At the end of these great slides, the author compares all the different integrators presented. One way or another, they all fall short except for Improved Euler Integration and Runge Kutta 4 Integration, which both pass all tests.
I suppose I should mention that I am working on a 2D game that isn't very physics intensive. I'm just curious as to where Improved Euler Integration would fall short and RK4 would have to be used instead.
My game consists mostly of simple gravity (jumping and falling), movement along the X and Y axes, and bounding box collision. Is it worthwhile to implement RK4 or would Improved Euler be sufficient? I see many discussions where Euler Integration's users are chastised, but from what I can see, Improved Euler is quivalent in simple 2D matters. I imagine it'd also be faster.
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