Ellipse

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • ellipse 10ellipse 13Ellipse (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)

Where:

 Units English Metric
\(\large{ x }\) = horizontal coordinate of a point on the ellipse - -
\(\large{ y }\) = vertical coordinate of a point on the ellipse - -
\(\large{ a }\) = 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 - -

 

Ellipse Area formula

\(\large{ A = \pi \;a\; b }\)   

Where:

 Units 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 \(\large{3.141 592 653 ...}\)

 

Ellipse Circumference formula

\(\large{ C =  2\;\pi \;  \sqrt{ \frac{ a^2 \;+\; b^2 }{ 2 }  }   }\)   

Where:

 Units 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 \(\large{3.141 592 653 ...}\)

 

Ellipse eccentricity formula

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

Where:

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

 

Ellipse Foci formula

\(\large{ c^2 = a^2 - b^2  }\)   

Where:

 Units English Metric
\(\large{ c }\) = length center to focus \(\large{ in }\)  \(\large{ mm }\)  
\(\large{ a }\) = length semi-major axis \(\large{ in }\) \(\large{ mm }\) 
\(\large{ b }\) = length semi-minor axis \(\large{ in }\) \(\large{ mm }\) 
\(\large{ F }\) and \(\large{ G }\) = focus \(\large{ in }\) \(\large{ mm }\)  

 

Ellipse Perimeter formula

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

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

Where:

 Units 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 \(\large{3.141 592 653 ...}\)

 

Ellipse Latus Rectum formula

\(\large{ a = \frac{ 2 \; b^2 }{ a } }\)   

Where:

 Units English Metric
\(\large{ L }\) = Latus rectum - -
\(\large{ a }\) = length semi-major axis \(\large{ in }\) \(\large{ mm }\)
\(\large{ b }\) = length semi-minor axis \(\large{ in }\) \(\large{ mm }\)

 

Ellipse Semi-major Axis Length formula

\(\large{ a = \frac{A}{\pi \; b} }\)   

Where:

 Units 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 \(\large{3.141 592 653 ...}\)

 

Ellipse Semi-minor Axis Length formula

\(\large{ b = \frac{A}{\pi \; a} }\)   

Where:

 Units 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 \(\large{3.141 592 653 ...}\)

 

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