@Carl Love Thank you for this. I'm busy now with another project but sometime within the next few days I'll access your post and see if I can adapt it as a higher-level plotting command.

@Rouben Rostamian  Thank you. 

As I replied to Carl Love, I'm guessing your answer implies that there is no way for a procedure invoked by a plot command to pass anything other than the function to be plotted. Is this true?

@Carl Love I'm guessing your answer implies that there is no way for a procedure invoked by a plot command to pass anything other than the function to be plotted. 

@Rouben Rostamian  I very much appreciate that you've taken the time to show me the energy/forces way to find the coaster loop curve.

In the worksheet below, I have corrected my original energy/forces ode equations (which are explicated more clearly in your worksheet) and then use Kepler's equation to reach an identical curve.

Amazing that there seem to be three different approaches yielding the correct curve!

Roller_coaster_loop_2.mw

@Rouben Rostamian  In the worksheet below, in the second ODE equation, I corrected the gravity vector. Then I applied Carl's change. The coaster loops now proceed horizontally but successive loops increase in size.

The second odeplot shows velocity increasing in successive loops, which seems to indicate a fault in the first ODE equation describing the system's energy.

Can you detect the error?

 Roller_coaster_with_constant_g_force_2.mw

@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.

Dual_cones.mw

@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.

DEBUG_test.mw

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

Reverse_spin_cueball.mw

@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