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+# 曲线网格基础
+
+本教程介绍如何使用 `meshkernel` 库生成曲线网格。
+
+[返回示例目录](index.md)
+
+以下保留原笔记本的代码和已保存输出；转换过程中未重新执行代码。
+
+首先导入所需的库。
+
+```python
+import matplotlib.pyplot as plt
+import numpy as np
+
+from meshkernel import (
+    CurvilinearParameters,
+    MakeGridParameters,
+    GeometryList,
+    MeshKernel,
+    SplinesToCurvilinearParameters,
+    OrthogonalizationParameters,
+)
+```
+
+定义一个函数，使用 `curvilinear_compute_transfinite_from_splines` 生成曲线网格，并创建包含该网格的 `MeshKernel` 实例：
+
+- 首先创建用于生成曲线网格的样条曲线，各条样条曲线用 `-999.0` 分隔。
+- 在新的 `CurvilinearParameters` 实例中设置 m、n 方向的划分数。
+- 创建一个新的 `MeshKernel` 实例。
+- 使用超限插值算法生成曲线网格。
+
+```python
+def create_mk_instance_with_curvilinear_grid_from_transfinite_method():
+    r"""创建包含曲线网格的 MeshKernel 实例。"""
+    mk = MeshKernel()
+
+    separator = -999.0
+    splines_x = np.array(
+        [
+            2.0,
+            4.0,
+            7.0,
+            separator,
+            -1.0,
+            1.0,
+            5.0,
+            separator,
+            3.0,
+            -2.0,
+            separator,
+            7.0,
+            4.0,
+        ],
+        dtype=np.double,
+    )
+    splines_y = np.array(
+        [
+            1.0,
+            3.0,
+            4.0,
+            separator,
+            4.0,
+            6.0,
+            7.0,
+            separator,
+            1.0,
+            6.0,
+            separator,
+            3.0,
+            8.0,
+        ],
+        dtype=np.double,
+    )
+    splines = GeometryList(splines_x, splines_y)
+
+    curvilinear_parameters = CurvilinearParameters()
+    curvilinear_parameters.n_refinement = 10
+    curvilinear_parameters.m_refinement = 10
+
+    mk.curvilinear_compute_transfinite_from_splines(splines, curvilinear_parameters)
+
+    return mk
+```
+
+定义一个用于创建矩形曲线网格的函数。
+
+```python
+def create_mk_instance_with_a_rectangular_curvilinear_grid(num_columns=3, num_rows=3):
+    r"""创建包含矩形曲线网格的 MeshKernel 实例的局部函数。"""
+    mk = MeshKernel()
+
+    # 创建 MakeGridParameters 实例并设置参数值
+    make_grid_parameters = MakeGridParameters()
+    make_grid_parameters.num_columns = num_columns
+    make_grid_parameters.num_rows = num_rows
+    make_grid_parameters.angle = 0.0
+    make_grid_parameters.origin_x = 0.0
+    make_grid_parameters.origin_y = 0.0
+    make_grid_parameters.block_size_x = 10.0
+    make_grid_parameters.block_size_y = 10.0
+
+    mk.curvilinear_compute_rectangular_grid(make_grid_parameters)
+
+    return mk
+```
+
+## 使用超限插值法生成曲线网格
+
+```python
+curvilinear_grid_transfinite = (
+    create_mk_instance_with_curvilinear_grid_from_transfinite_method().curvilineargrid_get()
+)
+```
+
+绘制结果。
+
+```python
+fig, ax = plt.subplots()
+curvilinear_grid_transfinite.plot_edges(ax)
+```
+
+![单元格 10 的绘图输出](images/04_curvilineargrid_basics_10_0.png)
+
+## 使用推进前沿法生成曲线网格
+
+定义生成曲线网格的过程，使用 `curvilinear_compute_orthogonal_from_splines` 方法创建包含网格的 `MeshKernel` 实例：
+
+- 首先创建用于生成曲线网格的样条曲线，各条样条曲线用 `-999.0` 分隔。
+- 在新的 `CurvilinearParameters` 实例中设置 m、n 方向的划分数。
+- 该算法还需要设置从样条曲线生成曲线网格所需的附加参数。
+- 然后使用推进前沿算法生成曲线网格。
+
+```python
+mk = MeshKernel()
+
+separator = -999.0
+splines_x = np.array([-1.0, 2.0, 6.0, separator, 3.0, -2.0, separator], dtype=np.double)
+splines_y = np.array([2.0, 5.0, 6.0, separator, 1.0, 6.0, separator], dtype=np.double)
+
+splines_values = np.zeros_like(splines_x)
+splines = GeometryList(splines_x, splines_y, splines_values)
+
+curvilinearParameters = CurvilinearParameters()
+curvilinearParameters.n_refinement = 10
+curvilinearParameters.m_refinement = 10
+
+splinesToCurvilinearParameters = SplinesToCurvilinearParameters()
+splinesToCurvilinearParameters.aspect_ratio = 1.0
+splinesToCurvilinearParameters.aspect_ratio_grow_factor = 1.0
+splinesToCurvilinearParameters.average_width = 0.2
+splinesToCurvilinearParameters.nodes_on_top_of_each_other_tolerance = 1e-4
+splinesToCurvilinearParameters.min_cosine_crossing_angles = 0.95
+splinesToCurvilinearParameters.check_front_collisions = 0
+splinesToCurvilinearParameters.curvature_adapted_grid_spacing = 1
+splinesToCurvilinearParameters.remove_skinny_triangles = 1
+
+mk.curvilinear_compute_orthogonal_from_splines(
+    splines, curvilinearParameters, splinesToCurvilinearParameters
+)
+
+curvilinear_grid_orthogonal = mk.curvilineargrid_get()
+```
+
+绘制结果。
+
+```python
+fig, ax = plt.subplots()
+curvilinear_grid_orthogonal.plot_edges(ax)
+```
+
+![单元格 15 的绘图输出](images/04_curvilineargrid_basics_15_0.png)
+
+## 曲线网格加密与粗化
+
+加密前的网格。
+
+```python
+mk = create_mk_instance_with_curvilinear_grid_from_transfinite_method()
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 18 的绘图输出](images/04_curvilineargrid_basics_18_0.png)
+
+在两个选定点之间，为每一行添加两条水平网格线进行加密，并绘制结果。
+
+```python
+mk.curvilinear_refine(2.299, 4.612, 3.074, 3.684, 2)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 20 的绘图输出](images/04_curvilineargrid_basics_20_0.png)
+
+删除相同行中的网格线，进行粗化。
+
+```python
+mk.curvilinear_refine(2.299, 4.612, 3.074, 3.684, -2)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 22 的绘图输出](images/04_curvilineargrid_basics_22_0.png)
+
+## 创建矩形网格
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid()
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 24 的绘图输出](images/04_curvilineargrid_basics_24_0.png)
+
+也可以根据多边形生成矩形网格。多边形必须闭合。
+
+```python
+node_x = np.array([2.5, 5.5, 3.5, 0.5, 2.5], dtype=np.double)
+node_y = np.array([0.5, 3.0, 5.0, 2.5, 0.5], dtype=np.double)
+geometry_list = GeometryList(node_x, node_y)
+```
+
+```python
+make_grid_parameters = MakeGridParameters()
+make_grid_parameters.num_columns = 10
+make_grid_parameters.num_rows = 10
+make_grid_parameters.angle = 0.0
+make_grid_parameters.origin_x = 0.0
+make_grid_parameters.origin_y = 0.0
+make_grid_parameters.block_size_x = 0.2
+make_grid_parameters.block_size_y = 0.2
+```
+
+```python
+mk.curvilinear_compute_rectangular_grid_from_polygon(
+    make_grid_parameters, geometry_list
+)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 28 的绘图输出](images/04_curvilineargrid_basics_28_0.png)
+
+## 利用多边形边界上的节点生成曲线网格
+
+定义多边形并生成曲线网格。
+
+```python
+node_x = np.array([2, 4, 6, 7, 8, 8, 8, 8, 7, 5, 3, 2, 2, 2, 2], dtype=np.double)
+node_y = np.array([1, 1, 1, 1, 1, 1.2, 4, 6, 6, 6, 6, 6, 5, 3, 1], dtype=np.double)
+geometry_list = GeometryList(node_x, node_y)
+mk = MeshKernel()
+mk.curvilinear_compute_transfinite_from_polygon(geometry_list, 0, 4, 7, False)
+```
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 32 的绘图输出](images/04_curvilineargrid_basics_32_0.png)
+
+## 曲线网格正交化
+
+移动一个节点，使网格不再正交，并绘制结果。
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid()
+mk.curvilinear_move_node(10.0, 20.0, 18.0, 12.0)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 35 的绘图输出](images/04_curvilineargrid_basics_35_0.png)
+
+执行正交化。
+
+```python
+# 在正交化前检查节点位置
+orthogonalization_parameters = OrthogonalizationParameters()
+orthogonalization_parameters.outer_iterations = 1
+orthogonalization_parameters.boundary_iterations = 25
+orthogonalization_parameters.inner_iterations = 25
+orthogonalization_parameters.orthogonalization_to_smoothing_factor = 0.95
+
+# 初始化曲线网格正交化算法
+# 设置要正交化的网格块（本例指定网格的左下角和右上角）
+mk.curvilinear_orthogonalize(orthogonalization_parameters, 0.0, 0.0, 30.0, 30.0)
+```
+
+绘制正交化后的结果。
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 39 的绘图输出](images/04_curvilineargrid_basics_39_0.png)
+
+## 固定一条网格线进行曲线网格正交化
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid()
+mk.curvilinear_move_node(10.0, 20.0, 18.0, 12.0)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 41 的绘图输出](images/04_curvilineargrid_basics_41_0.png)
+
+执行正交化，同时固定被移动节点所在的垂直网格线。
+
+```python
+# 在正交化前检查节点位置
+orthogonalization_parameters = OrthogonalizationParameters()
+orthogonalization_parameters.outer_iterations = 1
+orthogonalization_parameters.boundary_iterations = 25
+orthogonalization_parameters.inner_iterations = 25
+orthogonalization_parameters.orthogonalization_to_smoothing_factor = 0.95
+
+# 固定被移动节点所在的垂直网格线
+mk.curvilinear_frozen_line_add(10.0, 0.0, 10.0, 30.0)
+
+# 执行正交化
+# 初始化曲线网格正交化算法
+# 设置要正交化的网格块（本例指定网格的左下角和右上角）
+mk.curvilinear_orthogonalize(orthogonalization_parameters, 0.0, 0.0, 30.0, 30.0)
+```
+
+绘制正交化后的结果。
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 45 的绘图输出](images/04_curvilineargrid_basics_45_0.png)
+
+## 曲线网格平滑
+
+移动一个节点，使网格不再平滑，并绘制结果。
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid()
+mk.curvilinear_move_node(10.0, 20.0, 18.0, 12.0)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 48 的绘图输出](images/04_curvilineargrid_basics_48_0.png)
+
+执行平滑。
+
+```python
+mk.curvilinear_smoothing(10, 0.0, 0.0, 30.0, 30.0)
+```
+
+绘制平滑后的结果。
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 52 的绘图输出](images/04_curvilineargrid_basics_52_0.png)
+
+## 曲线网格定向平滑
+
+移动一个节点，使网格不再平滑，并绘制结果。
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid()
+mk.curvilinear_move_node(10.0, 20.0, 18.0, 12.0)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 55 的绘图输出](images/04_curvilineargrid_basics_55_0.png)
+
+执行定向平滑。
+
+```python
+mk.curvilinear_smoothing_directional(
+    10,  # 平滑迭代次数
+    10.0,
+    0.0,
+    10.0,
+    30.0,  # 用于定义平滑方向的网格线坐标
+    0.0,
+    0.0,
+    30.0,
+    30.0,
+)  # 要平滑的网格块角点
+```
+
+绘制定向平滑后的结果。
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 59 的绘图输出](images/04_curvilineargrid_basics_59_0.png)
+
+## 曲线网格线平移
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid(5, 5)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 61 的绘图输出](images/04_curvilineargrid_basics_61_0.png)
+
+初始化网格线平移操作，并设置要移动的网格线。
+
+```python
+mk.curvilinear_initialize_line_shift()
+mk.curvilinear_set_line_line_shift(0.0, 0.0, 0.0, 50.0)
+```
+
+设置用于分配网格线位移的网格块。
+
+```python
+mk.curvilinear_set_block_line_shift(0.0, 0.0, 20.0, 50.0)
+```
+
+将曲线网格左侧的所有节点向左移动。
+
+```python
+mk.curvilinear_move_node_line_shift(0.0, 0.0, -50.0, 0.0)
+mk.curvilinear_move_node_line_shift(0.0, 10.0, -50.0, 10.0)
+mk.curvilinear_move_node_line_shift(0.0, 20.0, -50.0, 20.0)
+mk.curvilinear_move_node_line_shift(0.0, 30.0, -50.0, 30.0)
+mk.curvilinear_move_node_line_shift(0.0, 40.0, -50.0, 40.0)
+mk.curvilinear_move_node_line_shift(0.0, 50.0, -50.0, 50.0)
+```
+
+执行网格线平移。前面指定网格块以外的节点不会移动。
+
+```python
+mk.curvilinear_line_shift()
+```
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 70 的绘图输出](images/04_curvilineargrid_basics_70_0.png)
+
+## 在曲线网格中插入网格面
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid(5, 5)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 72 的绘图输出](images/04_curvilineargrid_basics_72_0.png)
+
+插入两个网格面。
+
+```python
+mk.curvilinear_insert_face(-10.0, 5.0)
+mk.curvilinear_insert_face(-5.0, 10.0)
+```
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 75 的绘图输出](images/04_curvilineargrid_basics_75_0.png)
+
+## 删除曲线网格节点
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid(5, 5)
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 77 的绘图输出](images/04_curvilineargrid_basics_77_0.png)
+
+删除角点节点。
+
+```python
+mk.curvilinear_delete_node(0.0, 0.0)
+```
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 80 的绘图输出](images/04_curvilineargrid_basics_80_0.png)
+
+## 曲线网格线吸引与排斥
+
+将网格块内的节点向指定网格线吸引。
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid(5, 5)
+mk.curvilinear_line_attraction_repulsion(
+    1.0,  # 正值表示排斥网格线，此处为列宽的 1 倍
+    30.0,
+    0.0,
+    30.0,
+    50.0,  # 网格线坐标
+    10.0,
+    0.0,
+    50.0,
+    50.0,
+)  # 受影响的网格块
+```
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 84 的绘图输出](images/04_curvilineargrid_basics_84_0.png)
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid(5, 5)
+mk.curvilinear_line_attraction_repulsion(
+    -1.0,  # 负值表示吸引网格线，此处为列宽的 0.5 倍
+    30.0,
+    0.0,
+    30.0,
+    50.0,  # 网格线坐标
+    10.0,
+    0.0,
+    50.0,
+    50.0,
+)  # 受影响的网格块
+```
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 86 的绘图输出](images/04_curvilineargrid_basics_86_0.png)
+
+## 曲线网格线镜像扩展
+
+以两倍列宽对左侧网格线进行镜像扩展。
+
+```python
+mk = create_mk_instance_with_a_rectangular_curvilinear_grid(5, 5)
+# 镜像系数、要镜像的网格线数量以及目标网格线
+mk.curvilinear_line_mirror(2.0, 1, 0.0, 0.0, 0.0, 50.0)
+```
+
+```python
+curvilinear_grid = mk.curvilineargrid_get()
+fig, ax = plt.subplots()
+curvilinear_grid.plot_edges(ax)
+```
+
+![单元格 90 的绘图输出](images/04_curvilineargrid_basics_90_0.png)