Conic section
In mathematics, a conic section is a curve that is formed by the intersection of the surface of an infinite cone and an infinite plane.
The most well-known members of this family of two-dimensional curves are the circle and the ellipse. These arise when the intersection is a closed curve[?]: the circle is a special case of the ellipse in which the plane is exactly perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the section is called a parabola. Finally, if the intersection is an open curve, and the plane isn't parallel to a generator line of the cone, the figure is a hyperbola.
The degenerate cases, where the plane passes through the vertex of the cone, resulting in an intersection figure of a point, a straight line or a pair of lines, are not considered as conic sections.
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. If the equation is of the form
- <math>ax^2 + 2hxy + by^2 +2gx + 2fy + c = 0</math>
then:
- If h2 = ab, this equation represents a parabola.
- If h2 < ab, this equation represents an ellipse.
- If h2 > ab, this equation represents a hyperbola.
- If a = b and h = 0, it represents a circle.
- If a + b = 0, it represents a rectangular hyperbola.
An alternative definition of conic sections starts with a point F (the focus), a line L not containing F (the directrix) and a positive number e (the eccentricity). The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. The case of a circle needs to be treated specially: one takes e = 0 and imagines the directrix infinitely far removed from the focus.
The eccentricity of a conic section is thus a measure of how far it deviates from being circular.
Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas.
External links
- Special plane curves: Conic sections (http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html)
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