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  Ellipse Homework Help - K-12 Grade Level, College Level Geometry Mathematics

Introduction of Ellipse

In mathematics, an Ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circle is special case of ellipses, obtained when the cutting plane is orthogonal to the cone's axis. Ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

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Ellipses are closed curves and are the bounded case of the conic sections the curve that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas. Ellipse arises from the intersection of a right circular cylinder with a plane that is not parallel to the cylinder's main axis of symmetry. Ellipse also arises as image of a circle under parallel projection and the bounded cases of perspective projection, which are just intersections of the projective cone with the plane of projection. In addition it is the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency.

Equations of Ellipse

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The equation of an ellipse given where major and minor axes coincide with the Cartesian axes is

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Any noncircular ellipse is known as squashed circle. If we describe an ellipse twice as long as it is wide, and illustrate the circle cantered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse. Thus the area covered by an ellipse is very easy to calculate-- it's measured as the lengths of elliptic arcs that are hard.

Measurement of Focus

The total distance from the center C to either focus is given f = ae, and it can be expressed in terms of the major and minor radii:

f   =  (a2 - b2)1/2

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