Mathcad Community Challenge July 2022 - Area of a Spherical Triangle

[#1]

This month's challenge is another geometry-based challenge - but it's non-Euclidean geometry. Given the following three points on a circle of radius 10 units, calculate the area of the spherical triangle:

(5.339, 6.948, 4.819)

(-4.018, -3.703, 8.375)

(6.455, -6.679, -3.705)

(Note that the point coordinates are rounded to three decimal places.)

Optional:

• Can you write a program or function that calculates the area for any three points on the surface?
• Is there a symbolic evaluation for three generic points (a,b,c), (d,e,f), and (g,h,k)?

Here is a 3D model in Creo 9 of the sphere and points:

Find the Mathcad Community Challenge Guidelines here!

Edit: the coordinates of the second point were incorrect. Wildly incorrect. I have no idea how I wrote down the wrong numbers.

[ppal]

Are the numbers correct?  I thought that the length of the vector for all three should be 10??

[terryhendicott]

Hi,

Thought I would get the ball rolling.

The coordinates supplied for three points on the sphere are correct the origin of the sphere is not at 0,0,0.

Enclosed is my solution and 3D model checking of the procedure to calculate the area.

Fulfilled the first option with a program that solves for area given three points on the sphere and a radius.

Used the program in some test cases.

Cheers

Terry

220704-Spherical.mcdx

[terryhendicott]

Hi,

Spotted a couple of things so would like to repost solution.

Cheers

Terry

220705-Spherical v2.mcdx

[terryhendicott]

Hi,

The second optional part of the challenge was to get the spherical triangular area in terms of a,b,c,d,e,f,g,h,k being the nine xyz coordinates of the three definition points.

Here it is using an alternative to Girard's Theorem called L'Huilier's Theorem that is easier to apply.

It is a long definition so needs to be seen in draft mode.

Cheers

Terry

220705-LHiliers-Theorem.mcdx

[terryhendicott]

Hi,

One last post that defines the problem in four equations and solves problem with four equations.

Cheers

Terry 

220705-Four Equations.mcdx

[DaveMartin]

You are correct. The second point should be (-4.018, -3.703, 8.375). I have no idea why I had written down the y- and z- values so far off.

[DaveMartin]

As people realized, I wrote down the values for the second point incorrectly. The sphere has a radius of 10 and is centered at the origin. The original post has been edited to contain the correct coordinates. Thanks for catching my mistake.

For verification, I have attached the original model that I used to construct the problem. I believe I created in Creo 9.

spherical_triangle.zip

[ppal]

As pointed out by Terry - still works with the original data but the center is as follows:

[terryhendicott]

Hi,

Having the sphere at the origin simplifies solution. 

Solved by Girard's and L'Hilier's theorems.

Cheers

Terry

220706-LHiliers-Theorem.mcdx

220706-Girards Theorem.mcdx

[ppal]

Here is my shot at it.

July Spherical Triangle.mcdx

[ppal]

The equation cleaned up a bit.

July Spherical Triangle.mcdx

[terryhendicott]

Hi,

ppal has provided equation that leads to a neat symbolically derived function in terms of the input coordinates of unit circle

220715-SymbolicSolution.mcdx

[ttokoro]

[ttokoro]

[ttokoro]

r=0.5 Blue flat star has 0.28065 and Red sphere star has 0.29736 of area.

[ppal]

FINAL - No more.

Compacted.

SPHERE TRIANG.mcdx

[ppal]

To myself:

or maybe this looks better?

[terryhendicott]

Hi ttokoro,

Could you upload the worksheet that generates the 3D image of the sphere and spherical triangle.

I would like to know how you have drawn the orange great circle segments please?

Cheers

Terry

[JE_9022802]

227 units-squared.  Not sure if I'm using the Spherical Trig correctly.  -Johan Eppinga

Eppinga_AreaOnCircularTriangleCh 20JULY22.mcdx

[terryhendicott]

Hi ttokoro

No need I got it.

[ttokoro]

Fill the triangle on the sphere requires more Mathcad power to plot all points in red. 

By the way I made this 3D plot both by Mathcad Prime8 and Maple2022, both show ABC rotation is counterclockwise and original 3D model has clockwise ABC rotation.

[ppal]

or

[ppal]

[DJNewman]

Quick reminder to everyone that July (and this challenge) ends soon!

[ttokoro]

It should be 2*Pi. Terry's sheet shows 2*Pi.

[OH_10381595]

My try

triangulo_esfera.mcdx

[ppal]

Excellent. But with the lack of comments it looks like:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1311997/pdf/jaba00061-0143a.pdf

[DJNewman]

Oh my gosh, that's the first time I've seen that, so I was laughing out loud.

Anyway, @OH_10381595 , welcome to the PTC Community and thanks for your contribution. July has passed, but we still appreciate your participation.

That said, Mathcad Prime has a lot of documentation features, and per the challenge guidelines, following along with your worksheet would be more of a pleasure if you incorporated some of them, like text boxes.

[MJ_9796426]

Some of the elements building up to calculation of the area can be useful too. Here is a version that shows those details.

sta.mcdx

[DJNewman]

We published Dave's solution blog article! And distributed badges. Thanks to everyone who participated!

https://www.mathcad.com/en/blogs/community-challenge-spherical-triangle

(Dave gave a hint that in September we'll have something from the electrical engineering domain.)

Quick reminder that we'd really like to see people's worksheets and not just screenshots of worksheets!