Smooth Subdivision Surfaces based on Triangles

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https://people.eecs.berkeley.edu/~sequin/CS284/LECT09/L11.html
Some Errata in C.T. Loop:
"Smooth Subdivision Surfaces based on Triangles"

Page 23, para 2:
The new vertices are NOT "the midpoints of of the line segments
connecting a faces centroid to each of its vertices (Fig.3.1)."
This would create a subdivision mask of

10 ---- 2
| |
| |
2 ---- 2

They are the centroids of the subfaces formed by the face centroid,
a corner vertex, and the two mid-edge points next to the corner.

Page 27, eqn (3.2)
The coefficient of the R-term should be 2/N.

???
In Eqn. 4.7 and 4.8 terms should be V^(l-1) instead of V^l

[kanition]

证明终于找到了
http://www.cad.zju.edu.cn/home/zhx/GM/007/00-bscs2.pdf
搜索关键词
B-spline two scale relation
https://en.wikipedia.org/wiki/Spline_wavelet

[kanition]

@xhp-hust-2018-2011

[kanition]

import numpy as np
from scipy.special import comb
from matplotlib import pyplot as plt


def n0(x: float, r=0) -> float:
    if r != 0:
        raise ValueError(f'r={r} in n0')
    if 0 <= x < 1:
        return 1.0
    else:
        return 0.0


def nr(x: float, r: int, i: int = 0):
    if not isinstance(r, int) or r < 0:
        raise ValueError(f'wrong r={r}')
    if r == 0:
        return n0(x - i)
    else:
        y = ((x - i) * nr(x - i, r - 1) + (r + 1 - (x - i)) * nr(x - i - 1, r - 1)) / r
        return y


def main():
    x = np.arange(-1, 10, 0.02)
    r = 3
    y1 = [nr(u, r) for u in x]
    c1 = [u / r * nr(u, r - 1) for u in x]
    c2 = [(r + 1 - u) / r * nr(u - 1, r - 1) for u in x]
    y3 = [nr(r + 1 - u, r) for u in x]
    y2 = [sum(1 / np.power(2, r) * comb(r + 1, i) * nr(2 * u - i, r) for i in range(r + 2)) for u in x]
    plt.axis('equal')
    plt.plot(x, y1, 'b')
    plt.plot(x, c1, 'm')
    plt.plot(x, c2, 'm')
    plt.plot(x, y2, 'r')
    plt.plot(x, y3, 'c')
    for i in range(r + 2):
        y = [1 / np.power(2, r) * comb(r + 1, i) * nr(2 * u - i, r) for u in x]
        z = [nr(2 * u - i, r) for u in x]
        plt.plot(x, y, 'g')
        plt.plot(x, z, 'y')
    plt.show()


if __name__ == '__main__':
    main()

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