Hi Jonathan, The nice thing about the "methodology" that I'm attempting, is that it doesn't require calculus so much as it requires that I be able to determine the area of a triangle, and solve for the third leg of the triangle in question.
Here is what I was thinking... The planet's position will always give you TWO sides of the triangle. Why? Because the orbit itself is defined to be an ellipse with a known formula (Look up the formulas for Ellipses using Polar Co-ordinates to see what I'm talking about). As a consequence, any two points on the orbit will be a known distance to the center of the sun. As a consequence of this, simple trigonometry allows us to solve for the third side of the triangle if you know the two of three sides of the triangle. Knowing all three sides of the triangle that equals a given area value, will in turn, permit one to determine any of the interior angles using Sine, Co-sine, and tangents. So. Step One: Determine the smallest unit of time that is desired for purposes of the simulation. Step Two: Determine the area of orbit itself Step Three: divide the total area of the Ellipse/orbit by the smallest unit of time being used in the simulation. Step Four: Determine what the triangle's measurements must be to equal the area of the triangle for the smallest sweep in step 3. Step Five: Determine WHERE the planet is now after the given amount of time based on the calculations from Steps 1-4. Presto. Planet is in the approximate vicinity of the third junction of the triangle (the other two junctions being the sun itself, and the last known position of the planet in question. THAT was what I was hoping to discover as a solution without having to actively use calculus outright. I know it emulates derivatives to a degree, but it doesn't REQUIRE it. :) -----Original Message----- From: [email protected] [mailto:[email protected]] On Behalf Of Jon Lang Sent: Monday, October 10, 2011 1:29 AM To: The GURPSnet mailing list Subject: Re: [gurps] Planetary movement and checking the Math FWIW, the problem that he outlined in his first post is one that I wouldn't mind having an answer to myself: given a starting position and velocity relative to a primary star of known mass, provide a straightforward, easy-to-use formula that will tell you where you're going to be after a certain amount of time has passed. Thing is, he mentioned that he doesn't like advanced math, and in particular has largely forgotten everything he ever learned about calculus. So to solve this problem, he's effectively trying to reinvent calculus from the ground up - which is the right way to go, because you need calculus to solve this for anything other than very short time spans. And when times are that short, you can effectively treat it as straight-line movement. Trying to figure out a solution to this problem is a worthwhile exercise for those who enjoy advanced math; for those who don't, it will quickly turn into an exercise of beating your head into a wall. -- Jonathan "Dataweaver" Lang _______________________________________________ GurpsNet-L mailing list <[email protected]> http://mail.sjgames.com/mailman/listinfo/gurpsnet-l _______________________________________________ GurpsNet-L mailing list <[email protected]> http://mail.sjgames.com/mailman/listinfo/gurpsnet-l
