Calculus Animations,Graphics and Lecture Notes

Conic Sections

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This page consists of Animations showing how the conic sections are generated and lecture notes on the derivation of their algebraic equations.

The conic sections are the curves of intersection of a double cone and a plane.

In the following Animation We start with a horizontal plane intersecting the double cone in a circle. As the plane rotates it the curve of intersection becomes an ellipse followed by a parabola and finally a hyperbola as the plane approaches the verticaL.

Double Cone

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

Animation Circle

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.

Animation Ellipse

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.

Animation Parabola

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.

The second 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



The Smart Bunny-A very short story by Kurt Vonnegut Jr.

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