Question

我需要旋转凹多边形以最小化其高度。我正在考虑找到一条直线，该直线是该多边形的最小直径，然后旋转它以使直线与Y轴平行。

我的问题是如何找到这样一条线？或者，是否有其他算法可以旋转多边形以使其高度最小化？

提前致谢。

Answer 1

 ARRAY points := {P1, P2, ..., PN};

 points.delete(middle vertices of any collinear sequence of three points);

 REAL p_a := index of vertex with minimum y-coordinate;
 REAL p_b := index of vertex with maximum y-coordinate;

 REAL rotated_angle := 0;
 REAL min_width := INFINITY;

 VECTOR caliper_a(1,0);    // Caliper A points along the positive x-axis
 VECTOR caliper_b(-1,0);   // Caliper B points along the negative x-axis

 WHILE rotated_angle < PI

   // Determine the angle between each caliper and the next adjacent edge in the polygon
   VECTOR edge_a(points[p_a + 1].x - points[p_a].x, points[p_a + 1].y - points[p_a].y);
   VECTOR edge_b(points[p_b + 1].x - points[p_b].x, points[p_b + 1].y - points[p_b].y);
   REAL angle_a := angle(edge_a, caliper_a);
   REAL angle_b := angle(edge_b, caliper_b);
   REAL width := 0;

   // Rotate the calipers by the smaller of these angles
   caliper_a.rotate(min(angle_a, angle_b));
   caliper_b.rotate(min(angle_a, angle_b));

   IF angle_a < angle_b
     p_a++;  // This index should wrap around to the beginning of the array once it hits the end
     width = caliper_a.distance(points[p_b]);
   ELSE
     p_b++;  // This index should wrap around to the beginning of the array once it hits the end
     width = caliper_b.distance(points[p_a]);
   END IF

   rotated_angle = rotated_angle + min(angle_a, angle_b);

   IF (width < min_width)
     min_width = width;

   END IF
 END WHILE

 RETURN min_width;

请参阅http://en.wikipedia.org/wiki/Rotating_calipers

另请查看：http://www.mathworks.com/matlabcentral/fileexchange/7844-geom2d/content/geom2d/polygons2d/minimumCaliperDiameter.m

注意：这两个问题都解决了凸的问题。要解决凹面的情况，您只需通过执行以下操作将输入从凸起转换为凸面：

1. Compute convex hull of your concave polygon.
2. Run algorithm above on convex polygon.
3. Once minimum height found, rotate.
4. Transform back to concave polygon.

凸包在Python中很常见。您可以使用：http://docs.scipy.org/doc/scipy-dev/reference/generated/scipy.spatial.ConvexHull.html

如果你不习惯使用这个库，你可以使用这个算法将凹形转换为凸形:(这并不意味着使用，只是非常糟糕的伪代码，你可以理解如何计算凸包）

Traverse vertices of concave polygon in clockwise order
_prev = starting vertex of traversal
_middle = second vertex in traversal
FOR _next vertex in traversal:
  IF _prev -> _middle -> _next make right turn (i.e. concave part)
     Connect _prev and _next and set _middle = _next
  ELSE
     Shift _prev to _middle and _middle to _next.

换句话说，你只需删除凹陷部分：）

将多边形旋转到最小高度

1 个答案: