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61e89b0 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | import numpy as np
from scipy.spatial import ConvexHull
from scipy.optimize import linprog
# ---------- Small helpers ----------
def orthonormal_basis_from_up(up):
up = np.asarray(up, dtype=float)
up = up / (np.linalg.norm(up) + 1e-12)
# pick any vector not parallel to up
t = np.array([1.0, 0.0, 0.0]) if abs(up[0]) < 0.9 else np.array([0.0, 1.0, 0.0])
u0 = t - (t @ up) * up
u0 /= (np.linalg.norm(u0) + 1e-12)
v0 = np.cross(up, u0)
v0 /= (np.linalg.norm(v0) + 1e-12)
return u0, v0, up
def project_to_plane(points, u0, v0):
U = np.stack([u0, v0], axis=1) # 3x2
return points @ U # (N,2), coordinates in (u0,v0)
def hull_halfspaces_2d(P2):
hull = ConvexHull(P2)
# equations: for 2D, each row [a,b,c] with a*x + b*y + c == 0 on edge, <= 0 inside
A = hull.equations[:, :2]
b = -hull.equations[:, 2]
return A, b, hull
def solve_rect_lp(A, b, R2, alpha):
"""
Maximize alpha^T h, subject to A c + |A R2| h <= b, h>=0.
Vars: [c_x, c_y, h_x, h_y]
"""
S = np.abs(A @ R2) # (m,2)
A_ub = np.hstack([A, S]) # (m,4)
b_ub = b.copy()
bounds = [(-np.inf, np.inf), (-np.inf, np.inf), (0, np.inf), (0, np.inf)]
c_vec = np.array([0.0, 0.0, -alpha[0], -alpha[1]])
res = linprog(c=c_vec, A_ub=A_ub, b_ub=b_ub, bounds=bounds, method="highs")
if not res.success:
return None
c2 = res.x[:2]
h2 = res.x[2:]
return c2, h2
def rect_vertices_3d(center3, hx, hy, u_dir, v_dir):
signs = np.array([[-1,-1],[ -1, 1],[ 1, 1],[ 1,-1]], float)
verts2 = signs * np.array([hx, hy])
verts3 = center3[None,:] + verts2[:,0:1]*u_dir[None,:] + verts2[:,1:2]*v_dir[None,:]
return verts3 # 4x3 (rectangle corners in plane, no vertical thickness)
# ---------- Main: max inscribed rectangle footprint + height ----------
def max_inscribed_rectangle_2d_box(
centers_np,
up=np.array([0.0, -1.0, 0.0]),
ground_level=0.0,
n_angles=181, # sample 0..90° inclusive (exploits symmetry); doubled internally
n_weights=16, # LPs per angle (sweep trade-offs)
seed=0
):
"""
Finds the maximum-area rectangle (any in-plane orientation) inside the 2D projection
of `centers_np` onto the plane orthogonal to `up`. Height is then set from ground_level
to the highest camera along +up.
Returns dict with:
center (3,), half_sizes (hx,hy,hz/2), sizes (3,),
R (3x3), quat_xyzw (4,), footprint_vertices (4,3), box_vertices (8,3), area, volume
"""
rng = np.random.default_rng(seed)
u0, v0, up = orthonormal_basis_from_up(up)
# 1) 2D projection and hull halfspaces in (u0, v0) coords
P2 = project_to_plane(centers_np, u0, v0) # (N,2)
A2, b2, hull2 = hull_halfspaces_2d(P2)
# 2) Search over angles theta in [0, pi/2) due to rectangle symmetry
thetas = np.linspace(0.0, 0.5*np.pi, n_angles)
best = {"area": -1.0}
# Simple Dirichlet weights over 2 dims => Beta; add a couple of axis-focused weights
weights = list(rng.dirichlet(np.ones(2), size=n_weights))
weights += [np.array([1.0, 0.0]), np.array([0.0, 1.0]), np.array([0.5, 0.5])]
for theta in thetas:
# local rectangle axes in (u0,v0) coordinates
ct, st = np.cos(theta), np.sin(theta)
R2 = np.array([[ct, -st],
[st, ct]], dtype=float) # maps local (x,y) to (u0,v0)
for alpha in weights:
sol = solve_rect_lp(A2, b2, R2, alpha)
if sol is None:
continue
c2, h2 = sol
area = float(4.0 * h2[0] * h2[1])
if area > best["area"]:
best = {"area": area, "theta": theta, "c2": c2, "h2": h2}
if best["area"] <= 0:
raise RuntimeError("Failed to inscribe a rectangle; check point configuration.")
# 3) Build 3D pose from best solution
theta = best["theta"]; ct, st = np.cos(theta), np.sin(theta)
u_dir = ct * u0 + st * v0 # rectangle local X in world
v_dir = -st * u0 + ct * v0 # rectangle local Y in world (right-handed with up)
c2 = best["c2"]; hx, hy = best["h2"]
center_plane = c2[0]*u0 + c2[1]*v0
# Height: ground -> highest camera
cam_h = centers_np @ up
H = float(np.max(cam_h) - ground_level)
H = max(H, 1e-9)
center3 = center_plane + (ground_level + 0.5*H) * up
# Rotation matrix with columns = local axes (X=u_dir, Y=v_dir, Z=up)
R_box = np.column_stack([u_dir, v_dir, up])
# Quaternion xyzw (from rotmat)
# Manual conversion (no SciPy quaternion dependency):
def rotmat_to_quat_xyzw(M):
t = np.trace(M)
if t > 0:
s = np.sqrt(t+1.0)*2
w = 0.25*s
x = (M[2,1]-M[1,2])/s
y = (M[0,2]-M[2,0])/s
z = (M[1,0]-M[0,1])/s
else:
i = np.argmax([M[0,0], M[1,1], M[2,2]])
if i == 0:
s = np.sqrt(1.0 + M[0,0] - M[1,1] - M[2,2]) * 2
w = (M[2,1] - M[1,2]) / s
x = 0.25 * s
y = (M[0,1] + M[1,0]) / s
z = (M[0,2] + M[2,0]) / s
elif i == 1:
s = np.sqrt(1.0 + M[1,1] - M[0,0] - M[2,2]) * 2
w = (M[0,2] - M[2,0]) / s
x = (M[0,1] + M[1,0]) / s
y = 0.25 * s
z = (M[1,2] + M[2,1]) / s
else:
s = np.sqrt(1.0 + M[2,2] - M[0,0] - M[1,1]) * 2
w = (M[1,0] - M[0,1]) / s
x = (M[0,2] + M[2,0]) / s
y = (M[1,2] + M[2,1]) / s
z = 0.25 * s
return np.array([x, y, z, w], dtype=float)
quat_xyzw = rotmat_to_quat_xyzw(R_box)
# Vertices (footprint & full 3D box)
footprint4 = rect_vertices_3d(center_plane, hx, hy, u_dir, v_dir) # 4x3 at ground plane height=0 (in plane coords)
# 8 box corners:
rect4_top = footprint4 + H * up
verts8 = np.vstack([footprint4, rect4_top])
sizes3 = np.array([2*hx, 2*hy, H], dtype=float)
return {
"center": center3,
"half_sizes": np.array([hx, hy, 0.5*H], dtype=float),
"sizes": sizes3,
"R": R_box,
"quat_xyzw": quat_xyzw,
"footprint_vertices": footprint4, # 4x3 (bottom rectangle)
"box_vertices": verts8, # 8x3
"area": float(4*hx*hy),
"volume": float((2*hx)*(2*hy)*H),
"up": up,
"u_dir": u_dir,
"v_dir": v_dir,
}
# -------- Example usage --------
# result = max_inscribed_rectangle_2d_box(
# centers_np,
# up=np.array([0, 1, 0]), # your "up"
# ground_level=0.0, # if your ground plane is y=0 (for example)
# n_angles=181,
# n_weights=24,
# seed=42
# )
# c = result["center"]; sizes = result["sizes"]; R = result["R"]; q = result["quat_xyzw"]
# print("center:", c, "sizes (W,D,H):", sizes)
def point_in_convex_hull_2d(points_xy, query_xy, tol=1e-12):
"""
points_xy: (N,2) cloud
query_xy: (...,2) points to test
Returns: boolean array with shape query_xy.shape[:-1]
"""
hull = ConvexHull(points_xy)
# hull.equations: rows [a, b, c] with a*x + b*y + c == 0 on edge, <= 0 inside
A = hull.equations[:, :2]
c = hull.equations[:, 2]
q = np.atleast_2d(query_xy) # (M,2)
vals = (A @ q.T) + c[:, None] # (num_edges, M)
inside = np.all(vals <= tol, axis=0) # inside if all halfspaces satisfied
return inside.reshape(query_xy.shape[:-1]) |