凸多边形三角剖分伪代码及代码实现

凸多边形三角剖分，可以使用分治法来实现。伪代码如下：

function TriangulateConvexPolygon(polygon):
    if polygon.size() == 3:
        return [polygon];
    else:
        // 找到多边形的中心点
        center = findCenter(polygon);
        // 找到离中心点最远的边
        farthestEdge = findFarthestEdge(polygon, center);
        // 将多边形沿着这条边分成两个子多边形
        subPolygons = splitPolygon(polygon, farthestEdge);
        // 递归地对子多边形进行三角剖分
        leftTriangles = TriangulateConvexPolygon(subPolygons[0]);
        rightTriangles = TriangulateConvexPolygon(subPolygons[1]);
        // 合并子多边形的三角剖分结果
        return mergeTriangles(leftTriangles, rightTriangles);

其中，`findCenter` 函数可以找到多边形的中心点；`findFarthestEdge` 函数可以找到离中心点最远的边；`splitPolygon` 函数可以将多边形沿着指定的边分成两个子多边形；`mergeTriangles` 函数可以将两个子多边形的三角剖分结果合并起来。

以下是一个 Python 实现的示例代码：

from typing import List

class Point:
    def __init__(self, x: float, y: float):
        self.x = x
        self.y = y

class Edge:
    def __init__(self, p1: Point, p2: Point):
        self.p1 = p1
        self.p2 = p2

class Triangle:
    def __init__(self, p1: Point, p2: Point, p3: Point):
        self.p1 = p1
        self.p2 = p2
        self.p3 = p3

def TriangulateConvexPolygon(polygon: List[Point]) -> List[Triangle]:
    if len(polygon) == 3:
        return [Triangle(polygon[0], polygon[1], polygon[2])]
    else:
        center = findCenter(polygon)
        farthestEdge = findFarthestEdge(polygon, center)
        subPolygons = splitPolygon(polygon, farthestEdge)
        leftTriangles = TriangulateConvexPolygon(subPolygons[0])
        rightTriangles = TriangulateConvexPolygon(subPolygons[1])
        return mergeTriangles(leftTriangles, rightTriangles)

def findCenter(polygon: List[Point]) -> Point:
    n = len(polygon)
    cx = sum(p.x for p in polygon) / n
    cy = sum(p.y for p in polygon) / n
    return Point(cx, cy)

def findFarthestEdge(polygon: List[Point], center: Point) -> Edge:
    farthestEdge = None
    maxDist = -1
    for i in range(len(polygon)):
        j = (i + 1) % len(polygon)
        edge = Edge(polygon[i], polygon[j])
        dist = distance(center, edge)
        if dist > maxDist:
            farthestEdge = edge
            maxDist = dist
    return farthestEdge

def splitPolygon(polygon: List[Point], edge: Edge) -> List[List[Point]]:
    i = polygon.index(edge.p1)
    j = polygon.index(edge.p2)
    if i < j:
        subPolygon1 = polygon[i:j+1]
        subPolygon2 = polygon[j:] + polygon[:i+1]
    else:
        subPolygon1 = polygon[j:i+1]
        subPolygon2 = polygon[i:] + polygon[:j+1]
    return [subPolygon1, subPolygon2]

def distance(point: Point, edge: Edge) -> float:
    x1, y1 = edge.p1.x, edge.p1.y
    x2, y2 = edge.p2.x, edge.p2.y
    x0, y0 = point.x, point.y
    dx = x2 - x1
    dy = y2 - y1
    if dx == 0 and dy == 0:
        return ((x1 - x0) ** 2 + (y1 - y0) ** 2) ** 0.5
    t = ((x0 - x1) * dx + (y0 - y1) * dy) / (dx * dx + dy * dy)
    if t < 0:
        return ((x1 - x0) ** 2 + (y1 - y0) ** 2) ** 0.5
    elif t > 1:
        return ((x2 - x0) ** 2 + (y2 - y0) ** 2) ** 0.5
    else:
        xp = x1 + t * dx
        yp = y1 + t * dy
        return ((xp - x0) ** 2 + (yp - y0) ** 2) ** 0.5

def mergeTriangles(triangles1: List[Triangle], triangles2: List[Triangle]) -> List[Triangle]:
    mergedTriangles = []
    for t in triangles1 + triangles2:
        if isInPolygon(t.p1, triangles2) and isInPolygon(t.p2, triangles2) and isInPolygon(t.p3, triangles2):
            continue
        else:
            mergedTriangles.append(t)
    return mergedTriangles

def isInPolygon(point: Point, triangles: List[Triangle]) -> bool:
    for t in triangles:
        if isInTriangle(point, t):
            return True
    return False

def isInTriangle(point: Point, triangle: Triangle) -> bool:
    p1, p2, p3 = triangle.p1, triangle.p2, triangle.p3
    d1 = sign(point, p1, p2)
    d2 = sign(point, p2, p3)
    d3 = sign(point, p3, p1)
    has_neg = (d1 < 0) or (d2 < 0) or (d3 < 0)
    has_pos = (d1 > 0) or (d2 > 0) or (d3 > 0)
    return not (has_neg and has_pos)

def sign(p1: Point, p2: Point, p3: Point) -> float:
    return (p1.x - p3.x) * (p2.y - p3.y) - (p2.x - p3.x) * (p1.y - p3.y)

# 示例用法
polygon = [Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1)]
triangles = TriangulateConvexPolygon(polygon)
for t in triangles:
    print(t.p1.x, t.p1.y, t.p2.x, t.p2.y, t.p3.x, t.p3.y)

注意，以上实现假设输入的多边形是凸多边形。如果输入的是非凸多边形，则需要先对其进行分解成若干个凸多边形，再分别进行三角剖分。