> "I'd argue the smoothstep version looks better" Why would this be? I would have thought the theoretically correct sinc version would look nicer.
For a fixed well-defined mathematical problem, you might be able to solve it optimally or approximately. One perspective is to treat the problem as given and immutable and then try to compute an exact or optimal solution.
But often the original problem statement is fairly arbitrary, based on a bunch of guesses or simplifications, and you might be able to get a better result by changing the problem definition (perhaps unfortunately making it much messier to solve exactly) and then solving the new problem statement approximately.
What's the actual problem we're trying to solve here? Generate something that looks visually pleasing. Why is an expression involving cosine the natural way to define that problem statement mathematically? There's likely a lot of freedom to here to vary our problem definition.
It might be interesting to start with the smoothstep multiplied result and take the derivative and look at how that differs from a normal cosine, and ponder why that might produce a more pleasing result than a cosine.
For a fixed well-defined mathematical problem, you might be able to solve it optimally or approximately. One perspective is to treat the problem as given and immutable and then try to compute an exact or optimal solution.
But often the original problem statement is fairly arbitrary, based on a bunch of guesses or simplifications, and you might be able to get a better result by changing the problem definition (perhaps unfortunately making it much messier to solve exactly) and then solving the new problem statement approximately.
What's the actual problem we're trying to solve here? Generate something that looks visually pleasing. Why is an expression involving cosine the natural way to define that problem statement mathematically? There's likely a lot of freedom to here to vary our problem definition.
It might be interesting to start with the smoothstep multiplied result and take the derivative and look at how that differs from a normal cosine, and ponder why that might produce a more pleasing result than a cosine.