conic section

There are a number of different shapes that can be made from taking a plane from a full cone. A full cone is the shape of an infinite diabolo: imagine this diagram in 3-D.

\   /
 \ /
  X
 / \
/   \

If a slice is taken horizontally, then a circle occurs. Generic formula: x²+y²=r (radius)

\     /         __
_\___/_        /  \
  \ /          |  |
   X           \__/ 
  / \
 /   \

If a slice is taken obliquely through one of the two halves, an ellipse can be seen. Generic formula: ax²+by²=c

\    _/_
 \__//
_/\ /          _____ 
   X          /     \
  / \         \_____/
 /   \
/     \

If a slice is taken parallel with one of the sides of the cone, a parabola occurs, which extends upwards to infinity. (sample formula: y=nx²+c)

\   / /    |   ^^   | 
 \ / /     |        |   
  X /       \      /   
 / X         |    |
/ / \         \__/
 /   \         
/     \ 

If a slice is taken through both cones, a hyperbola occurs. (can't remember formula)

\    |/       \         /
 \   |         \       /
  \ /|          \_   _/
   X |            \_/
  / \|
 /   |             _
/    |\          _/ \_
                /     \
               /       \
              /         \

This is a rectangular hyperbola with all angles tending to 45 degrees, due to the vertical line.

Conic sections are formed by a double right circular cone that is intersected by a plane. A double right cone is essentially a right cone that has a congruent right cone extending upwards from the first one.

When a plane is passed through these cones, one of seven things will happen, depending on the (smallest) angle the plane makes with the axis of the cone and whether the plane passes through the vertex of the cone:

1. If the plane intersects the axis of the cone at a right angle, the intersection will form a circle.
2. If the plane intersects the axis at an angle larger than the angle the cone makes with the axis, the intersection will form an ellipse.
3. If the plane intersects the axis at an angle equal to the angle the cone makes with the axis, the intersection will form a parabola.
4. If the plane intersects the axis at an angle smaller than the angle the cone makes with the axis, the intersection will form a hyperbola.
5. A point (degenerate circle or ellipse) is formed when the plane passes through the vertex at an angle greater than that of the cone.
6. A line (degenerate parabola) is formed when the plane passes through the vertex and along the sides of the two cones.
7. Intersecting lines (degenerate hyperbola) will form when a plane passes through the vertex of the the two cones at an angle less than the angle of the cones.

Without loss of generality, place vertex of the cone at the origin and assume the cone makes an angle α with the z axis. The locus of points is
x² + y² - (tan(α)*z)² = 0.

Assume that the plane, P, intersects the z axis at z0, is parallel with the x axis, and makes an angle β with the y axis. The locus of points is
z = z0 + cot(β)*y.
Combining the two equations will give the locus of points of the intersection, the conic section.

For the circle, the plane P is parallel to the x-y plane β =π/2, so z = z0 and
x² + y² - (tan(α)*z0)² = 0
which is the standard formula for a circle with a radius tan(α)*z0.

For the parabola, the plane P is parallel with the edge of the cone, α = β and z = z0 + cot(α)*y. Substituting this expression for z in the formula for the cone gives
x² + y² - (tan(α)*(z0 + cot(β)*y))² = 0. which simplifies to
2*a*tan(α))*y + a²*tan(α)² = x², one of the formulas for a parabola.
We would like to have the formula expressed in the coordinate system of the plane P which to avoid y' I'll refer to as (x,w). Substitute w * csc(α) for y to get
2*a*sec(α))*w + a²*tan(α)² = x².

For the ellipse, the angle of the plane P is α < β < π/2. Again, substitute for z to give
x² + y² - (tan(α)*(z0+y*cot(β))² = 0

The rest of the math for this and the hyperbola is, as they say, left as an exercise. (exercise, left as an)

Equations for Conic Sections

Rectangular Form

The general form of the equation for any conic section is:

  Ax²+Bxy+Cy²+Dx+Ey+F=0

where x, y, A, B, C, D, E, and F are all real numbers. To find the type of conic, use this:

  B²-4AC>0 : hyperbola (or its degenerate case of 2 intersecting lines)
  B²-4AC=0 : parabola (or one of its degenerate cases: 2 parallel lines, a single line, or nothing)
  B²-4AC<0 : ellipse if A≠C or circle if A=C (or the degenerate case of a point or nothing)

If a B term is present, this indicates that the axes of the graph have been rotated, and need to be rotated back in order to make the equation easier to work with. This is done by eliminating the xy term. First, use the equation

  (A-C)/B = cot(2θ)

to solve for θ, then use the equations

  x=x′cos(θ)-y′sin(θ)  and  y=x′sin(θ)+y′cos(θ)

to put the general form equation in terms of x′ and y′. This will eliminate the xy term.

With the equation now in the form

  A′x′²+C′y′²+D′x′+E′y′+F′=0

more information can be found by putting that equation into the standard form for each conic section.

Polar Form

The standard form of the equation of a conic section in polar form with one focus located and the pole is

  r=(ep)/(1±ecos(θ) : vertical directrix
or
  r=(ep)/(1±esin(θ) : horizontal directrix

where r, e, and p are real numbers and θ goes from [0,2π) or [0°,360°). e is called the eccentricity, and is used to find the type of conic like this:

  e<1 : ellipse (e=0 is a circle)
  e=1 : parabola
  e>1 : hyperbola

The directrix of the conic will be p units away from the focus, in the positive direction if the denominator is added, and in the negative direction if the denominator is subtracted.

For more information, see the writeups on the individual conic sections:

• Parabola
• Hyperbola
• Ellipse
• Circle