Ellipse Calculation

Tangent, Curvature, Osculating Circles

Richard Hitt
Department of Mathematics and Statistics
University of South Alabama

General Setup

We just juse the 3-space setup even though we could specialize to the 2-dimensional formulas.

mag[g_] := 
      (Sqrt[g[[1]]^2 + 
          g[[2]]^2 + 
          g[[3]]^2])tangent[g_] := (∂_t) (g/mag)[(∂_t) g]normal[g_] := 
      (∂_t) tangent[g]/mag[(∂_t) tangent[g]]k[g_]  :=  
      mag[Cross[(∂_t) g, (∂_t) (∂_t) g]]/
        (((mag[(∂_t) g]))^3)

Example

Now for the ellipse example.

r[t_] := {2  Cos[t], 3  Sin[t], 0}

First we compute the unit tangent vector function.

tangent[r[t]]

Next we can calculate the unit normal.  Note that differentiating the above formula for the unit tangent would be tedious.  Dividing by the magnitude of the derivative might not be fun either.

normal[r[t]]

Lets let Mathematica simplify this.

Simplify[%]

Well, that's a little better.

Now for the curvature.

k[r[t]]

Simplify[%]

k[r[t]] /. t -> 0

k[r[t]] /. t -> Pi/2

c1 = Circle[{2 - 9/2, 0}, 9/2]c2 = Circle[{0, 3 - 4/3}, 4/3]

g1 = ParametricPlot[{2  Cos[t], 3  Sin[t]}, {t, 0, 2  Pi}]

Show[{g1, Graphics[{c1, c2}]}, AspectRatio -> Automatic]