/// <summary>
/// Returns the rotation of the supplied counter clockwise enumerated
/// convex polygon that results in the minimum area enclosing box.
/// If multiple rotations are within epsilon in their area, the one
/// that is closest to an axis-aligned rotation (0, 90, 180, 270) is
/// returned. O(n).
/// </summary>
public static M22d ComputeMinAreaEnclosingBoxRotation(
this Polygon2d polygon, double epsilon = 1e-6)
{
polygon = polygon.WithoutMultiplePoints(epsilon);
var pc = polygon.PointCount;
if (pc < 2)
{
return(M22d.Identity);
}
var ea = polygon.GetEdgeArray();
ea.Apply(v => v.Normalized);
int i0 = 0, i1 = 0;
int i2 = 0, i3 = 0;
var min = polygon[0]; var max = polygon[0];
for (int pi = 1; pi < pc; pi++)
{
var p = polygon[pi];
if (p.Y < min.Y)
{
i0 = pi; min.Y = p.Y;
}
else if (p.Y > max.Y)
{
i2 = pi; max.Y = p.Y;
}
if (p.X > max.X)
{
i1 = pi; max.X = p.X;
}
else if (p.X < min.X)
{
i3 = pi; min.X = p.X;
}
}
V2d p0 = polygon[i0], e0 = ea[i0], p1 = polygon[i1], e1 = ea[i1];
V2d p2 = polygon[i2], e2 = ea[i2], p3 = polygon[i3], e3 = ea[i3];
int end0 = (i0 + 1) % pc, end1 = (i1 + 1) % pc;
int end2 = (i2 + 1) % pc, end3 = (i3 + 1) % pc;
var dir = V2d.XAxis;
var best = dir;
var bestArea = double.MaxValue;
var bestValue = double.MaxValue;
while (true)
{
var s0 = Fun.FastAtan2(e0.Dot90(dir), e0.Dot(dir));
var s1 = Fun.FastAtan2(e1.Dot180(dir), e1.Dot90(dir));
var s2 = Fun.FastAtan2(e2.Dot270(dir), e2.Dot180(dir));
var s3 = Fun.FastAtan2(e3.Dot(dir), e3.Dot270(dir));
int si, si01, si23; double s01, s23;
if (s0 < s1)
{
s01 = s0; si01 = 0;
}
else
{
s01 = s1; si01 = 1;
}
if (s2 < s3)
{
s23 = s2; si23 = 2;
}
else
{
s23 = s3; si23 = 3;
}
if (s01 < s23)
{
si = si01;
}
else
{
si = si23;
}
if (si == 0)
{
dir = ea[i0];
}
else if (si == 1)
{
dir = ea[i1].Rot270;
}
else if (si == 2)
{
dir = ea[i2].Rot180;
}
else
{
dir = ea[i3].Rot90;
}
double sx = (p2 - p0).Dot90(dir), sy = (p1 - p3).Dot(dir);
double area = sx * sy;
double value = Fun.Min(Fun.Abs(dir.X), Fun.Abs(dir.Y));
if (area < bestArea - epsilon ||
(area < bestArea + epsilon && value < bestValue))
{
bestArea = area; bestValue = value; best = dir;
}
if (si == 0)
{
if (++i0 >= pc)
{
i0 -= pc;
}
if (i0 == end1)
{
break;
}
p0 = polygon[i0]; e0 = ea[i0];
}
else if (si == 1)
{
if (++i1 >= pc)
{
i1 -= pc;
}
if (i1 == end2)
{
break;
}
p1 = polygon[i1]; e1 = ea[i1];
}
else if (si == 2)
{
if (++i2 >= pc)
{
i2 -= pc;
}
if (i2 == end3)
{
break;
}
p2 = polygon[i2]; e2 = ea[i2];
}
else
{
if (++i3 >= pc)
{
i3 -= pc;
}
if (i3 == end0)
{
break;
}
p3 = polygon[i3]; e3 = ea[i3];
}
}
return(new M22d(best.X, best.Y, -best.Y, best.X));
}