I'm working with opengl to create a sunrise and sunset based on the actual times. I'm having trouble coming up with a sin/cos formula that goes from -1 to 1 to -1 for any given sunrise and sunset time with sunrise and sunset equaling -1 and high noon equaling 1.
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EDITED LIKE CRAZY: Yeah, use two separate equations. For each you have some requirements: $$f(0)=1$$ $$f(12)=1$$ $$f(s)=-1, f'(s)=0$$ Since you'll switch functions every 12 hours, you don't need something like $f'(12)=0$ because it's guaranteed to be a maximum anyway when you switch to the decreasing function on the other half. It's not smooth, but it is continuous, and since I assume the effect will be minimal I doubt it would be noticed much. The thing is that you don't really get customizable flowy behaviour with sines/cosines (you could technically use fourier transforms, but that seems wildly excessive). With 4 conditions though, a cubic can be specified to flow smoothly like this: $$f(x)=ax^3+bx^2+cx+d$$ $$f'(x)=3ax^2+2bx+c$$ It will start at 1, dip down to -1 at the specified time, and come back up to 1. I'm letting the 12 hour period be over [0,1] instead of [0,12] (so $t_{day}=12x$) to simplify calculations. The first requirement means $d=1$. The second means $$a+b+c=0$$ The requirements on sunset are $$as^3+bs^2+cs=-2$$ $$3as^2+2bs+c=0$$ solving this system of equations (and converting $x$ to $t$) gives a cubic describing amount of sunset in terms of time: $$At^3+Bt^2+Ct+D$$ where $$A=\frac{3456(2s-1)}{s^2(s-1)^2}$$ $$B=\frac{288(3s^2-1)}{s^2(s-1)^2}$$ $$C=\frac{24(3s-2)}{s^2(s-1)^2}$$ $$D=1$$ |
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