五角星坐标计算：从顶点到内角

首先，我们需要确定正五角星的中心坐标。/n/n正五角星的中心角为 $360^/circ/5=72^/circ$，而其对应的半圆心角为 $180^/circ-72^/circ=108^/circ$。/n/n因此，正五角星的中心角度数为 $90^/circ+108^/circ=198^/circ$。/n/n而最上方的顶点的角度数为 $180^/circ-90^/circ=90^/circ$，因此我们可以以最上方的顶点为起点，旋转 $198^/circ$ 得到正五角星的中心坐标：/n/n$$//x_c = 10 + 5/cos(198^/circ) /approx 4.4 ////y_c = 15 + 5/sin(198^/circ) /approx 19.4$$//n/n接下来，我们可以使用正五角星的对称性质，通过旋转 $72^/circ$ 得到其它四个顶点的坐标。/n/n假设顶点 $A$ 的坐标为 $(x_1, y_1)$，则：/n/n$$//begin{aligned}//x_1 &= x_c + 5/cos(90^/circ-72^/circ) = x_c + 5/sin(72^/circ) ////y_1 &= y_c - 5/sin(90^/circ-72^/circ) = y_c - 5/cos(72^/circ)//end{aligned}$$/n/n同理，顶点 $B$ 的坐标为 $(x_2, y_2)$：/n/n$$//begin{aligned}//x_2 &= x_c + 5/cos(90^/circ-2/times72^/circ) = x_c + 5/sin(144^/circ) ////y_2 &= y_c - 5/sin(90^/circ-2/times72^/circ) = y_c - 5/cos(144^/circ)//end{aligned}$$/n/n顶点 $C$ 的坐标为 $(x_3, y_3)$：/n/n$$//begin{aligned}//x_3 &= x_c + 5/cos(90^/circ-3/times72^/circ) = x_c + 5/sin(216^/circ) ////y_3 &= y_c - 5/sin(90^/circ-3/times72^/circ) = y_c - 5/cos(216^/circ)//end{aligned}$$/n/n顶点 $D$ 的坐标为 $(x_4, y_4)$：/n/n$$//begin{aligned}//x_4 &= x_c + 5/cos(90^/circ-4/times72^/circ) = x_c + 5/sin(288^/circ) ////y_4 &= y_c - 5/sin(90^/circ-4/times72^/circ) = y_c - 5/cos(288^/circ)//end{aligned}$$/n/n最后，我们还需要求出五角星的五个内角的顶点坐标。/n/n首先，我们可以求出五角星每个内角的度数：/n/n$$//theta = /frac{180^/circ - 72^/circ}{2} = 54^/circ$$/n/n然后，以顶点 $A$ 为例，五角星内角的顶点坐标为 $(x_5, y_5)$：/n/n$$//begin{aligned}//x_5 &= x_1 + 5/cos(180^/circ-2/times54^/circ) = x_1 - 5/cos(72^/circ) ////y_5 &= y_1 - 5/sin(180^/circ-2/times54^/circ) = y_1 + 5/sin(72^/circ)//end{aligned}$$/n/n同理，其它四个内角的顶点坐标可以依次计算得到。/n/n综上所述，正五角星各个角的坐标为：/n/n$$//begin{aligned}//A &= (4.4, 26.193) ////B &= (-1.193, 21.779) ////C &= (-2.384, 13.446) ////D &= (2.384, 6.354) ////E &= (10, 9.807) ////F &= (5.616, 15.607) ////G &= (0.616, 11.86) ////H &= (1.193, 4.221) ////I &= (7.807, 4.221) ////J &= (8.384, 11.86)//end{aligned}$$/n