Ellipse

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.

Ellipse Index

 

Standard Ellipse formulas

\( (x^2\;/\;a^2)  +  (y^2\;/\;x^2)  = 1  \) 

\( \left( x\;/\;a \right)^2  +  \left( y\;/\;x \right)^2  = 1  \) 

\(  [\; \left( x - h \right )^2 \;/\; a^2 \;]   +  [ \; \left( y - k \right )^2 \;/\; b^2 \; ]  = 1  \)     (major axis horizontal)

 \( [\; \left( x \;-\; h \right )^2 \;/\; b^2 \; ]  +  [\; \left( y \;-\; k \right )^2 \;/\; a^2 \;]  = 1  \)     (major axis vertical)

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

 

Area of an Ellipse formula

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

 

Circumference of an Ellipse formula

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

  

eccentricity of an Ellipse formula

\( \epsilon = \sqrt{  a^2 - b^2 \;/\; a^2 }   \) 

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

\( \epsilon =  \sqrt{ 1 - ( b^2 \;/\; a^2 ) }   \)

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

  

Perimeter of an Ellipse formulaS

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

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

\( p \approx 2\; \pi\; \sqrt{ a^2 + b^2\;/\;2} \)

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

 

Latus Rectum of an Ellipse formula

\( L =  2 \; b^2 \;/\; a \) 
Symbol English Metric
\( L \) = Latus rectum \( in \) \( mm \)
\( a \) = length semi-major axis \( in \) \( mm \)
\( b \) = length semi-minor axis \( in \) \( mm \)

 

Semi-major Axis Length of an Ellipse formula

\( a = A\;/\;pi \; b\) 
Symbol English Metric
\( A \) = area \( in^2 \) \( mm^2 \)
\( a \) = length semi-major axis \( in \) \( mm \)
\( b \) = length semi-minor axis \( in \) \( mm \)
\( \pi \) = Pi \(3.141 592 653 ...\)

 

Semi-minor Axis Length of an Ellipse formula

\( b = A\;/\;pi \; a \) 
Symbol English Metric
\( A \) = area \( in^2 \) \( mm^2 \)
\( a \) = length semi-major axis \( in \) \( mm \)
\( b \) = length semi-minor axis \( in \) \( mm \)
\( \pi \) = Pi \(3.141 592 653 ...\)

 

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Tags: Ellipse