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
dt2		 =  dT
ds		 æ
ç
è		 ds
dt		 ö
÷
ø		 2 +T d2s
dt2
= k æ
ç
è		 ds
dt		 ö
÷
ø		 2 N+ d2s
dt2		 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.
    2. For the given b, construct b(t) = bei2t,b¢(t) and b¢¢(t).

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

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

      		5.
    5. Find the projection of r¢¢ upon the normal line. From the projection complete the right-angle triangle.

      		6.
    6. Trace the curve r. Draw the circle of curvature. Finish with the animation.

      		7.
  2. Construct the circle of curvature for a complex curve of the form
    3. r(t) = aeit+bei2t+cei3t,0 £t £ 2p.
  3. 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¢¢).
  4. 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.

    		5.
  5. 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.

    		6.
exp13.tex