So..........in relationship to the ellipse...........that naturally forms by the two overlapping circles.........................as with any ellipse........u have a major axis and a minor one............in other words............a longer line and a shorter line.......................................a long axis and a short one............u know that one axis is 1.................why?  It is the same as the equilateral triangle......along the X axis...........from - 1/2 to 1/2...............................




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]

{{{annotations}}}

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:

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