The Conic section reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Conic section

In mathematics, a conic section is a two-dimensional curved locus of points, formed by the intersection of a cone and a plane. Should the plane pass through the cone's vertex, a degenerate conic will be formed.

The most well-known members, of this family, 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 is not 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

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.

The conic sections were given their names by Apollonius of Perga.

In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations.

Table of contents

1 Derivation 1.1 Derivation of the Parabola 1.2 Derivation of the Ellipse 1.3 Derivation of the Hyperbola 2 See also 3 External links

Derivation

Let there be a cone whose axis is the z-axis. Let its vertex be the origin. The equation for the cone is

where

and θ is the angle which the generators of the cone make with respect to the axis. Notice that this cone is actually a pair of cones: one cone standing upside down on the vertex of the other cone.

Let there be a plane with a slope running along the x direction but which is level along the y direction. Its equation is

where

and φ is the angle of the plane with respect to the x-y plane.

We are interested in finding the intersection of the cone and the plane, which means that equations (1) and (2) shall be combined. Both equations can be solved for z and then equate the two values of z. Solving equation (1) for z yields

therefore

Square both sides and expand the squared binomial on the right side,

Grouping by variables yields

Derivation of the Parabola

The parabola is obtained when the slope of the plane is equal to the slope of the generators of the cone. When these two slopes are equal, then the angles θ and φ become complementary. This implies that

therefore

Substituting equation (4) into equation (3) makes the first term in equation (3) vanish, and the remaining equation is

Multiply both sides by a²,

then solve for x,

Equation (5) describes a parabola whose axis is parallel to the x-axis. Other versions of equation (5) can be obtained by rotating the plane around the z-axis.

Derivation of the Ellipse

An ellipse happens when the angles θ and φ, when added together, do not measure up to a right angle:

which implies that the tangent of the sum of these two angles is positive.

But a trigonometric identity states that

therefore

but m + a is positive, since the summands are given to be positive, so inequation (6) is positive if the denominator is also positive:

From inequation (7) we can deduce

Let us start out again from equation (3),

but this time the coefficient of the x² term does not vanish but is instead positive. Solve for y,

This would clearly describe an ellipse were it not for the second term under the radical, the 2 m b x: it would be the equation of a circle which has been stretched proportionally along the directions of the x-axis and the y-axis. Equation (8) is an ellipse but it is not obvious, so it will be rearranged further until it is obvious. Complete the square under the radical,

Group together the b² terms,

Divide by a then square both sides,

The x has a coefficient. It is desired to pull this coefficient out by factoring it out of the second term which is a square,

Further rearrangements of constants finally leads to

The coefficient of the y term is positive (for an ellipse). Renaming of coefficients and constants leads to

which is clearly the equation of an ellipse. That is, equation (9) describes a circle of radius R and center (C,0) which is then stretched vertically by a factor of . The second term on the left side (the x term) has no coefficient but is a square, so that it must be positive. The radius is a product of squares, so it must also be positive. The first term on the left side(the y term) has a coefficient which is positive, so the equation describes an ellipse.

Derivation of the Hyperbola

The hyperbola happens when the angles θ and φ add up to an obtuse angle, which is greater than a right angle. The tangent of an obtuse angle is negative. All the inequations which were valid for the ellipse become reversed. Therefore

Otherwise the equation for the hyperbola is the same as equation (9) for the ellipse, except that the coefficient A of the y term is negative. The sign change is enough to convert an ellipse into a hyperbola. This is because the equation of a real ellipse contains an imaginary hyperbola, and the equation of a real hyperbola contains an imaginary ellipse (see imaginary number). The sign change of coefficient A causes real and imaginary values of the function y=f(x) equivalent to equation (9) to swap.

See also

• Matrix representation of conic sections
• Quadratic function

External links

• Special plane curves: Conic sections

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