The circle is defined as the set of all pts equidistant
from a fixed pt called the center. The radius is the distance from any pt on the circle to the center
The ellipse is defined as the set of all pts (x,y) such that
the sum of the distances from (x,y) to 2 fixed pts, called the foci, is a constant.
The parabola is defined as the set of all pts (x,y)
such that the distance to a fixed point called the focus is equal to the distance from (x,y) to a fixed line called the directrix.
Parabolic Mirrors and headlights. The next 2 animations
show that for a parabolic mirror incoming light rays striking the mirror are reflected to the focus.
animation Shows that light emanating from a source at the focus relfect off the mirror in parallel light rays.
Incoming Light Rays
Outgoing Light Rays
The hyperbola is defined as the set of all pts (x,y)
such that the difference in the distances from (x,y) to 2 fixed pts called the foci is a constant.
Animation Hyperbola Right Branch
Animation Hyperbola Left Branch
The Algebraic Equations of the Conic Sections all come from the distance formulas.
For the circle The distance from any pt (x,y) to the center (h,k) is r.
We obtain (x-h)2
+ (y-k)2 = r2 .
A common parametric form (not unique) is x(t)
= cos(t) y(t) = sin(t).
The derivations of the equations for the other conic sections are a little
more difficult and the lecture notes are in the following downloads.
Lecture Notes Ellipse
Lecture Notes Parabola
Lecture Notes Hyperbola