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16-Pearl

## Mathcad Community Challenge July 2022 - Area of a Spherical Triangle

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:

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

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com
30 REPLIES 30
16-Pearl
(To:DaveMartin)

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

16-Pearl
(To:ppal)

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.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com
19-Tanzanite
(To:DaveMartin)

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

19-Tanzanite
(To:terryhendicott)

Hi,

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

Cheers

Terry

19-Tanzanite
(To: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

19-Tanzanite
(To:terryhendicott)

Hi,

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

Cheers

Terry

16-Pearl
(To: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.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com
19-Tanzanite
(To:DaveMartin)

Hi,

Having the sphere at the origin simplifies solution.

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

Cheers

Terry

16-Pearl
(To:DaveMartin)

Here is my shot at it.

19-Tanzanite
(To:ppal)

Hi,

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

18-Opal
(To:ttokoro)

18-Opal
(To:ttokoro)

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

19-Tanzanite
(To:ttokoro)

Hi ttokoro

No need I got it.

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.

19-Tanzanite
(To:ttokoro)

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

16-Pearl
(To:DaveMartin)

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

16-Pearl
(To:DaveMartin)

The equation cleaned up a bit.

16-Pearl
(To:DaveMartin)

FINAL - No more.

Compacted.

16-Pearl
(To:ppal)

To myself:

or maybe this looks better?

16-Pearl
(To:ppal)

or

16-Pearl
(To:ppal)

18-Opal
(To:ppal)

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

1-Newbie
(To:DaveMartin)

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

16-Pearl
(To:DaveMartin)

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

I manage the Creo and PTC Mathcad YouTube channels for PTC, as well as all PTC Mathcad marketing in general.
1-Newbie
(To:DaveMartin)

My try

16-Pearl
(To:OH_10381595)

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

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

16-Pearl
(To:ppal)

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.

I manage the Creo and PTC Mathcad YouTube channels for PTC, as well as all PTC Mathcad marketing in general.
1-Newbie
(To:DaveMartin)

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

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