This month we have another 3D geometry challenge. We are projecting squares onto cylinders and spheres and measuring the perimeter. Create a worksheet that calculates any of the following:
Challenge 1: Perimeter
Calculate the perimeter of a square with a side length of 50 onto the following:
- A sphere of diameter 100.
- A cylinder of diameter 100 and length 100.
Clarifications:
- Note that the square is projected from a plane that is parallel to a plane tangent to the sphere or cylinder. It is projected in a normal direction from the tangent plane as in the accompanying images.
- For the sphere, the square is centered on the point of tangency for the plane to the sphere.
- For the cylinder, the square is centered on the middle of the cylinder and the tangent point.
- If the intention is unclear, please refer to the images or post a comment where the community can help explain the scenario to you.
Challenge 2: Area
Calculate the area of the above projected squares.
Challenge 3: Function or Program
Write a function or program that computes the perimeter or area of the projected square where the inputs are the length of a side of the square and the diameter of the sphere / cylinder.
Can you incorporate error checking in situations where the projected square goes beyond the boundaries of the square or cylinder?
Bonus Challenge – Graphing:
Use the XY Plot, 3D Plot, or Chart Component to depict the change in perimeter or area as a function of the length of one side of the square.
Bonus Challenge – Advanced Input Controls:
Incorporate advanced input controls (e.g., sliders, radio buttons, etc.) to allow people to change the inputs (e.g., length of the side of a square, diameter of the cylinder / sphere) to change the results.
You can submit partial worksheets. For example, you can solve the first challenge and submit additional worksheets later in the month as you solve later challenges.
3D CAD models created in Creo Parametric 7 have been attached to this challenge if anyone wants to verify their calculations.
Find the Mathcad Community Challenge Guidelines here!
Hmm, a little more than a quarter of July has already passed and still no contribution has been received.
Then I would like to open the round dance.
Incidently, I would not say that we are practicing non-Euclidean geometry here. 😉
EDIT: BTW, how about adding a Gnomonic projection?
Prime 11 sheet and pdf print attached
New version of my sheet.
Added the Gnomonic projection and a few bits and pieces here and there.
I would really wish the sliders to work more smoothly and precisely and not wait for subsequent calculations before displaying another change. Subsequent calculations that depend on them should be canceled when the slider is moved.
Not sure if a stereographic projection (central projection with center ( 0 | -d/2 | 0) ) would be worth dealing with...
Dear Werner_E
I try to plot your last image. How about the perimeter or area of the projected square?
@AlanStevens wrote:
I think that in a gnomonic projection the projection point is from the centre of the sphere, as in your earlier diagram, not from the far surface.
Correct, but the gnonomic projection is already implemented in the last (second) version of my worksheet. I added the "heigth", the distance of the square plane to the sphere as a third input parameter, but of course putting the square just in a tangential plane would suffice to cover all cases.
The same would apply to the sterographic projection which I only mentioned as being a possible additional task. There was then the corresponding drawing. However, I have not (yet?) worked on this variant.
