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
d^{2}s
dt^{2}
=
dT
ds
æ
ç
è
ds
dt
ö
÷
ø
2
+T
d^{2}s
dt^{2}
= k
æ
ç
è
ds
dt
ö
÷
ø
2
N+
d^{2}s
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)]) ^{2}N of the acceleration and
the speed [(ds)/( dt)] are known. Recall that in this configuration
of a right triangle
we have
q^{2} = pr.
Construct the circle of curvature of the complex curve
r = ae^{it}+be^{2it},0
£t
£ 2p
where a,b are given complex scalars.
Let a variable point on the circle centered at 0 containing a be
denote by a(t) = ae^{it}.Construct a^{¢}(t)
and a^{¢¢}(t)
by rotation.
For the given b, construct b(t) = be^{i}^{2t},b^{¢}(t)
and b^{¢¢}(t).
Construct the vector sums
r(t)
=
a(t)+b(t),
r^{¢}(t)
=
a^{¢}(t)+b^{¢}(t),
r^{¢¢}(t)
=
a^{¢¢}(t)+b^{¢¢}(t).
Join the line segment 0r¢and draw
its perpendicular through 0. This gives the direction of the normal.
Find the projection of r^{¢¢}
upon the normal line. From the projection complete the right-angle triangle.
Trace the curve r. Draw the circle of curvature. Finish with the
animation.
Construct the circle of curvature for a complex curve of the form
r(t) = ae^{it}+be^{i}^{2t}+ce^{i}^{3t},0
£t
£ 2p.
Construct the circle of curvature of the square of a circle, not necessarily
centered at 0. Recall that if z = w^{2} then z^{¢}
= 2w·w^{¢}
and z^{¢¢} = 2(w^{¢}·w^{¢}+w·w^{¢¢}).
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.
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.