两个多边形相交，其闵可夫斯基差必过原点
python
import tkinter as tk
import math
import numpy as np
def idx2char(i):
"""0→A,1→B,2→C... 返回大写字母"""
return chr(ord('A') + i)
class PolyDrawApp:
def __init__(self, root):
self.root = root
self.root.title("闵可夫斯基差｜Y轴向上为正(右上正)")
self.root.geometry("1000x750")
self.is_drawing = False
self.current_points = []
self.close_threshold = 8
# polygons: {"points_canvas":[[x,y],...], "item_id":canvas_id, "vertex_text_ids":[]}
self.polygons = []
self.max_poly_count = 2
self.drag_poly_idx = None
self.drag_start_mx = 0
self.drag_start_my = 0
self.minkowski_item = None
self.minkowski_text_ids = [] # 点标记 A1A2
self.minkowski_formula_ids = [] # 计算公式文本
frame_top = tk.Frame(root)
frame_top.pack(fill=tk.X, padx=5, pady=5)
self.btn_create = tk.Button(frame_top, text="创建多边形", command=self.start_create)
self.btn_create.pack(side=tk.LEFT)
self.btn_clear = tk.Button(frame_top, text="清空全部", command=self.clear_all)
self.btn_clear.pack(side=tk.LEFT, padx=10)
self.canvas = tk.Canvas(root, bg="white")
self.canvas.pack(fill=tk.BOTH, expand=True, padx=5, pady=5)
self.canvas.bind("<Button-1>", self.on_canvas_click)
self.canvas.bind("<B1-Motion>", self.on_drag_move)
self.canvas.bind("<ButtonRelease-1>", self.on_drag_release)
self.root.update()
self.redraw_axis()
def redraw_axis(self):
self.canvas.delete("axis")
w = self.canvas.winfo_width()
h = self.canvas.winfo_height()
self.ox = w / 2 # 画布原点像素x
self.oy = h / 2 # 画布原点像素y
# X轴
self.canvas.create_line(0, self.oy, w, self.oy, fill="#aaaaaa", dash=(4,2), tags="axis")
# Y轴
self.canvas.create_line(self.ox, 0, self.ox, h, fill="#aaaaaa", dash=(4,2), tags="axis")
# 原点O标记
self.canvas.create_oval(self.ox-5,self.oy-5,self.ox+5,self.oy+5,fill="black",tags="axis")
self.canvas.create_text(self.ox+12,self.oy+12,text="O",font=("Consolas",11),tags="axis")
# X Y箭头标签
self.canvas.create_text(w-15, self.oy+12, text="+X", font=("Consolas",10), tags="axis")
self.canvas.create_text(self.ox+12,15, text="+Y", font=("Consolas",10), tags="axis")
def canvas_to_math(self,cx,cy):
"""画布像素坐标 →数学坐标系（Y向上为正）"""
mx = cx - self.ox
my = self.oy - cy
return mx, my
def math_to_canvas(self,mx,my):
"""数学坐标 →画布像素"""
cx = mx + self.ox
cy = self.oy - my
return cx, cy
def start_create(self):
if len(self.polygons) >= self.max_poly_count:
print("已经存在2个多边形，不能继续创建！")
return
self.is_drawing = True
self.current_points.clear()
print("开始绘制多边形，点击顶点，点回第一个顶点闭合")
self.canvas.delete("temp")
def on_canvas_click(self, event):
x, y = event.x, event.y
hit_idx = self.pick_polygon(x, y)
if hit_idx is not None and not self.is_drawing:
self.drag_poly_idx = hit_idx
self.drag_start_mx = event.x
self.drag_start_my = event.y
return
if not self.is_drawing:
return
if len(self.current_points) >= 1:
sx, sy = self.current_points[0]
dist = math.hypot(x - sx, y - sy)
if dist < self.close_threshold and len(self.current_points) >=3:
self.finish_polygon()
return
self.current_points.append([x, y])
self.canvas.create_oval(x-4, y-4, x+4, y+4, fill="red", outline="red", tags="temp")
if len(self.current_points) >=2:
x0,y0 = self.current_points[-2]
self.canvas.create_line(x0,y0, x,y, fill="blue", width=2, tags="temp")
def finish_polygon(self):
pts_canvas = self.current_points.copy()
item_id = self.canvas.create_polygon(pts_canvas, outline="blue", fill="#cce5ff", width=2)
poly_index = len(self.polygons)+1 # 1 /2
text_ids = []
for i,(px,py) in enumerate(pts_canvas):
char = idx2char(i)
label = f"{char}{poly_index}"
tid = self.canvas.create_text(px+8, py-8, text=label, font=("Consolas",10), fill="#0000bb")
text_ids.append(tid)
self.polygons.append({
"points_canvas": pts_canvas,
"item_id": item_id,
"vertex_text_ids": text_ids
})
self.is_drawing = False
self.current_points.clear()
self.canvas.delete("temp")
print(f"已完成多边形{poly_index}，当前数量:{len(self.polygons)}")
if len(self.polygons)==2:
self.draw_minkowski_diff()
def pick_polygon(self, mx, my):
for idx,poly in enumerate(self.polygons):
pts = poly["points_canvas"]
if self.point_in_polygon(mx,my,pts):
return idx
return None
@staticmethod
def point_in_polygon(x,y,polygon):
inside=False
n=len(polygon)
for i in range(n):
j=(i+1)%n
xi,yi=polygon[i]
xj,yj=polygon[j]
intersect=((yi>y)!=(yj>y))
if intersect:
x_intersect= ((y-yi)*(xj-xi))/(yj-yi)+xi
if x<x_intersect:
inside=not inside
return inside
def on_drag_move(self, event):
if self.drag_poly_idx is None:
return
dx = event.x - self.drag_start_mx
dy = event.y - self.drag_start_my
poly = self.polygons[self.drag_poly_idx]
self.canvas.move(poly["item_id"], dx, dy)
for tid in poly["vertex_text_ids"]:
self.canvas.move(tid, dx, dy)
# 更新画布坐标
for p in poly["points_canvas"]:
p[0] += dx
p[1] += dy
self.drag_start_mx = event.x
self.drag_start_my = event.y
if len(self.polygons)==2:
self.draw_minkowski_diff()
def on_drag_release(self, event):
self.drag_poly_idx = None
self.drag_start_mx = 0
self.drag_start_my = 0
def get_math_poly(self,poly_data):
"""把多边形canvas像素点列表转为数学坐标系点列表(Y向上为正)"""
pts_canvas = poly_data["points_canvas"]
math_pts = []
for cx,cy in pts_canvas:
mx,my = self.canvas_to_math(cx,cy)
math_pts.append([mx,my])
return math_pts
def minkowski_difference(self, A_math, B_math):
"""A⊖B = a−b，返回 (ia,ib,a_pt,b_pt,diff_pt)，全部数学坐标"""
diff_points = []
for ia,a in enumerate(A_math):
for ib,b in enumerate(B_math):
diff = [a[0]-b[0], a[1]-b[1]]
diff_points.append((ia, ib, a, b, diff))
return diff_points
def draw_minkowski_diff(self):
if self.minkowski_item is not None:
self.canvas.delete(self.minkowski_item)
for tid in self.minkowski_text_ids:
self.canvas.delete(tid)
for tid in self.minkowski_formula_ids:
self.canvas.delete(tid)
self.minkowski_text_ids.clear()
self.minkowski_formula_ids.clear()
A_math = self.get_math_poly(self.polygons[0])
B_math = self.get_math_poly(self.polygons[1])
diff_all = self.minkowski_difference(A_math,B_math)
diff_pts_only = [d[4] for d in diff_all]
if len(diff_pts_only)<3:
return
hull_pts = self.convex_hull(diff_pts_only)
# 凸包数学坐标转画布
draw_hull = [self.math_to_canvas(p[0],p[1]) for p in hull_pts]
self.minkowski_item = self.canvas.create_polygon(draw_hull, outline="#c82423", fill="#ffcccc", width=2)
for ia, ib, a_pt, b_pt, diff_pt in diff_all:
# diff_pt 数学坐标
px_canvas, py_canvas = self.math_to_canvas(diff_pt[0], diff_pt[1])
charA = idx2char(ia)
charB = idx2char(ib)
label_name = f"{charA}1{charB}2"
ax, ay = round(a_pt[0],1), round(a_pt[1],1)
bx, by = round(b_pt[0],1), round(b_pt[1],1)
dx, dy = round(diff_pt[0],1), round(diff_pt[1],1)
formula_text = f"{label_name}={charA}1-{charB}2=({ax},{ay})-({bx},{by})=({dx},{dy})"
tid_name = self.canvas.create_text(px_canvas+6, py_canvas-8, text=label_name, font=("Consolas",9), fill="#bb0000")
tid_formula = self.canvas.create_text(px_canvas+6, py_canvas+8, text=formula_text, font=("Consolas",7), fill="#600000")
self.minkowski_text_ids.append(tid_name)
self.minkowski_formula_ids.append(tid_formula)
@staticmethod
def convex_hull(points):
pts = sorted(points)
lower = []
for p in pts:
while len(lower)>=2:
a = np.array(lower[-2])
b = np.array(lower[-1])
c = np.array(p)
cross = np.cross(b-a, c-a)
if cross <=0:
lower.pop()
else:
break
lower.append(p)
upper = []
for p in reversed(pts):
while len(upper)>=2:
a = np.array(upper[-2])
b = np.array(upper[-1])
c = np.array(p)
cross = np.cross(b-a, c-a)
if cross <=0:
upper.pop()
else:
break
upper.append(p)
full = lower[:-1] + upper[:-1]
if not full:
return pts[:3]
return full
def clear_all(self):
self.canvas.delete(tk.ALL)
self.polygons.clear()
self.is_drawing = False
self.current_points.clear()
self.minkowski_item = None
self.minkowski_text_ids.clear()
self.minkowski_formula_ids.clear()
self.drag_poly_idx = None
self.redraw_axis()
if __name__ == "__main__":
win = tk.Tk()
app = PolyDrawApp(win)
win.mainloop()