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We have a computationally expensive function of a large set of data and an angle, that is known to result in a sine wave: $$ f(\text{data}, \theta) \approx a \sin (\theta + b) + c$$ We want to find the constants $a$, $b$, and $c$, executing the function as few times as possible. |
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Yes, you can do it in 3 points, considering $f(0)$, $f(\pi)$, and $f(\frac{\pi}{2})$. To find $c$: $$ \begin{aligned} f(0) + f(\pi) & = a\sin(0 + b) + a\sin(\pi + b) + 2c \\ & = a\sin b - a\sin b + 2c \\ & = 2c \\ \implies c & = \frac{f(0) + f(\pi)}{2} \end{aligned} $$ To find $a$: $$ \begin{aligned} (f(0) - c)^2 + (f(\tfrac{\pi}{2}) - c)^2 & = a^2\sin^2(0 + b) + a^2\sin^2(\tfrac{\pi}{2} + b) \\ & = a^2 \left(\sin^2(0 + b) + \sin^2(\tfrac{\pi}{2} + b)\right)\\ & = a^2 \left(\sin^2 b + \cos^2 b\right) \\ & = a^2 \\ \implies a & = \sqrt{(f(0) - c)^2 + (f(\tfrac{\pi}{2}) - c)^2} \end{aligned} $$ $b$ is now trivial to find: $$ \begin{aligned} f(0) &= a\sin b + c \\ \implies f(0) - c &= a\sin b \\ \implies \frac{f(0) - c}{a} &= \sin b \\ \implies b &= \sin^{-1}\frac{f(0) - c}{a} \end{aligned} $$ |
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