If I'm remembering this right: Length of Side ONE (call it A) is the known distance from Sun's center to the planet at time Zero. Length of Side Two (call it B) is the known distance from the Sun's center to the planet at Time X (what ever that time might be). The computer would have to iterate the angle formed by A and B by a given amount of seconds (ie there are 60 minutes to a degree, and 60 seconds to a minute in angles). So, call the included angle between Sides A and Sides B as angle theta:
Formula to determine length C is: C^2 = A^2 + B^2 - 2ab x cos (theta) The formula for determining the area of a non-right triangle is: (see http://en.wikipedia.org/wiki/Heron%27s_formula ) Area = ((A^2 + B^2 + C^2) - 2*(A^4+B^4+C^4))^.5)/4 Consequently? If I only have TWO sides to work with, but I already know what the Area has to be in advance? I can solve for the missing Side just using Heron's formula. I don't even have to go through "iterations" to find the right included angle. Hmmm. Let's try this out in a worked example using Earth's orbit, and see how that works out. -----Original Message----- From: [email protected] [mailto:[email protected]] On Behalf Of Captain Joy Sent: Tuesday, October 11, 2011 1:19 AM To: The GURPSnet mailing list Subject: Fwd: [gurps] Planetary movement and checking the Math On Oct 10, 2011, at 6:02 PM, Alaconius <[email protected]> wrote: > 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. Absolutely true for a right triangle, which is not what you have. But, you may be able do do what you want given the two sides and the angle between them. Determining the length of the third side will not be as simple as using a single trig function or the pythagorean theorem, but it's not rocket science either. (Yes, I'm hilarious.) -- Captain Joy [email protected] _______________________________________________ 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
