Ellipse

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • An ellipse is a two-dimensional figure with 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".
  • The major axis is always the longest axis in an ellipse.
  • The minor axis is always the shortest axis in an ellipse.

ellipse 5Standard Ellipse Formula

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

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

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

Where:

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

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

\(\large{ a }\) = length semi-minor axis

\(\large{ b }\) = length semi-minor axis

\(\large{ h }\) and \(\large{ k }\) = center point of ellipse

ellipse 4Area of an Ellipse formula

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

Where:

\(\large{ A }\) = area

\(\large{ a }\) = length semi-major axis

\(\large{ b }\) = length semi-minor axis

\(\large{ \pi }\) = Pi

ellipse 3cFoci of an Ellipse formula

\(\large{ c^2 = a^2 - b^2  }\)      

Where:

\(\large{ c }\) = length center to focus

\(\large{ a }\) = length semi-major axis

\(\large{ b }\) = length semi-minor axis

\(\large{ F }\) and \(\large{ G }\) = focus

ellipse 4Perimeter of an Ellipse formula

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

Where:

\(\large{ p }\) = perimeter

\(\large{ a }\) = length semi-major axis

\(\large{ b }\) = length semi-minor axis

\(\large{ \pi }\) = Pi

ellipse 1ellipse 2Semi-major and Semi-minor Axis of an Ellipse formula

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

\(\large{ a = \frac {l} {1- \epsilon^2} }\)

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

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

Where:

\(\large{ a }\) = length semi-major axis

\(\large{ b }\) = length semi-minor axis

\(\large{ A }\) = area

\(\large{ \pi }\) = Pi

\(\large{ l }\) = semi-latus rectum

\(\large{ \epsilon }\)  (Greek symbol epsilon) = eccentricity