f02458b2-a356-4a8b-8ff9-a91cbc14bbb1 21 4 5 decimal

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vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

The Hermite polyline "Smoothlet" ...

... is a gorgeous piece of applied maths for smoothing data with little OFFset in the convex hull. It runs fast, populates at will. XW is the data set from K0(u) in De Boor Carbon α 12. The procedure produces duplicates in the return carriage. For a clean export, you will have to strip from repeated values. All segments are equally populated in the "t range 0..1". Is it a good idea to spline the XW ? ---------------------------- Answer: NOT at all, ordinary cubic spline does NOT smooth ! As mentionned in "De Boor Carbon α 12", Hermite does not need extra "Supsmooth" .

t0 1 time :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

XW 0.175 317.303 0.3325 217.0355 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 1.4 143.2053 1.575 173.5314 1.75 155.6779 1.925 105.1301 2.1 56.8338 2.275 34.1471 2.45 51.1141 2.625 114.8755 2.8 214.0454 2.975 306.0658 3.15 341.7215 3.325 304.952 3.5 214.1093 3.675 111.3089 3.85 42.8756 4.025 15.8375 4.2 7.5659 242 mat :

X XW 1 col :

Y XW 2 col :

r 2 :

X X r RepEnd :

Y Y r RepEnd :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

Hermite utilities

Hermite PolyLine t P 16 / t 3^- 3 t 2^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 mat * :

Hermite PolyLine 0...1

x P 1 el x P 2 el x P 3 el x P 4 el 4 1 sys

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

t 01 0.1 range :

<= populate for more points at will

j runs 1..rows(X)-(r+1)

U j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP : for ct ct 1 + : 21 line for Hy 41 line :

Each "j" segment is collected in matrix of 't' rows x 'j' cols. Each U,V is Unwrapped in column, augmented in 'xy', then stripped of repeated pairs for export.

xy U Unwrap V Unwrap augment :

xy xy Strip :

xw XW 1 col XW 2 col . 10 black plotG :

1 time t0 - 9 sec *

Hermite Smoothlet to "Carbon α 12"

xw xy 2 1 sys

49 pts * DeBoor 275 pts * Hermite 21 mat ~ 7 minutes 10 sec 21 mat ≡

Naturally, you could force the convex hull [not proposed].

jmGiraud: 2015/10/04