{pmatrix}}\,dr\,d\theta=\det{\begin{pmatrix}a\cos\theta&-ra\sin\theta\\ b\sin\theta&rb\cos\theta\end{pmatrix}}\,dr\,d\theta=abr\,dr\,d\theta}}$. We now integrate over the ellipse to find the area: ${\displaystyle{\displaystyle A_{\text{ellipse}}=\int_{\text{ellipse}}dA_{\text{ellipse}}=\iint_{\text{ellipse}}abr\,dr\,d\theta=ab\int_{0}^{2\pi}\int_{0}^{1}r\,dr\,d\theta=ab\pi}}$.

 المحيط

The circumference ${\displaystyle{\displaystyle C}}$ of an ellipse is:

${\displaystyle{\displaystyle C=4aE(e)}}$

where again a is the length of the semi-major axis and e is the eccentricity and where the function ${\displaystyle{\displaystyle E}}$ is the complete elliptic integral of the second kind. This may be evaluated directly using the Carlson symmetric form as illustrated by the following python code (this converges quadratically):

def EllipseCircumference(a, b):
   """
   Compute the circumference of an ellipse with semi-axes a and b.
   Require a >= 0 and b >= 0.  Relative accuracy is about 0.5^53.
   """
   import math
   x, y = max(a, b), min(a, b)
   digits = 53; tol = math.sqrt(math.pow(0.5, digits))
   if digits * y < tol * x: return 4 * x
   s = 0; m = 1
   while x - y > tol * y:
      x, y = 0.5 * (x + y), math.sqrt(x * y)
      m *= 2; s += m * math.pow(x - y, 2)
   return math.pi * (math.pow(a + b, 2) - s) / (x + y)

The exact infinite series is:

${\displaystyle{\displaystyle C=2\pi a\left[{1-\left({1\over 2}\right)^{2}e^{2}-\left({1\cdot 3\over 2\cdot 4}\right)^{2}{e^{4}\over 3}-\left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^{2}{e^{6}\over 5}-\cdots}\right]}}$

or

${\displaystyle{\displaystyle C=2\pi a\left[1-\sum_{n=1}^{\infty}\left({\frac{(2n-1)!!}{2^{n}n!}}\right)^{2}{\frac{e^{2n}}{2n-1}}\right],}}$

where ${\displaystyle{\displaystyle n!!}}$ is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of ${\displaystyle{\displaystyle h=(a-b)^{2}/(a+b)^{2}}}$, Ivory and Bessel derived an expression which converges much more rapidly,

${\displaystyle{\displaystyle C=\pi(a+b)\left[1+\sum_{n=1}^{\infty}\left({\frac{(2n-1)!!}{2^{n}n!}}\right)^{2}{\frac{h^{n}}{(2n-1)^{2}}}\right].}}$

A good approximation is Ramanujan's:

${\displaystyle{\displaystyle C\approx\pi\left[3(a+b)-{\sqrt{(3a+b)(a+3b)}}\right]=\pi\left[3(a+b)-{\sqrt{10ab+3(a^{2}+b^{2})}}\right]}}$

and a better approximation is

${\displaystyle{\displaystyle C\approx\pi\left(a+b\right)\left(1+{\frac{3h}{10+{\sqrt{4-3h}}}}\right).\!\,}}$

For the special case where the minor axis is half the major axis, these become:

${\displaystyle{\displaystyle C\approx{\frac{\pi a(9-{\sqrt{35}})}{2}}}}$

or, as an estimate of the better approximation,

${\displaystyle{\displaystyle C\approx{\frac{a}{2}}{\sqrt{93+{\frac{1}{2}}{\sqrt{3}}}}}}$

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.

انظر أيضاً: Meridian arc § Meridian distance on the ellipsoid

The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

 الأوتار

The midpoints of a set of parallel chords of an ellipse are collinear.^:p.147

 Latus rectum

The chords of an ellipse which are perpendicular to the major axis and pass through one of its foci are called the latera recta of the ellipse. The length of each latus rectum is 2b²/a.

 الانحناء

The curvature is ${\displaystyle{\displaystyle{\frac{1}{a^{2}b^{2}}}\left({\frac{x^{2}}{a^{4}}}+{\frac{y^{2}}{b^{4}}}\right)^{-{\frac{3}{2}}}}}$

 المعادلة الجبرية

جبريا, القطع الناقص هو منحنى في المستوى الكارتيزي معرف بالمعادلة:

${\displaystyle{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}}$

بحيث ان جميع المعاملات حقيقية و بحيث ${\displaystyle{\displaystyle B^{2}<4AC}}$. وجود أكثر من حل لقيم معينة لx, يعرف زوجا من النقاط (x, y1) و (x, y2) تقع على القطع الناقص.

ولإيجاد القانون العام للقطع الناقص, نستعمل التعريف التالي:

${\displaystyle{\displaystyle PS=e.PM}}$

حيث:

- P هي نقطة (x,y) تقع على القطع

- S البؤرة

- e معامل الاختلاف المركزي ( e<1 )

- و m هي مسقط العمودي ل P على الدليل

و يعبر القانون (أو المعادلة) على كون نسبة المسافة بين النقطة والبؤرة و المسافة بين النقطة والدليل ثابثة وتساوي معامل الاختلاف المركزي e.

 في حساب المثلثات

 الشكل المتغير العام

An ellipse in general position can be expressed parametrically as the path of a point ${\displaystyle{\displaystyle(X(t),Y(t))}}$, where

${\displaystyle{\displaystyle X(t)=X_{c}+a\,\cos t\,\cos\varphi-b\,\sin t\,\sin\varphi}}$

${\displaystyle{\displaystyle Y(t)=Y_{c}+a\,\cos t\,\sin\varphi+b\,\sin t\,\cos\varphi}}$

as the parameter t varies from 0 to 2π. Here ${\displaystyle{\displaystyle(X_{c},Y_{c})}}$ is the center of the ellipse, and ${\displaystyle{\displaystyle\varphi}}$ is the angle between the ${\displaystyle{\displaystyle X}}$-axis and the major axis of the ellipse.

 الشكل المتغير في الوضع التعريفي

Parametric equation for the ellipse (red) in canonical position. The eccentric anomaly t is the angle of the blue line with the X-axis.

For an ellipse in canonical position (center at origin, major axis along the X-axis), the equation simplifies to

${\displaystyle{\displaystyle X(t)=a\,\cos t}}$

${\displaystyle{\displaystyle Y(t)=b\,\sin t}}$

Note that the parameter t (called the eccentric anomaly in astronomy) is not the angle of ${\displaystyle{\displaystyle(X(t),Y(t))}}$ with the X-axis.

${\displaystyle{\displaystyle-\cot\phi={\frac{a}{b}}\tan t={\frac{\tan\theta}{(1-g)^{2}}}={\frac{\tan\theta}{1-e^{2}}},}}$

${\displaystyle{\displaystyle-\tan t={\frac{b}{a}}\cot\phi={\sqrt{(1-e^{2})}}\cot\phi=(1-g)\cot\phi={\frac{-\tan\theta}{\sqrt{(1-e^{2})}}}=-{\frac{a}{b}}\tan\theta.}}$

 الشكل القطبي بالنسبة للمركز

Polar coordinates centered at the center.

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate ${\displaystyle{\displaystyle\theta}}$ measured from the major axis, the ellipse's equation is

${\displaystyle{\displaystyle r(\theta)={\frac{ab}{\sqrt{(b\cos\theta)^{2}+(a\sin\theta)^{2}}}}}}$

 الشكل القطبي بالنسبة للبؤرة

Polar coordinates centered at focus.

If instead we use polar coordinates with the origin at one focus, with the angular coordinate ${\displaystyle{\displaystyle\theta=0}}$ still measured from the major axis, the ellipse's equation is

${\displaystyle{\displaystyle r(\theta)={\frac{a(1-e^{2})}{1\pm e\cos\theta}}}}$

where the sign in the denominator is negative if the reference direction ${\displaystyle{\displaystyle\theta=0}}$ points towards the center (as illustrated on the right), and positive if that direction points away from the center.

In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate ${\displaystyle{\displaystyle\phi}}$, the polar form is

${\displaystyle{\displaystyle r={\frac{a(1-e^{2})}{1-e\cos(\theta-\phi)}}.}}$

Semi-latus rectum.

 الشكل القطبي العام

The following equation on the polar coordinates (r, θ) describes a general ellipse with semidiameters a and b, centered at a point (r₀, θ₀), with the a axis rotated by φ relative to the polar axis:

${\displaystyle{\displaystyle r(\theta)={\frac{P(\theta)+Q(\theta)}{R(\theta)}}}}$

حيث

${\displaystyle{\displaystyle P(\theta)=r_{0}\left[\left(b^{2}-a^{2}\right)\cos\left(\theta+\theta_{0}-2\varphi\right)+\left(a^{2}+b^{2}\right)\cos\left(\theta-\theta_{0}\right)\right]}}$

${\displaystyle{\displaystyle Q(\theta)={\sqrt{2}}ab{\sqrt{R(\theta)-2r_{0}^{2}\sin^{2}\left(\theta-\theta_{0}\right)}}}}$

${\displaystyle{\displaystyle R(\theta)=\left(b^{2}-a^{2}\right)\cos(2\theta-2\varphi)+a^{2}+b^{2}}}$

 Angular eccentricity

The angular eccentricity ${\displaystyle{\displaystyle\alpha}}$ is the angle whose sine is the eccentricity e; that is,

${\displaystyle{\displaystyle\alpha=\sin^{-1}(e)=\cos^{-1}\left({\frac{b}{a}}\right)=2\tan^{-1}\left({\sqrt{\frac{a-b}{a+b}}}\right);\,\!}}$

 القطوع الناقصة في الفيزياء

 مدارات الكواكب

•  مقالة مفصلة: مدار إهليلجي

For elliptical orbits, useful relations involving the eccentricity ${\displaystyle{\displaystyle e}}$ are:

${\displaystyle{\displaystyle e={\frac{r_{a}-r_{p}}{r_{a}+r_{p}}}={\frac{r_{a}-r_{p}}{2a}}}}$

${\displaystyle{\displaystyle r_{a}=(1+e)a}}$

${\displaystyle{\displaystyle r_{p}=(1-e)a}}$

حيث

• ${\displaystyle{\displaystyle r_{a}}}$ is the radius at apoapsis (the farthest distance)
• ${\displaystyle{\displaystyle r_{p}}}$ is the radius at periapsis (the closest distance)
• ${\displaystyle{\displaystyle a}}$ is the length of the semi-major axis

Also, in terms of ${\displaystyle{\displaystyle r_{a}}}$ and ${\displaystyle{\displaystyle r_{p}}}$, the semi-major axis ${\displaystyle{\displaystyle a}}$ is their arithmetic mean, the semi-minor axis ${\displaystyle{\displaystyle b}}$ is their geometric mean, and the semi-latus rectum ${\displaystyle{\displaystyle l}}$ is their harmonic mean. In other words,

${\displaystyle{\displaystyle a={\frac{r_{a}+r_{p}}{2}}}}$

${\displaystyle{\displaystyle b={\sqrt[2]{r_{a}\cdot r_{p}}}}}$

${\displaystyle{\displaystyle l={\frac{2}{{\frac{1}{r_{a}}}+{\frac{1}{r_{p}}}}}={\frac{2r_{a}r_{p}}{r_{a}+r_{p}}}}}$.

 انظر أيضاً

• Apollonius of Perga, the classical authority
• Cartesian oval, a generalization of the ellipse
• Circumconic and inconic
• Conic section
• Ellipsoid, a higher dimensional analog of an ellipse
• Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolae
• Elliptical distribution, in statistics
• Elliptic partial differential equation
• Great ellipse
• Hyperbola
• Kepler's laws of planetary motion
• Matrix representation of conic sections
• n-ellipse, a generalization of the ellipse for n foci
• Oval
• Parabola
• Proofs involving the ellipse
• Spheroid, the ellipsoid obtained by rotating an ellipse about its major or minor axis
• Steiner circumellipse, the unique ellipse circumscribing a triangle and sharing its centroid
• Steiner inellipse, the unique ellipse inscribed in a triangle with tangencies at the sides' midpoints
• Superellipse, a generalization of an ellipse that can look more rectangular or more "pointy"
• True, eccentric, and mean anomaly
• Geodesics on an ellipsoid

 المراجع

• Besant, W.H. (1907). "Chapter III. The Ellipse". Conic Sections. London: George Bell and Sons. p. 50. {{cite book}}: Invalid |ref=harv (help)
• Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990). Fundamentals of College Algebra (3rd ed.). Scott Foresman/Little. p. 381. ISBN 0-673-38638-4.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Coxeter, H.S.M. (1969). Introduction to Geometry (2nd ed.). New York: Wiley. pp. 115–9.
• Ellipse at Planetmath
• Eric W. Weisstein, Ellipse at MathWorld.

 وصلات خارجية

• Video: How to draw Ellipse
• Apollonius' Derivation of the Ellipse at Convergence
• The Shape and History of The Ellipse in Washington, D.C. by Clark Kimberling
• Collection of animated ellipse demonstrations. Ellipse, axes, semi-axes, area, perimeter, tangent, foci.
• Eric W. Weisstein, Ellipse as hypotrochoid at MathWorld.
• Ivanov, A.B. (2001), "Ellipse", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104