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These are replies submitted by Earl

@Carl Love Thank you for the lesson. I hope Rouben Rostamian can correct the physics of my ODE equations so I can practice on them the technique you have taught me.

@Rouben Rostamian  Thank you! Now I can proceed to animate your solution.

By the way. I was inspired by Problem 14.6 in Mark Levi's Why Cats Land On Their Feet, Princeton University Press.

He notes that the tangent vector angle with the horizontal, theta, satisfies Kepler's equation;

theta - g/G*sin(theta) = cs where G, a multiple of g, is the constant force on the rider, s is arclength and c is a constant related to the car's speed at the lowest point of the track and is a sizing factor for the track.

Regarding Carl Love's reply, please let know of any fault with my ODE equations.  

@dharr Polar coordinates throughout the worksheet run into the same error message.

@Carl Love After reading your reference to Gauss-Bonnet in MathWorld it seems that corner angles are only relevant to geodesic triangles. However my dual cones worksheet does not contain geodesic triangles.

Curiously, experimenting with passing different parameters to procedure DualCones shows that if the radius and height of the upper cone are equal, then the G-B formula does yield 2*Pi. There are two examples of this in the uploaded worksheet.


@Carl Love Thank you, I'll try adjusting my calculation.

@Kitonum My only knowledge of the Gauss-Bonnet theorem and the inspiration for my worksheet come from topic 10.2 in the book The Mathematical Mechanic by Mark Levi. Unfortunately, nowhere in his text or diagrams does he state that his restricted version of the theorem (which you quoted in your reply) applies only to dual cones with their common vertex at the sphere's centre.

As a programming challenge I create and display dual cones whose z coordinate of their common vertex depends on the parameters passed to the procedure which creates them.

Thank you for your reply. 

@Carl Love Please see my response to Kitonum. My knowledge of the Gauss-Bonnet theorem is too thin to answer your question. 

@Joe Riel I tried to create a smaller example but DEBUG in it did not display its values in the worksheet, so below is the full worksheet in which I found the problem. The DEBUG values at the worksheet's bottom were left there after using continue to display 3 debug windows.

I use DEBUG at the top level to show values during several loop iterations and to halt loop execution.


The uploaded worksheet below (in Maple 2016) contains an ODE with three events. These examples of event coding may help your understanding.


@Kitonum It seems the volume of this body also equals 1/24

plot3d(x*y, x = 0 .. -1/3, y = 0 .. -x+1, style = surface, filled);

since int(int(x*y, y = 0 .. -x+1), x = 0 .. -1/3) = 1/24

@Rouben Rostamian  Once again I am in your debt. Is it true for any two 3D surfaces which are tangent to each other that their normals at their common point of tangency are collinear?

@Rouben Rostamian  Thank you for taking the time to answer me. Since my math skills are about middle undergraduate this subject is interesting but far over my head.

@Carl Love When I execute the 4th method I receive "Error (in Last) too many levels of recursion".

The following code works in Maple 2016

Last := proc (k::posint)

procname(k) := 4*thisproc(k-1)+k :

end proc:

Last(0) := 1:

`seq`(Last(k), k = 1 .. 10);

@Christopher2222 are enlightening. It is fascinating that this question has drawn the interest of a number of early calculus stars.

@Rouben Rostamian  your reference and the animation. Christopher 2222's references are also interesting.

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