Ellipse

Written by Jerry Ratzlaff on . Posted in Plane Geometry

Ellipse

An ellipse is a flat plane curve that when add togeather any two distances from any point on the ellipse to each of the foci will always equal the same. 

Standard Ellipseellipse 5

Standard Ellipse Formula

\( \frac {x^2}{a^2}  \;+\;  \frac {y^2}{x^2}  \;=\; 1  \)         

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

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

Where:

\( x \) = horizontal coordinate of a point on the ellipse

\( y \) = vertical coordinate of a point on the ellipse

\( a \) = length semi-minor axis

\( b \) = length semi-minor axis

\( h \) and \( k \) = center point of ellipse

Area of an Ellipseellipse 4

Area of an Ellipse formula

\(A = \pi a b \)          \( area  \;=\;  Pi  \;\;x\;\;  axis \; a  \;\;x\;\; axis \; b \)

Where:

\(A\) = area

\(a\) = length semi-major axis

\(b\) = length semi-minor axis

\(\pi\) = Pi

Foci of an Ellipseellipse 3c

Foci is a point used to define the conic section.  \(F\) and \(G\) seperately are called "focus", both togeather are called "foci".

Foci of an Ellipse formula

\(c^2 = a^2 - b^2  \)          \( center \; to \; focus^2  \;=\;  semi-major \; axis^2  \;-\;  semi-minor \; axis^2  \)        

Where:

\(c\) = length center to focus

\(a\) = length semi-major axis

\(b\) = length semi-minor axis

\(F\) and \(G\) = focus

Perimeter of an Ellipseellipse 4

The 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".

Perimeter of an Ellipse formula

\(p \approx 2\pi \sqrt { \frac{1}{2} \left(a^2 + b^2 \right) } \)          \( perimeter   \;\approx\;   2 \;\;x\;\;  Pi \; \sqrt { \; \frac{1}{2} \; \left( \; axis \; a^2 \;+\; axis \; b^2 \; \right) } \)

Where:

\(p\) = perimeter

\(a\) = length semi-major axis

\(b\) = length semi-minor axis

\(\pi\) = Pi

Semi-major and Semi-minor Axis of an Ellipseellipse 1ellipse 2

The major axis is always the longest axis in an ellipse.

The minor axis is always the shortest axis in an ellipse.

Semi-major and Semi-minor Axis of an Ellipse formula

\(a = \frac{A}{\pi b} \)          \( axis \; a \;=\;  \frac{ area }{ Pi \;\;x\;\; axis \; b} \)

\(a = \frac {l} {1- \epsilon^2} \)          \( axis \; a \;=\;  \frac { semi-latus \; rectum } { 1 \;-\; eccentricity^2} \)

\(b = \frac{A}{\pi a} \)          \( axis \; b \;=\; \frac{ area }{ Pi  \;\;x\;\; axis \; a} \)

\(b = a \sqrt {1- \epsilon^2} \)          \( axis \; b \;=\; semi-major \; axis \; \sqrt { \; 1 \;-\; eccentricity^2} \)

Where:

\(a\) = length semi-major axis

\(b\) = length semi-minor axis

\(A\) = area

\(\pi\) = Pi

\(l\) = semi-latus rectum

\(\epsilon\) (Greek symbol epsilon) = eccentricity

 

Tags: Equations for Area Equations for Perimeter