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   Ellipse & Parabola Formulas   

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also see Defining Geometric Figures

"A conic (or conic section) is a plane curve that can be obtained by intersecting a cone with a plane that does not go through the vertex of the cone. There are three possibilities, depending on the relative position of the cone and the plane. If no line of the cone is parallel to the plane, the intersection is a closed curve, called an ellipse. If one line of the cone is parallel to the plane, the intersection is an open curve whose two ends are asymptotically parallel; this is called a parabola. Finally, there may be two lines in the cone parallel to the plane; the curve in this case has two open pieces, and is called a hyperbola." (See Conics, The Geometry Center.)

 Ellipse

   
Semi-axes: a, b

Eccentricity: e = sqrt(a2-b2)/a
Area: K
Circumference: C
 

    K = Pi ab

C = 4aE, where E is an elliptic integral with k = e, which can be used to derive the following formulas:

C = (a+b)[1 + x2/4 + x4/64 + ...],
where x = (a-b)/(a+b)

C = (a+b)(1 + 3x2/[10 + sqrt(4 - 3x2)]), approximately


 Segment of a Parabola

   
Height: h
Chord length: c
Area: K
Length: s
 
    s = c[1+2(2h/c)2/3-2(2h/c)4/5+...]
s = sqrt[4h2+c2/4]+[c2/(8h)]
     ln[(2h+sqrt[4h2+c2/4])/(c/2)]

K = 2ch/3
K = 4T/3, where T is the area of
     the triangle formed by the chord
     and the point of tangency of a
     tangent to the parabola parallel
     to the chord  

 


To read more about ellipses and parabolas, visit:

The Geometry Center: Ellipse,  Parabola
J. Wilson Coe: Assignments
Paul Bourke: Classic Curves

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