将多边形旋转到最小高度

Question

我需要旋转一个凹多边形来最小化它的高度。我正在考虑找到一条该多边形的最小直径的线，然后将其旋转以使该线与 Y 轴平行。

我的问题是如何找到这样的线？或者，是否有任何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;

见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情况。要解决 concave 的情况，您只需通过执行以下操作将输入从 conve 转换为 conve：

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你的凹形转换为凸形：（这不是要使用的，只是非常糟糕的伪代码让你了解凸包是如何计算的）

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，您只需去除凹入部分:)