# 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. 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