Mathematical Experiment 13


According to calculus, the velocity vector of a curve r is given by
v dr
dt
dr
ds
ds
dt
= T ds
dt
while the acceleration vector is given by
a dv
dt
dT
dt
ds
dt
+T d2s
dt
2
dT
ds
æ
ç
è
ds
dt
ö
÷
ø
2 +T d2s
dt
2
= k æ
ç
è
ds
dt
ö
÷
ø
2 N+ d2s
dt
2
T
Here T is the unit tangent and N is the unit normal, s is the length function, k is the curvature. Therefore, the radius of curvature

r = [1/( k)] can be constructed when the normal component k( [(ds)/( dt)]) 2N of the acceleration and the speed [(ds)/( dt)] are known. Recall that in this configuration of a right triangle


we have
q2 = pr
  1. Construct the circle of curvature of the complex curve
  2. r = aeit+be2it,0 £t £ 2p
    where a,b are given complex scalars.
    1. Let a variable point on the circle centered at 0 containing a be denote by a(t) = aeit.Construct a¢(t) and a¢¢(t) by rotation.

    2. For the given b, construct b(t) = bei2t,b¢(t) and b¢¢(t).

    3. Construct the vector sums
    4. r(t
      a(t)+b(t), 
      r¢(t
      a¢(t)+b¢(t), 
      r¢¢(t
      a¢¢(t)+b¢¢(t).

    5. Join the line segment 0r¢and draw its perpendicular through 0. This gives the direction of the normal.

    6. Find the projection of r¢¢ upon the normal line. From the projection complete the right-angle triangle.

    7. Trace the curve r. Draw the circle of curvature. Finish with the animation.

  3. Construct the circle of curvature for a complex curve of the form
  4. r(t) = aeit+bei2t+cei3t,0 £t £ 2p.
  5. Construct the circle of curvature of the square of a circle, not necessarily centered at 0. Recall that if z = w2 then z¢ = 2w·w¢ and z¢¢ = 2(w¢·w¢+w·w¢¢).
  6. Illustrate Poncelet's porism: If two circles are so related that a polygon can be inscribed to one and circumscribed to the other, then infinitely many polygons can be so drawn, and each diagonal of the variable polygon is tangent to a fixed circle.

  7. Illustrate Poncelet's theorem: If polygon moves with its vertices on a fixed circle, and each of its sides with one exception is known to touch a fixed circle of a coaxal system including the first circle, then the remaining side touches a fixed circle of the system, and each diagonal connectors touches a circle of the system.


exp13.tex