Ellipse. Formulas, elements and properties of an ellipse

Definition of ellipse Elements of ellipse Properties of ellipse Equations of ellipse Inscribed circle (incircle) of ellipse Exscribed circle (excircle) of ellipse Area of ellipse Area of the ellipse segment Circumference of ellipse Arc of 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 (M_n) on the curve.

F₁M₁ + F₂M₁ = F₁M₂ + F₂M₂ = A₁A₂ = const


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 (М_n) at ellipse.

R =	ab	=	b
	√a²sin²φ + b²cos²φ		√1 - 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 (x_M, y_M):

1 =	xx_M	+	yy_M
	a²		b²

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 ∙ AB		AB ∙ BC		BC ∙ 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
	a²		b²

If the center of the ellipse O is located in point with coordinates (x_o, y_o), than:

1 =	(x - x_o)²	+	(y - y_o)²
	a²		b²

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	(	x	√	a² - x² + a² ∙ arcsin	x	)
	2		a					a

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 ≈ 4	πab + (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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