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Ellipse. Formulas, elements and properties of an ellipse

Definition of an ellipse

Definition.
Ellipse is a closed curve around two different points (focal points F1 and F2) in a plane such that the sum of the distances from the two focal points is constant for every point (Mn) on the curve.

F1M1 + F2M1 = F1M2 + F2M2 = A1A2 = const


Ellipse Image Ellipse Image
Fig.1 Fig.2

Elements of ellipse

F1 и F2 - focal points of ellipse
Axes of ellipse.

А1А2 = 2a - major axis (larger direct that crosses focal points F1 and F2)

B1B2 = 2b - minor axis (smaller direct that perpendicular to major axis and intersect it at the center of the ellipse О)

a - semi-major axis

b - semi-minor axis

O - center of the ellipse

Vertices of ellipse A1, A2, B1, B2
Diameter of ellipse - is any straight line segment that passes through the center of the ellipse and whose endpoints lie on the ellipse.
Linear eccentricity c - is the distance from the focal point to the center of the ellipse.
Eccentricity of ellipse e is the ratio of the linear eccentricity c to the length of the semi-major axis a. The eccentricity of an ellipse always be 0 < e < 1, the eccentricity of the circle is e = 0, the eccentricity of the parabola is e = 1, the eccentricity of the hyperbola is e > 1.
e = c
a
Focal radius of ellipse r1, r2 is a distances from point on ellipse to focal point.
Radius of an ellipse R - is a distance from ellipse the center to point (Мn) at ellipse.
R = ab = b
a2sin2φ + b2cos2φ1 - e2cos2φ

де e - eccentricity, а φ - the angles within the radius (R) and major axis A1A2.
Focal parameter of ellipse p - is the focal radius that perpendicular to ma axis:
p = b2
a
Flattening (ellipticity) of ellipse (oblateness) k is a measure of the compression of a circle along a semi-minor axis. Since the semi-minor axis of the ellipse is always smaller then semi-major axis, then k < 1, and for circle k = 1:
f = a - b
a

f = 1 - √1 - e2

where e - eccentricity.
Directrix of ellipse (1 - k ) is a line parallel to the minor axis and no touch to the ellipse. The distance from any point M on the ellipse to the focus F is a constant fraction of that points perpendicular distance to the directrix, resulting in the equality p/e.

Properties of ellipse

1. Angle betwin tangent and focal radius r1 is equal to angle betwin tangent and focal radius r2 (Fig. 2, point М3).
2. Equations of tangent in point М with coordinates (xM, yM):
1 = xxM + yyM
a2b2
3. If two parallel lines cross the ellipse than the distance that conect the midle point of a segment this lines will cross the center of the ellipse.
4. Evolute is the asteroid that stretched along the long axis.
5. If the inscribe the ellipse with foci F1 and F2 in any triangle ∆ ABC than:

1 =  F1A ∙ F2A + F1B ∙ F2B + F1C ∙ F2C
CA ∙ ABAB ∙ BCBC ∙ CA

Equations of ellipse

Canonical equations of ellipse:

The ellipse can be described in Cartesian coordinate system if the center O of the ellipse locate in origin coordinate axis, than:
1 = x2 + y2
a2b2
If the center of the ellipse O is located in point with coordinates (xo, yo), than:
1 = (x - xo)2 + (y - yo)2
a2b2

Parametric equations of ellipse:

{x = a cos α  де 0 ≦ α < 2π
y = b sin α

Inscribed circle of ellipse

A circle inscribe in the ellipse touch to two vertices B1 and B2. Thereby, inradius r the same like semi-minor axis OB1:
r = b

Exscribed circle of ellipse

A circle exscribe the ellipse touch to two vertices A1 and A2. Thereby, exradius R the same like semi-major axis OA1:
R = a

Area of ellipse

Formulas definition of ellipse area:
S = πab

Area of ellipse segment

Arc segment area at the left side of chord with coordinates (x, y) and (x, -y):
S = πab - b(xa2 - x2 + a2 ∙ arcsinx)
2aa

Circumference of ellipse (perimeter approximation)

The circumference (C) of ellipse is very difficult to calculate. Below formula an approximation that is within about ~0,63% of the true value:
C ≈ 4πab + (a - b)2
a + b

Arc of ellipse

Formulas definition length of an arc of an ellipse:

1. Parametric form the length of an arc of an ellipse in terms of semi-major axisa and semi-minor axis b:
t2
l = a2sin2t + b2cos2t  dt
t1
2. Parametric form the length of an arc of an ellipse in terms of semi-major axis a and eccentricity e:
t2
l = 1 - e2cos2t  dt,    e < 1
t1

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