How can I smoothly transform one periodic function into another if the period time is allowed to differ?
Let us visit the very simplest example I can think of : Two audio signals pure sine waves.
Note A ($440$ Hz) and C ($523$ Hz). There are probably many ways we can morph first into the other.
Own work: Maybe the most straight forward solution would be : $$\sin((\alpha(t) f_1 + (1-\alpha(t))f_2)t )\\ t \to \alpha(t)\\ t\in [0,1]\\\alpha(t) \in [0,1]$$ And $\alpha$ monotonically smoothly increasing.
This one is nice in the sense that we don't need to leave the set of $t\to \sin(ft)$ functions.
Maybe there exist other smooth solutions?
How could we express those?
We can consider any smooth path that connects $f_0$ to $f_1$ in the complex plane. Assume $f_1 > f_0$. Let's simplify to the setting $$\sin(\alpha(x)t) \ \ \ \ t \in (-\pi,\pi] $$ $$ x \in [0,1] \ \ \ \ \alpha(0)=f_1, \, \alpha(1)=f_0 $$ An interesting example is the semicircle $\alpha(x)=0.5((f_1-f_0)e^{i\pi x}+f_0+f_1)$. Here is what happens during the transition when $f_0=1$ and $f_1=3$:
which is cooler if animated:
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