That was Euclid's idea to make an equilateral triangle...........have two circles interconnect.............where each circle's edge meets the center of the other..........

So I know that the tip of the top equilateral triangle is where the top of the ellipse is that is formed by the intersection of the two circles.........kinda like a Venn diagram............................but with two circle's instead of 3...................

Mathematical definitions and properties[edit]

In Euclidean geometry[edit]

Definition[edit]

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the directrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).
The equivalence of these definitions can be proved using the Dandelin spheres.

Equations[edit]

The equation of an ellipse whose major axis is the ${\displaystyle x}$ axis and minor axis is the ${\displaystyle y}$ axis is

${\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1.}$

(1)

This equation is a direct consequence of the definition from the two focal points.^[14] This equation means an ellipse is a unit circle scaled by a factor of ${\displaystyle a}$ in the ${\displaystyle x}$ direction and a factor of ${\displaystyle b}$ in the ${\displaystyle y}$ direction.
The trigonometric parametric formula

${\displaystyle {\begin{aligned}x(t)&=a\cos t,\\y(t)&=b\sin t\end{aligned}}}$

( 2)

is equivalent to (1). Substituting ${\displaystyle a\cos t}$ for ${\displaystyle x}$ and ${\displaystyle b\sin t}$ for ${\displaystyle y}$ in (1) yields the basic trigonometric identity ${\displaystyle \cos ^{2}t+\sin ^{2}t=1.}$

Focus[edit]

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Relationship between the linear eccentricity, major axis and minor axis, exemplified by the (5, 12, 13) Pythagorean triple

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

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

The sum of the distances from any point P = P(x,y) on the ellipse to those two foci is constant and equal to the major axis length:

${\displaystyle PF_{1}+PF_{2}={\sqrt {(x+f)^{2}+y^{2}}}+{\sqrt {(x-f)^{2}+y^{2}}}=2a.}$

This is just a mathematical formulation of the definition in the first sentence of this article.

Eccentricity[edit]

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

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

(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 ${\displaystyle g=1-{\frac {b}{a}}=1-{\sqrt {1-e^{2}}},}$

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

Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections. Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.

Directrix[edit]

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right, in which the ellipse is centered at the origin. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix, resulting in the equality e = PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse.
Besides the well-known ratio e = f/a, where f is the distance from the center to the focus and a is the distance from the center to the farthest vertices (most sharply curved points of the ellipse), it is also true that e = a/d, where d is the distance from the center to the directrix.

Circular directrix[edit]

The ellipse can also be defined as the set of points that are equidistant from one focus and a circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle equals the ellipse's major axis, so the focus and the entire ellipse are inside the directrix circle.

Ellipse as hypotrochoid[edit]

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, as shown in the adjacent image.

Area[edit]

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

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

where ${\displaystyle a}$ and ${\displaystyle b}$ are the lengths of the semi-major and semi-minor axes, respectively. The area formula ${\displaystyle \pi ab}$ is intuitive: start with a circle of radius ${\displaystyle b}$ (so its area is ${\displaystyle \pi b^{2}}$) and stretch it by a factor ${\displaystyle a/b}$ to make an ellipse. This scales the area by the same factor: ${\displaystyle \pi b^{2}(a/b)=\pi ab.}$ It is also easy to rigorously prove the area formula using integration as follows. Equation (1) can be rewritten as ${\displaystyle y(x)=b{\sqrt {1-x^{2}/a^{2}}}.}$ For ${\displaystyle x\in [-a,a],}$ this curve is the top half of the ellipse. So twice the integral of ${\displaystyle y(x)}$ over the interval ${\displaystyle [-a,a]}$ will be the area of the ellipse:

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

The second integral is the area of a circle of radius ${\displaystyle a,}$ that is, ${\displaystyle \pi a^{2}.}$ So

${\displaystyle A_{\text{ellipse}}={\frac {b}{a}}\pi a^{2}=\pi ab.}$

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

Circumference[edit]

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

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

where again ${\displaystyle a}$ is the length of the semi-major axis, ${\displaystyle e}$ is the eccentricity ${\displaystyle {\sqrt {1-b^{2}/a^{2}}},}$ and the function ${\displaystyle E}$ is the complete elliptic integral of the second kind,

${\displaystyle E(e)=\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta ,}$

which calculates the circumference of the ellipse in the first quadrant alone, and the formula for the circumference of an ellipse can thus be written

${\displaystyle C=4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta .}$

(3)

The arc length of an ellipse, in general, has no closed-form solution in terms of elementary functions. Elliptic integrals were motivated by this problem. Equation (3) may be evaluated directly using the Carlson symmetric form.^[15] This gives a succinct and quadratically converging iterative method for evaluating the circumference using the arithmetic-geometric mean.^[16]
The exact infinite series is:

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

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

where ${\displaystyle n!!}$ is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of ${\displaystyle h=(a-b)^{2}/(a+b)^{2},}$ Ivory^[17] and Bessel^[18] derived an expression that converges much more rapidly,

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

Ramanujan gives two good approximations for the circumference in §16 of "Modular Equations and Approximations to ${\displaystyle \pi }$";^[19] they are

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

and

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

The errors in these approximations, which were obtained empirically, are of order ${\displaystyle h^{3}}$ and ${\displaystyle h^{5},}$ respectively.
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.

See also: 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.^{[citation needed]}
Some lower and upper bounds on the circumference of the canonical ellipse ${\displaystyle x^{2}/a^{2}+y^{2}/b^{2}=1}$ with ${\displaystyle a\geq b}$ are^[20]

${\displaystyle C\leq 2\pi a,}$

${\displaystyle \pi (a+b)\leq C\leq 4(a+b),}$

${\displaystyle 4{\sqrt {a^{2}+b^{2}}}\leq C\leq {\sqrt {2}}\pi {\sqrt {a^{2}+b^{2}}}.}$

Here the upper bound ${\displaystyle 2\pi a}$ is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound ${\displaystyle 4{\sqrt {a^{2}+b^{2}}}}$ is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.

Chords[edit]

The midpoints of a set of parallel chords of an ellipse are collinear.^[21]:p.147

Latus rectum[edit]

The chords of an ellipse that 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.

Curvature[edit]

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

Angle bisection property[edit]

A local normal (perpendicular) to the ellipse at any point P on the ellipse bisects the angle ${\displaystyle \angle F_{1}PF_{2}}$ to the foci. This is evident graphically in the parallelogram method of construction, and can be proven analytically, for example by using the parametric form in canonical position, as given below.

Reflexive property[edit]

When a ray of light originating from one focus reflects off the inner surface of an ellipse, it always passes through the other focus. A ray of light coming from outside the ellipse toward a focus reflects off the ellipse directly away from the other focus.^{[22]:pp. 36ff.}

Tangent property[edit]

The angle containing part of the ellipse, formed at a point on the major axis by a tangent line to the ellipse and the major axis, has measure less than 45°.^{[22]:p. 26.}