conic section (idea)

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.