Example 38
Peano vs Hilbert curves
| P |
730 |
27.000000000000178 |
| Q |
731 |
27.052378280088075 |
| Fréchet |
0.2636469803670366 |
0 |
| VE Fréchet |
0.2636469803670366 |
92955 |
Animation: Fréchet morphing
This is the animation of the morphing computed
that is both continuous and monotone.
Animation: VE Retractable Fréchet
This is the animation of the VE retractable morphing. It is potentially not monotone (but it is continuous.
Free space diagram heatmap:
VE-Fréchet Retractable solution:
Monotonized...
Fréchet cont+monotone solution:
Discrete Fréchet
The resulting morphing - extended to continuous:
Specifically, to get a smooth animation, the
leash is shown as moving continuously, by
interpolating between the discrete locations.
The discrete retractable version
The discrete dynamic time warping
P # vertices: 730
P # vertices: 731
DFréchet iters : 533630
Retract DFréchet iters : 44097
Refinement for removing monotonicity
By introducing vertices in the middle of parts of the curves that are being traversed in the wrong direction, one can refine the solution, till effectively reaching the optimal monotone solution. This process is demonstrated below. As one can see, the error is negligible after a few (say four) iterations. After that, it becomes a bit pointless.
We emphasize that a monotone morphing can
always
be extracted, by monotonizing the current
solution.
This is easy and fast to do, and is the error
accounted for in the below graphics.
More info
Animation: Fréchet morphing as morphing
2025-03-08 21:31:40