Simulating Motion on the Surface of a Sphere

Ummm.....seems like the method is pre-determined or constrained so I hesitate to make broad suggestions....but....

This sort of problem is most easily worked out using vectors for (x,y) on a unit sphere using radians.

There is much to be lost in the transform into and out of your coordinate system, so be careful.

Uhm

It is the word *simulate* you use that makes me hesitate in a definite answer

Spherical geometry will allow you to easily *calculate* any position on a sphere.

*Simulating* motion on a sphere is simply laying out the series of positions you have calculated. This can be tackled one of two _{(that come to mind, don't let my answer constrain you)} ways:

1. Given a Start and End points, calculate all the points between at an increment that renders smoothly enough for your purpose. Throw all points into a matrix, step your way through them.

2. Given a Start and End points, calculate a rough bearing (a Rhumb line), calculate the next position along the Rhumb at the increment you wish, repeat.

Either way works, there are probably other ways that work.

The first is probably best given limited processing power, the second would require more brute force, but offers flexibility. (Like turning?)

But...for simulation purposes - usually gobs of this are done by the simulation engine itself, or done by a separate rendering engine; unless these are what you actually trying to develop.

AND one of the constraints of this system is it only works on an utterly smooth sphere.

GA from me too.

I've not done that stuff in so long, I'd have to spend some time in the books to get it back, let alone explain it with words! Well done.

Mike

cheers @ edignan and mikerho.

@ OP: it sounds scarier that it looks (or is it the other way around lol). ok since you know your radius, and starting and ending points, it reduces this problem down to a simple surface/line integral (which could be done using vectors as suggested before), but you still need to familiarize yourself with the spherical coordinate system, and how to use parametric representations of your surface (in this case it's a sphere = constant radius....see how the problem keeps reducing itself? :)). if you are unfamiliar with these concepts then as my uncle in Texas always says, "you are up sh!t creek without a puddle".

At any moment we can describe the velocity vector by it three components.

Go to http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx it provides a break down of surface integrals that will help with your quest. scroll down to "Example 2"

Hi to rajeshrocks25 ,

You will need to understand Spherical geometry before you are through.

Understanding of a problem and its solution(s) are key to an understanding of life and how to solve the next problem.

One very good boook is " Textbook on Sperical Astronomy" by W.M Smart.

ISBN 0 521 29180 1 Paperback published by Cambridge University Press.

The first few chapters are on Spherical trigonometry and the Celestial Sphere.

If you set your sights on a thorough understanding of Chapter 1 you will have gained a lot!

I was given a similar problem to do with Space Mechanics over 50 years ago and had to go away and find the answers. The maths has stayed with me over the years and I have had to reuse the knowlege for Space mechanics, Survey work and Astronomy at different times.

Good Luck, do not be afraid to get your mind dirty - it will stretch itself as you go along. And the next problem will be easier and you will learn a lot about life and yourself.

Sleepy

A sphere is a 3D object. You can't move on it with only x and y coordinates. You'll need z. You'll get "off" the sphere if z doesn't change. You'll also need equation for force acting on the moving object. If everything is equal on the moving object, it'll only move around equator on the sphere with constant speed. Or stop.

Pineapple,

thank you for that comment.

I have utilised Spherical Trig for Survey work and for the analysis of those results which

1 can allowfor non spherical surfaces <oblate earths for example>

2 can allow for z values at various points. heights above zero.

Now it is some 20 plus years ago that I last did this but the work was all professional and resulted in work that was accurate and usable. GPS style measurements were used <this was just pre GPS but utilised a pre existing SatNav> So inputs were x, y and z for each location.

I do not have access to the work these days so cannot quote the details. We could not have used this work for survey analysis if this had not been available.

Thanks for the thoughts,

Sleepy

Is this a homework problem? The reason I'm asking is not to rail on you, but a lot of times you can find a general solution on the web and work it to match your constraints. Sometimes it's the only way to learn how to solve some of the more challenging dynamics problems...

Also...you may want to use a spherical coordinate system...