# Ellipse

on . Posted in Plane Geometry

• Ellipse (a two-dimensional figure) is a conic section or a stretched circle.  It is a flat plane curve that when adding togeather any two distances from any point on the ellipse to each of the foci will always equal the same.
• Foci is a point used to define the conic section.  F and G seperately are called "focus", both togeather are called "foci".
• The perimeter of an ellipse formula is an approximation that is about 5% of the true value as long as "a" is no more than 3 times longer than "b".
• Latus rectum is a line drawn perpencicular to the transverse axis of the ellipse and is passing through the foci of the ellipse.
• The major axis is always the longest axis in an ellipse.
• The minor axis is always the shortest axis in an ellipse.

## Standard Ellipse formulas

$$\large{ \frac {x^2}{a^2} + \frac {y^2}{x^2} = 1 }$$

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

$$\large{ \frac { \left( x \;-\; h \right )^2 } { a^2 } + \frac { \left( y \;-\; k \right )^2 } { b^2 } = 1 }$$     (major axis horizontal)

$$\large{ \frac { \left( x \;-\; h \right )^2 } { b^2 } + \frac { \left( y \;-\; k \right )^2 } { a^2 } = 1 }$$     (major axis vertical)

Symbol English Metric
$$\large{ x }$$ = horizontal coordinate of a point on the ellipse $$\large{ in }$$ $$\large{ mm }$$
$$\large{ y }$$ = vertical coordinate of a point on the ellipse $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = length semi-major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ b }$$ = length semi-minor axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h }$$ and $$\large{ k }$$ = center point of ellipse $$\large{ in }$$ $$\large{ mm }$$

## Area of an Ellipse formula

$$\large{ A = \pi \;a\; b }$$
Symbol English Metric
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ a }$$ = length semi-major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ b }$$ = length semi-minor axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Circumference of an Ellipse formula

$$\large{ C = 2\;\pi \; \sqrt{ \frac{ a^2 \;+\; b^2 }{ 2 } } }$$
Symbol English Metric
$$\large{ C }$$ = circumference $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = length semi-major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ b }$$ = length semi-minor axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## eccentricity of an Ellipse formula

$$\large{ \epsilon = \sqrt{ \frac{ a^2 \;-\; b^2 }{ a^2 } } }$$

$$\large{ \epsilon = \left( \frac{ 1 \;-\; b^2 }{ a^2 } \right)^{0.5} }$$

$$\large{ \epsilon = \sqrt{ 1 - \frac{ b^2 }{ a^2 } } }$$

Symbol English Metric
$$\large{ \epsilon }$$  (Greek symbol epsilon) = eccentricity $$\large{ dimensionless }$$
$$\large{ a }$$ = one half of the ellipse's major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = one half of the ellipse's minor axis $$\large{ in }$$ $$\large{ mm }$$

This is an approximate perimeter of an ellipse formula.  There is no easy way to calculate the ellipse perimeter with high accuracy.

## Perimeter of an Ellipse formula

$$\large{ p \approx 2\; \pi\; \sqrt { \frac{1}{2}\; \left(a^2 + b^2 \right) } }$$

$$\large{ p \approx 2\; \pi\; \sqrt { \frac{a^2 \;+\; b^2}{2} } }$$

Symbol English Metric
$$\large{ p }$$ = perimeter approximation $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = length semi-major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ b }$$ = length semi-minor axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Latus Rectum of an Ellipse formula

$$\large{ L = \frac{ 2 \; b^2 }{ a } }$$
Symbol English Metric
$$\large{ L }$$ = Latus rectum $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = length semi-major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ b }$$ = length semi-minor axis $$\large{ in }$$ $$\large{ mm }$$

## Semi-major Axis Length of an Ellipse formula

$$\large{ a = \frac{A}{\pi \; b} }$$
Symbol English Metric
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ a }$$ = length semi-major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ b }$$ = length semi-minor axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Semi-minor Axis Length of an Ellipse formula

$$\large{ b = \frac{A}{\pi \; a} }$$
Symbol English Metric
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ a }$$ = length semi-major axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ b }$$ = length semi-minor axis $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$