- uniform for $\mu=0$
- chordal for $\mu=1$
- centripetal for $\mu=0.5$
To make it easier to imagine how these $\mu$ values will affect the shape of the spline, I made up a little story by looking the points as the racing game where the driver were required to pass these points in order to complete the race. Here we have 3 different drivers with different driving style.
Racer #1 drives with the same speed all the way, this is why he made sharp turn when he has to pass points that are closed to each other. This causes the ^ section which is called as cusp.
And then moving on to racer #2 who adjusts his speed extremely based from time to time by observing how far away the next point is. If the next point is near then he will slow down, otherwise if it is still far away then he will increase speed. This causes the last turn to be a late apex.
In racing, a driver will often use a "late apex," turning into the corner a little later than normal in order to straighten out the last part of the corner. This allows the driver to accelerate earlier and harder, gaining maximum speed down the next straight.Lastly racer #3 is also the type who adjusts speed based on how far away the next point is. But he does not make too much extreme changes of speed along the way. So here we can see that the driving path he made is the one passing those points elegantly.
The question now is which racers complete the trip the earliest?
They completed it in the same amount of time!