افتح القائمة الرئيسية

قطع ناقص

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.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

فهرست

عناصر القطع الناقص

 
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


طريقة متوازي الأضلاع

تقريبات القطوع الناقصة

الهندسة الإقليدية

التعريف

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

المعادلات

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is  

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:

 

Eccentricity

The eccentricity of the ellipse (commonly denoted as either e or  ) is

 

(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  

 

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   enclosed by an ellipse is:

 

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

An ellipse defined implicitly by   has area  .

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,  , and hence  , with horizontal intercepts at ± a, the area   can be computed as twice the integral of the positive square root:

 

The second integral is the area of a circle of radius  , i.e.,  ; thus we have  .

The area formula can also be proven in terms of polar coordinates using the coordinate transformation  

Any point inside the ellipse with x-intercept a and y-intercept b can be defined in terms of r and  , where   and  .

To define the area differential in such coordinates we use the Jacobian matrix of the coordinate transformation times  :  . We now integrate over the ellipse to find the area:  .


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

المحيط

The circumference   of an ellipse is:

 

where again a is the length of the semi-major axis and e is the eccentricity and where the function   is the complete elliptic integral of the second kind. This may be evaluated directly using the Carlson symmetric form[1] 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:

 

or

 

where   is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of  , Ivory[2] and Bessel[3] derived an expression which converges much more rapidly,

 

A good approximation is Ramanujan's:

 

and a better approximation is

 

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

 

or, as an estimate of the better approximation,

 

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.

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

الأوتار

The midpoints of a set of parallel chords of an ellipse are collinear.[4]: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  

المعادلة الجبرية

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

 

بحيث ان جميع المعاملات حقيقية و بحيث  . 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:

 

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  , where

 
 

as the parameter t varies from 0 to 2π. Here   is the center of the ellipse, and   is the angle between the  -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

 
 

Note that the parameter t (called the eccentric anomaly in astronomy) is not the angle of   with the X-axis.


 
 

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   measured from the major axis, the ellipse's equation is

 

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   still measured from the major axis, the ellipse's equation is

 

where the sign in the denominator is negative if the reference direction   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  , the polar form is

 
 
Semi-latus rectum.


الشكل القطبي العام

The following equation on the polar coordinates (rθ) describes a general ellipse with semidiameters a and b, centered at a point (r0θ0), with the a axis rotated by φ relative to the polar axis:[بحاجة لمصدر]

 

حيث

 
 
 

Angular eccentricity

The angular eccentricity   is the angle whose sine is the eccentricity e; that is,

 

درجات الحرية

القطوع الناقصة في الفيزياء

العاكسات الإهليلجية وعلم الصوت

مدارات الكواكب

For elliptical orbits, useful relations involving the eccentricity   are:

 
 
 

حيث

  •   is the radius at apoapsis (the farthest distance)
  •   is the radius at periapsis (the closest distance)
  •   is the length of the semi-major axis

Also, in terms of   and  , the semi-major axis   is their arithmetic mean, the semi-minor axis   is their geometric mean, and the semi-latus rectum   is their harmonic mean. In other words,

 
 
 .

المتأرجحات المتناغمة

انظر أيضاً

المراجع

  • Besant, W.H. (1907). "Chapter III. The Ellipse". Conic Sections. London: George Bell and Sons. p. 50.
  • 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.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.

الهامش

  1. ^ DOI:10.1007/BF02198293
    This citation will be automatically completed in the next few minutes. You can jump the queue or expand by hand
  2. ^ Ivory, J. (1798). "A new series for the rectification of the ellipsis". Transactions of the Royal Society of Edinburgh. 4: 177–190.
  3. ^ Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)". Astron. Nachr. 331 (8): 852–861. arXiv:0908.1824. doi:10.1002/asna.201011352. English translation of Astron. Nachr. 4, 241-254 (1825).
  4. ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.

وصلات خارجية