生成凹壳

http://blog.thehumangeo.com/2014/05/12/drawing-boundaries-in-python/

#!/usr/bin/env python
# -*- coding: utf-8 -*-

###############
#
# Code source modified from
# http://blog.thehumangeo.com/2014/05/12/drawing-boundaries-in-python/
#
###############


from shapely import geometry
from shapely.ops import cascaded_union, polygonize
from scipy.spatial import Delaunay
import numpy as np
import math


def alpha_shape(points, alpha):
    """
    Compute the alpha shape (concave hull) of a set
    of points.

    @param points: Iterable container of points.
    @param alpha: alpha value to influence the
        gooeyness of the border. Smaller numbers
        don't fall inward as much as larger numbers.
        Too large, and you lose everything!
    """
    if len(points) < 4:
        # When you have a triangle, there is no sense
        # in computing an alpha shape.
        return geometry.MultiPoint(list(points)).convex_hull

    def add_edge(edges, edge_points, coords, i, j):
        """
        Add a line between the i-th and j-th points,
        if not in the list already
        """
        if (i, j) in edges or (j, i) in edges:
            # already added
            return
        edges.add((i, j))
        edge_points.append(coords[[i, j]])

    coords = np.array([point.coords[0] for point in points])

    tri = Delaunay(coords)
    edges = set()
    edge_points = []
    # loop over triangles:
    # ia, ib, ic = indices of corner points of the
    # triangle
    for ia, ib, ic in tri.vertices:
        pa = coords[ia]
        pb = coords[ib]
        pc = coords[ic]

        # Lengths of sides of triangle
        a = math.sqrt((pa[0] - pb[0]) ** 2 + (pa[1] - pb[1]) ** 2)
        b = math.sqrt((pb[0] - pc[0]) ** 2 + (pb[1] - pc[1]) ** 2)
        c = math.sqrt((pc[0] - pa[0]) ** 2 + (pc[1] - pa[1]) ** 2)

        # Semiperimeter of triangle
        s = (a + b + c) / 2.0

        # Area of triangle by Heron's formula
        area = math.sqrt(s * (s - a) * (s - b) * (s - c))
        circum_r = a * b * c / (4.0 * area) #外接圆半径

        # Here's the radius filter.
        # print circum_r
        if circum_r < 1.0 / alpha:
            add_edge(edges, edge_points, coords, ia, ib)
            add_edge(edges, edge_points, coords, ib, ic)
            add_edge(edges, edge_points, coords, ic, ia)

    m = geometry.MultiLineString(edge_points)
    triangles = list(polygonize(m))
    return cascaded_union(triangles), edge_points

alpha = .4
concave_hull, edge_points = alpha_shape(new_points,  alpha=alpha)

plot_polygon(concave_hull)
_ = pl.plot(x,y,'o', color='#f16824')
plot_polygon(concave_hull.buffer(1))
_ = pl.plot(x,y,'o', color='#f16824')

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1、三角形面积=1/2*底*高（三边都可做底）
2、三角形面积=1/2absinC=1/2acsinB=1/2bcsinA
3、三角形面积=abc/4R(其中R是三角形外接圆半径）

a=2RsinA
b=2RsinB
c=2RsinC