空间度量是 GIS 分析的基础。Shapely 提供了丰富的度量函数，涵盖面积、长度、距离、
形状相似度等多种计算。这些函数基于平面几何（欧几里得空间），能够高效地计算几何对象的
各种度量属性。本章将全面介绍 Shapely 中的所有度量函数，包括数学含义、应用场景和使用方法。
area(geometry) 返回几何对象的面积。只有面状几何（Polygon / MultiPolygon）才有非零面积。
from shapely import area
from shapely.geometry import Polygon, Point, LineString
# 矩形面积
rect = Polygon([(0, 0), (4, 0), (4, 3), (0, 3)])
print(f"矩形面积: {area(rect)}") # 12.0
# 三角形面积
tri = Polygon([(0, 0), (6, 0), (3, 4)])
print(f"三角形面积: {area(tri)}") # 12.0
# 线和点的面积为 0
line = LineString([(0, 0), (1, 1)])
point = Point(0, 0)
print(f"线面积: {area(line)}") # 0.0
print(f"点面积: {area(point)}") # 0.0
from shapely import area
from shapely.geometry import Polygon
# 外环面积 100，内环面积 4
poly_with_hole = Polygon(
[(0, 0), (10, 0), (10, 10), (0, 10)],
[[(3, 3), (5, 3), (5, 5), (3, 5)]]
)
print(f"带洞多边形面积: {area(poly_with_hole)}") # 96.0
from shapely import area
from shapely.geometry import MultiPolygon, Polygon
mp = MultiPolygon([
Polygon([(0, 0), (1, 0), (1, 1), (0, 1)]),
Polygon([(2, 2), (4, 2), (4, 4), (2, 4)]),
])
print(f"MultiPolygon 面积: {area(mp)}") # 1 + 4 = 5.0
from shapely import area, box
import numpy as np
# 利用向量化进行批量计算
boxes = np.array([box(0, 0, i, i) for i in range(1, 6)])
areas = area(boxes)
print(f"批量面积: {areas}") # [1. 4. 9. 16. 25.]
length(geometry) 返回几何对象的长度。对于线状几何返回线长度，对于面状几何返回周长。
from shapely import length
from shapely.geometry import LineString, Polygon, Point
# 线长度
line = LineString([(0, 0), (3, 4)])
print(f"线长度: {length(line)}") # 5.0
# 折线长度
polyline = LineString([(0, 0), (3, 0), (3, 4)])
print(f"折线长度: {length(polyline)}") # 7.0
# 多边形周长
poly = Polygon([(0, 0), (4, 0), (4, 3), (0, 3)])
print(f"矩形周长: {length(poly)}") # 14.0
# 点没有长度
point = Point(0, 0)
print(f"点长度: {length(point)}") # 0.0
from shapely import length
from shapely.geometry import Point
import math
circle = Point(0, 0).buffer(5)
print(f"近似圆周长: {length(circle):.4f}")
print(f"理论圆周长: {2 * math.pi * 5:.4f}")
bounds(geometry) 返回几何对象的边界框，格式为 (minx, miny, maxx, maxy)。
from shapely import bounds
from shapely.geometry import Polygon, LineString, Point
poly = Polygon([(1, 2), (5, 1), (4, 6), (2, 5)])
print(f"多边形边界框: {bounds(poly)}") # (1.0, 1.0, 5.0, 6.0)
line = LineString([(0, 0), (3, 4)])
print(f"线边界框: {bounds(line)}") # (0.0, 0.0, 3.0, 4.0)
point = Point(3, 7)
print(f"点边界框: {bounds(point)}") # (3.0, 7.0, 3.0, 7.0)
from shapely import bounds, box
import numpy as np
geoms = np.array([
box(0, 0, 1, 1),
box(2, 3, 5, 6),
box(-1, -1, 4, 4),
])
bbox = bounds(geoms)
print(f"批量边界框:\n{bbox}")
total_bounds(geometries) 返回一组几何对象的总边界框。
from shapely import total_bounds, box
import numpy as np
geoms = np.array([
box(0, 0, 2, 2),
box(3, 1, 5, 4),
box(1, 3, 4, 6),
])
tb = total_bounds(geoms)
print(f"总边界框: {tb}") # (0.0, 0.0, 5.0, 6.0)
distance(a, b) 返回两个几何对象之间的最短欧几里得距离。
from shapely import distance
from shapely.geometry import Point, Polygon, LineString
# 两点之间的距离
p1 = Point(0, 0)
p2 = Point(3, 4)
print(f"两点距离: {distance(p1, p2)}") # 5.0
# 点到线的距离
line = LineString([(0, 1), (4, 1)])
print(f"点到线距离: {distance(Point(2, 3), line)}") # 2.0
# 点到多边形的距离
poly = Polygon([(2, 2), (6, 2), (6, 6), (2, 6)])
print(f"内部点距离: {distance(Point(3, 3), poly)}") # 0.0（在内部）
print(f"外部点距离: {distance(Point(0, 0), poly)}")
from shapely import distance
from shapely.geometry import Polygon
a = Polygon([(0, 0), (2, 0), (2, 2), (0, 2)])
b = Polygon([(1, 1), (3, 1), (3, 3), (1, 3)])
# 相交时距离为 0
print(f"相交几何体距离: {distance(a, b)}") # 0.0
from shapely import distance
from shapely.geometry import Point
import numpy as np
origin = Point(0, 0)
points = np.array([Point(i, i) for i in range(1, 6)])
# 向量化计算
distances = distance(origin, points)
print(f"批量距离: {distances}")
Hausdorff 距离是衡量两个几何对象形状相似度的指标。它定义为：对于 A 上的每个点，
找到距离 B 最近的点的距离，取所有这些距离中的最大值；对 B 做同样的操作；取两者中的较大值。
直觉上，如果 Hausdorff 距离为 d，那么任何一个几何体上的每个点，到另一个几何体的距离都不超过 d。
from shapely import hausdorff_distance
from shapely.geometry import LineString
line1 = LineString([(0, 0), (1, 0), (2, 0)])
line2 = LineString([(0, 1), (1, 1), (2, 1)])
hd = hausdorff_distance(line1, line2)
print(f"Hausdorff 距离: {hd}") # 1.0
from shapely import hausdorff_distance
from shapely.geometry import LineString
# 原始曲线
original = LineString([(0, 0), (1, 1), (2, 0), (3, 1), (4, 0)])
# 近似曲线
approx1 = LineString([(0, 0), (2, 0), (4, 0)])
approx2 = LineString([(0, 0.1), (1, 1.1), (2, 0.1), (3, 1.1), (4, 0.1)])
hd1 = hausdorff_distance(original, approx1)
hd2 = hausdorff_distance(original, approx2)
print(f"粗略近似 Hausdorff: {hd1:.3f}")
print(f"精细近似 Hausdorff: {hd2:.3f}")
densify 参数用于在计算前对几何体进行加密，以获得更精确的结果：
from shapely import hausdorff_distance
from shapely.geometry import LineString
line1 = LineString([(0, 0), (10, 0)])
line2 = LineString([(0, 1), (5, 1), (10, 0)])
# 不加密
hd_raw = hausdorff_distance(line1, line2)
print(f"不加密: {hd_raw:.3f}")
# 加密（densify 表示每段长度与总长度的比例）
hd_dense = hausdorff_distance(line1, line2, densify=0.1)
print(f"加密后: {hd_dense:.3f}")
Fréchet 距离（弗雷歇距离）是衡量两条曲线相似度的指标，直觉上类似于"遛狗距离"：
一个人沿着一条曲线走，他的狗沿着另一条曲线走，两者都只能前进不能后退，
Fréchet 距离就是在所有可能的行走方案中，最短狗绳的最小长度。
from shapely import frechet_distance
from shapely.geometry import LineString
line1 = LineString([(0, 0), (1, 0), (2, 0), (3, 0)])
line2 = LineString([(0, 1), (1, 1), (2, 1), (3, 1)])
fd = frechet_distance(line1, line2)
print(f"Fréchet 距离: {fd}") # 1.0
from shapely import frechet_distance, hausdorff_distance
from shapely.geometry import LineString
# 两条方向相反的线
line1 = LineString([(0, 0), (1, 0), (2, 0)])
line2 = LineString([(2, 1), (1, 1), (0, 1)])
hd = hausdorff_distance(line1, line2)
fd = frechet_distance(line1, line2)
print(f"Hausdorff 距离: {hd:.3f}") # 只看最近距离的最大值
print(f"Fréchet 距离: {fd:.3f}") # 考虑行进方向，通常更大
from shapely import frechet_distance
from shapely.geometry import LineString
line1 = LineString([(0, 0), (10, 0)])
line2 = LineString([(0, 0), (5, 5), (10, 0)])
fd = frechet_distance(line1, line2)
fd_dense = frechet_distance(line1, line2, densify=0.1)
print(f"Fréchet 距离: {fd:.3f}")
print(f"加密后 Fréchet: {fd_dense:.3f}")
from shapely import minimum_bounding_radius
from shapely.geometry import Polygon, MultiPoint
poly = Polygon([(0, 0), (4, 0), (4, 3), (0, 3)])
radius = minimum_bounding_radius(poly)
print(f"最小外接圆半径: {radius:.3f}") # 对角线一半
points = MultiPoint([(0, 0), (3, 0), (0, 4)])
radius_pts = minimum_bounding_radius(points)
print(f"三角形外接圆半径: {radius_pts:.3f}")
minimum_clearance(geometry) 返回几何对象中，移动任何顶点使几何体变为无效（如自相交）
所需的最小距离。这是衡量几何体"稳定性"的指标。
from shapely import minimum_clearance
from shapely.geometry import Polygon
# 接近退化的矩形
poly = Polygon([(0, 0), (10, 0), (10, 0.001), (0, 0.001)])
mc = minimum_clearance(poly)
print(f"最小间隙: {mc}") # 接近 0.001
# 正方形
square = Polygon([(0, 0), (1, 0), (1, 1), (0, 1)])
mc_sq = minimum_clearance(square)
print(f"正方形最小间隙: {mc_sq}") # 1.0
返回表示最小间隙的线段。
from shapely import minimum_clearance, minimum_clearance_line
from shapely.geometry import Polygon
poly = Polygon([(0, 0), (10, 0), (10, 5), (5, 5), (5, 4.9), (0, 5)])
mc = minimum_clearance(poly)
mc_line = minimum_clearance_line(poly)
print(f"最小间隙: {mc:.3f}")
print(f"最小间隙线: {mc_line}")
print(f"线长度: {mc_line.length:.3f}")
返回几何对象的最小宽度。
from shapely import minimum_width
from shapely.geometry import Polygon
# 窄矩形
narrow = Polygon([(0, 0), (10, 0), (10, 1), (0, 1)])
mw = minimum_width(narrow)
print(f"最小宽度: {mw.length}") # 1.0
# 三角形
tri = Polygon([(0, 0), (6, 0), (3, 2)])
mw_tri = minimum_width(tri)
print(f"三角形最小宽度: {mw_tri.length:.3f}")
返回几何对象内部的最大内切圆。
from shapely import maximum_inscribed_circle
from shapely.geometry import Polygon
# 正方形
square = Polygon([(0, 0), (10, 0), (10, 10), (0, 10)])
mic = maximum_inscribed_circle(square, tolerance=0.01)
print(f"最大内切圆类型: {mic.geom_type}") # LineString（圆心到边的线）
print(f"内切圆半径: {mic.length:.2f}") # 5.0
# L形
L_shape = Polygon([
(0, 0), (6, 0), (6, 2), (2, 2), (2, 6), (0, 6)
])
mic_L = maximum_inscribed_circle(L_shape, tolerance=0.01)
print(f"L形最大内切圆半径: {mic_L.length:.2f}")
返回连接两个几何对象的最短线段。
from shapely import shortest_line
from shapely.geometry import Point, Polygon, LineString
p = Point(0, 0)
poly = Polygon([(3, 3), (6, 3), (6, 6), (3, 6)])
sl = shortest_line(p, poly)
print(f"最短连线: {sl}")
print(f"长度: {sl.length:.3f}")
from shapely import shortest_line
from shapely.geometry import LineString
line1 = LineString([(0, 0), (4, 0)])
line2 = LineString([(1, 3), (3, 3)])
sl = shortest_line(line1, line2)
print(f"最短连线: {sl}")
print(f"起点: ({sl.coords[0][0]:.1f}, {sl.coords[0][1]:.1f})")
print(f"终点: ({sl.coords[1][0]:.1f}, {sl.coords[1][1]:.1f})")
from shapely import shortest_line
from shapely.geometry import Point, LineString
import numpy as np
road = LineString([(0, 0), (10, 0)])
houses = np.array([Point(i, j) for i, j in [(2, 3), (5, 1), (8, 4)]])
lines = shortest_line(houses, road)
for i, sl in enumerate(lines):
print(f"房屋{i} 到道路最短连线长度: {sl.length:.2f}")
Shapely 2.x 的度量函数支持 NumPy 数组输入，实现高效的批量计算：
from shapely import area, length, distance, bounds
from shapely.geometry import Point
import numpy as np
# 创建大量随机多边形
np.random.seed(42)
n = 1000
points = np.array([Point(x, y).buffer(r)
for x, y, r in zip(
np.random.uniform(0, 100, n),
np.random.uniform(0, 100, n),
np.random.uniform(0.5, 3, n),
)])
# 批量计算
areas = area(points)
perimeters = length(points)
print(f"面积统计: min={areas.min():.2f}, max={areas.max():.2f}, mean={areas.mean():.2f}")
print(f"周长统计: min={perimeters.min():.2f}, max={perimeters.max():.2f}")
from shapely import distance
from shapely.geometry import Point
import numpy as np
# 计算点之间的距离矩阵
pts = np.array([Point(0, 0), Point(1, 0), Point(0, 1), Point(1, 1)])
n = len(pts)
dist_matrix = np.zeros((n, n))
for i in range(n):
# 向量化计算一行
dist_matrix[i] = distance(pts[i], pts)
print("距离矩阵:")
print(np.round(dist_matrix, 3))
from shapely import distance
from shapely.geometry import Point
import numpy as np
# 设施位置
facilities = np.array([
Point(10, 20), Point(30, 10), Point(50, 40),
Point(20, 50), Point(40, 25),
])
# 查询点
query = Point(25, 30)
# 找到最近的设施
distances = distance(query, facilities)
nearest_idx = np.argmin(distances)
print(f"最近的设施索引: {nearest_idx}")
print(f"最近距离: {distances[nearest_idx]:.2f}")
from shapely import area, length
from shapely.geometry import Polygon, Point
import math
shapes = {
"正方形": Polygon([(0, 0), (1, 0), (1, 1), (0, 1)]),
"长矩形": Polygon([(0, 0), (10, 0), (10, 1), (0, 1)]),
"圆形": Point(0, 0).buffer(1),
"L形": Polygon([(0, 0), (2, 0), (2, 1), (1, 1), (1, 2), (0, 2)]),
}
print("形状紧凑度分析（4π × 面积 / 周长²）:")
for name, shape in shapes.items():
a = area(shape)
p = length(shape)
compactness = 4 * math.pi * a / (p * p)
print(f" {name}: 面积={a:.2f}, 周长={p:.2f}, 紧凑度={compactness:.4f}")
from shapely import area, length
from shapely.geometry import Polygon
import numpy as np
# 随机多边形
np.random.seed(0)
polygons = []
for _ in range(20):
cx, cy = np.random.uniform(0, 100, 2)
r = np.random.uniform(1, 10)
angles = np.sort(np.random.uniform(0, 2 * np.pi, 6))
coords = [(cx + r * np.cos(a), cy + r * np.sin(a)) for a in angles]
coords.append(coords[0])
try:
p = Polygon(coords)
if p.is_valid and not p.is_empty:
polygons.append(p)
except Exception:
pass
geoms = np.array(polygons)
areas = area(geoms)
perimeters = length(geoms)
print(f"有效多边形数: {len(geoms)}")
print(f"平均面积: {areas.mean():.2f}")
print(f"平均周长: {perimeters.mean():.2f}")
print(f"面积标准差: {areas.std():.2f}")
Shapely 只进行平面（笛卡尔坐标系）计算，不支持球面（地理坐标系）计算。
from shapely import distance
from shapely.geometry import Point
# 这是坐标差值，不是真实的地球表面距离！
beijing = Point(116.4, 39.9)
shanghai = Point(121.5, 31.2)
# 这个结果的单位是"度"，不是"米"或"千米"
d = distance(beijing, shanghai)
print(f"坐标距离: {d:.2f}（度，不是实际距离）")
# 如需计算真实地球距离，应该：
# 1. 将坐标投影到合适的平面坐标系（如 UTM）
# 2. 使用 pyproj 或 geopy 进行球面距离计算
from shapely import area
from shapely.geometry import Polygon
# 极小多边形可能有精度问题
tiny = Polygon([
(0, 0),
(1e-15, 0),
(1e-15, 1e-15),
(0, 1e-15),
])
print(f"极小面积: {area(tiny)}")
# 极大坐标也可能有精度问题
huge = Polygon([
(1e15, 1e15),
(1e15 + 1, 1e15),
(1e15 + 1, 1e15 + 1),
(1e15, 1e15 + 1),
])
print(f"极大坐标面积: {area(huge)}")
from shapely import area, length, distance, bounds
from shapely import wkt
empty = wkt.loads("GEOMETRYCOLLECTION EMPTY")
print(f"空几何面积: {area(empty)}") # 0.0
print(f"空几何长度: {length(empty)}") # 0.0
print(f"空几何边界框: {bounds(empty)}")
| 函数 | 说明 | 适用类型 |
|---|---|---|
area(geom) |
面积 | Polygon, MultiPolygon |
length(geom) |
长度/周长 | Line*, Polygon |
bounds(geom) |
边界框 | 所有类型 |
total_bounds(geoms) |
总边界框 | 几何数组 |
distance(a, b) |
最短距离 | 所有类型 |
hausdorff_distance(a, b) |
Hausdorff 距离 | 所有类型 |
frechet_distance(a, b) |
Fréchet 距离 | LineString |
minimum_bounding_radius(geom) |
最小外接圆半径 | 所有类型 |
minimum_clearance(geom) |
最小间隙 | 所有类型 |
minimum_clearance_line(geom) |
最小间隙线 | 所有类型 |
minimum_width(geom) |
最小宽度 | Polygon |
maximum_inscribed_circle(geom) |
最大内切圆 | Polygon |
shortest_line(a, b) |
最短连线 | 所有类型 |
本章全面介绍了 Shapely 的空间度量函数：
area、length、boundsdistance、hausdorff_distance、frechet_distanceminimum_clearance、minimum_width、maximum_inscribed_circleshortest_line关键要点：
下一章将介绍仿射变换，学习如何对几何对象进行平移、旋转、缩放等操作。
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。