# How to make a time-parametrization slower around a point but not too slow?

For animation purposes (see below) I need to use a parametrization which 'slows down' near a specific value;

More precisely, I am looking for a 'nice' monotonically increasing function $$r:[0,1] \to [0,1]$$ satisfying $$r(0)=0, r(1)=1,$$ whose derivative is relatively small around the point $$r^{-1}(\frac{1}{2})$$.

After some playing around with polynomials, I came up with the following example: $$r(t)=\frac{1}{2} \big((2 t - 1)^3 + 1\big).$$ This choice of $$r$$ satisfies $$r'(t)=0 \iff t=\frac{1}{2}=r^{-1}(\frac{1}{2})$$, so $$r'(r^{-1}(\frac{1}{2}))=0.$$

However, this is "too slow" for me - it turns out that having derivative zero makes the animation looks like it "stopped" (depending of course on the uniform discrete time-step size that I am using).

Question: How can I make an adaptation which makes the animation still slow but not "too slow" around $$r^{-1}(\frac{1}{2})$$?

Here is my animation: It is a shrinking disk - which starts at radius $$1$$, and shrinks to a point. After it reaches to $$r=\frac{1}{2}$$ it starts 'twisting' as well as shrinking (the twisting and shrinking occurs simultaneously beyond that stage). I want to focus attention on the part around where the twisting starts.

(If it's relevant I am working with Mathematica, but I guess there should be an analytic platform-independent solution to this problem.) • I don't think it is necessary for $r(t)$ to be smooth. Just make it piecewise linear. – Wolfgang Bangerth Feb 2 at 20:40
• Thanks, you are completely right. I thought that it might be possible to just add a small perturbative term to my original polynomial parametrization. I guess I can keep it smooth except around $\frac{1}{2}$, where I could make it piecewise-linear and glue. – Asaf Shachar Feb 3 at 11:31