Practical application of "bent axle theorem"

Now it can be told. (Then was OK, too, but I kept forgetting to ask.)

Recall the illustrations (and animation, if you paste the code into
Macsyma) www.ippi.com/rwg/axle.htm, depicting the circular motion of
the centers (hubs) of

Two toed-out wheels are rigidly connected (as if by a bent axle). They turn while
maintaining four points of contact with the rim of a circular manhole.

Theorem: there is some new axis about which the system rotates, despite the jostling
motion of the wheels as the points of contact oscillate through a
continuum of trapezoids.

This is (multidisciplinary inventor) Alan Adler's design for a spiffy
telescope mount!

Consider first the unbent case, when the rotation axis coincides with
the axle. The wheels (altitude rockers) and "manhole" (azimuth ring)
are teflon-covered bicycle rims. The axle runs perpendicular to (and
is interrupted by) the barrel of the telescope. If the midpoint of
the axle coincides with the center of gravity of the telescope-plus-
wheels, the system will be neutrally stable wrt elevation, and can be
adjusted for very low friction and very low effort.

However, this would restrict us to barrel length < rim diameter, since,
on high elevation, a protruding barrel would collide with the azimuth
ring. But only the mirror end would collide--the aperture/eyepiece end
could protrude beyond the altitude rockers, at the cost of throwing
forward the center of gravity.

For which we compensate by toeing in the rockers at the mirror end!

Paste into Macsyma expression input sections the following two commands

(c8) torotate(plotnum_animate,t,ri,ro,r):=plot_animate(e,0,2*%pi,torot(t,ri,ro,r,e),cyclic)$

(c20) TOROT(T,RI,RO,R,E) := BLOCK([Z : SQRT((R*TAN(T/2))^2+RO^2-RI^2),
plot_compile_functions:false],
PLOTSURF([[(COS(E)*(COS(T/2)*(RO*SIN(U)+R)-R*SEC(T/2))-SIN(E)*RO*COS(U))*(T/2-V)/T,
-(R*SIN(T/2)*(V+T/2)/COS(T/2)^2+SIN(T/2)*(RO*SIN(U)+R)*(T/2-V))/T,
((T/2-V)*(Z+SIN(E)*(COS(T/2)*(RO*SIN(U)+R)-R*SEC(T/2))+COS(E)*RO*COS(U))+(V+T/2)*Z)/T],
[(COS(E)*(COS(T/2)*(RO*SIN(U)+R)-R*SEC(T/2))-SIN(E)*RO*COS(U))*(T/2-V)/T,
(R*SIN(T/2)*(V+T/2)/COS(T/2)^2+SIN(T/2)*(RO*SIN(U)+R)*(T/2-V))/T,
((T/2-V)*(Z+SIN(E)*(COS(T/2)*(RO*SIN(U)+R)-R*SEC(T/2))+COS(E)*RO*COS(U))+(V+T/2)*Z)/T],
[COS(E)*((RO*SIN(U)+R)*COS(V)-R*SEC(T/2))-SIN(E)*RO*COS(U),(RO*SIN(U)+R)*SIN(V),
Z+SIN(E)*((RO*SIN(U)+R)*COS(V)-R*SEC(T/2))+COS(E)*RO*COS(U)],
RI*[COS(U),SIN(U),(V/T-1/2)/9]],U,0,2*%PI,V,-T/2,T/2))$

Then you can get three still views with the three commands

(c21) block([plotnum0:25,plotnum1:9],torot(%pi/3,13.5,12.5,13.5*cot(%pi/4),%pi/2))$

(c23) block([plotnum0:25,plotnum1:9],torot(%pi/3,13.5,12.5,13.5*cot(%pi/4),0))$

(c22) block([plotnum0:25,plotnum1:9],torot(%pi/3,13.5,12.5,13.5*cot(%pi/4),-%pi/2))$

To see it animated

(c19) block([plotnum0:25,plotnum1:3],torotate(52,%pi/3,13.5,12.5,13.5*cot(%pi/4)))$

Then click left on the graphic, and left on the (newly visible) rocket icon.

(c24) "<Saving as htm failed to produce animated GIF>"

Addendum: As stated, this theorem is messy to prove. But it's obvious if the wheels are the bases of

two noncoaxial cones with rims confined to a sphere.