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from __future__ import annotations
import math
from collections import defaultdict
import numpy as np
from scipy.optimize import least_squares
from statement_to_tikz.ir import (
AngleMeasure,
CircleDef,
Collinear,
EqualAngle,
EqualLength,
GeometryIR,
Intersection,
Length,
Midpoint,
OnCircle,
OnLine,
Parallel,
Perpendicular,
SolveMode,
SolvedScene,
)
DEFAULT_TOL = 1e-4
def _get(coords: dict[str, tuple[float, float]], name: str) -> np.ndarray:
return np.asarray(coords[name], dtype=float)
def _vec(a: np.ndarray, b: np.ndarray) -> np.ndarray:
return b - a
def _cross2(u: np.ndarray, v: np.ndarray) -> float:
return float(u[0] * v[1] - u[1] * v[0])
def _dot(u: np.ndarray, v: np.ndarray) -> float:
return float(np.dot(u, v))
def _norm(u: np.ndarray) -> float:
return float(np.linalg.norm(u))
def _angle_cos(a: np.ndarray, v: np.ndarray, b: np.ndarray) -> float:
u = a - v
w = b - v
nu, nw = _norm(u), _norm(w)
if nu < 1e-12 or nw < 1e-12:
return 1.0
return _dot(u, w) / (nu * nw)
def _circle_geometry(
ir: GeometryIR, coords: dict[str, tuple[float, float]], circle_id: str
) -> tuple[np.ndarray, float] | None:
circ = next((c for c in ir.circles if c.id == circle_id), None)
if circ is None:
return None
return _circle_center_radius_from_def(circ, coords)
def _circle_center_radius_from_def(
circ: CircleDef, coords: dict[str, tuple[float, float]]
) -> tuple[np.ndarray, float] | None:
if circ.center is not None and circ.center in coords:
c = _get(coords, circ.center)
if circ.radius is not None:
return c, float(circ.radius)
if circ.through:
for p in circ.through:
if p in coords:
return c, _norm(_get(coords, p) - c)
return c, 1.0
if circ.through and len(circ.through) >= 3:
pts = [p for p in circ.through[:3] if p in coords]
if len(pts) < 3:
return None
a, b, c = (_get(coords, p) for p in pts)
ax, ay = a
bx, by = b
cx, cy = c
d = 2 * (ax * (by - cy) + bx * (cy - ay) + cx * (ay - by))
if abs(d) < 1e-12:
return None
ux = (
(ax**2 + ay**2) * (by - cy)
+ (bx**2 + by**2) * (cy - ay)
+ (cx**2 + cy**2) * (ay - by)
) / d
uy = (
(ax**2 + ay**2) * (cx - bx)
+ (bx**2 + by**2) * (ax - cx)
+ (cx**2 + cy**2) * (bx - ax)
) / d
center = np.array([ux, uy])
return center, _norm(a - center)
return None
def constraint_residuals(
ir: GeometryIR, coords: dict[str, tuple[float, float]]
) -> list[float]:
"""Return list of scalar residuals for each constraint (may be multi-valued flattened)."""
r: list[float] = []
for c in ir.constraints:
if isinstance(c, EqualLength):
la = _norm(_get(coords, c.a2) - _get(coords, c.a1))
lb = _norm(_get(coords, c.b2) - _get(coords, c.b1))
r.append(la - lb)
elif isinstance(c, Length):
la = _norm(_get(coords, c.b) - _get(coords, c.a))
r.append(la - c.value)
elif isinstance(c, EqualAngle):
cos1 = _angle_cos(
_get(coords, c.a1), _get(coords, c.v1), _get(coords, c.b1)
)
cos2 = _angle_cos(
_get(coords, c.a2), _get(coords, c.v2), _get(coords, c.b2)
)
r.append(cos1 - cos2)
elif isinstance(c, AngleMeasure):
cos_t = math.cos(math.radians(c.degrees))
cos_m = _angle_cos(
_get(coords, c.a), _get(coords, c.vertex), _get(coords, c.b)
)
r.append(cos_m - cos_t)
elif isinstance(c, Perpendicular):
u = _vec(_get(coords, c.a1), _get(coords, c.a2))
v = _vec(_get(coords, c.b1), _get(coords, c.b2))
r.append(_dot(u, v))
elif isinstance(c, Parallel):
u = _vec(_get(coords, c.a1), _get(coords, c.a2))
v = _vec(_get(coords, c.b1), _get(coords, c.b2))
r.append(_cross2(u, v))
elif isinstance(c, OnLine):
a, b, p = _get(coords, c.a), _get(coords, c.b), _get(coords, c.point)
r.append(_cross2(b - a, p - a))
elif isinstance(c, OnCircle):
geom = _circle_geometry(ir, coords, c.circle)
if geom is None:
r.append(0.0)
else:
center, rad = geom
r.append(_norm(_get(coords, c.point) - center) - rad)
elif isinstance(c, Midpoint):
m = _get(coords, c.point)
mid = 0.5 * (_get(coords, c.a) + _get(coords, c.b))
r.extend([float(m[0] - mid[0]), float(m[1] - mid[1])])
elif isinstance(c, Collinear):
pts = [_get(coords, p) for p in c.points]
base = pts[1] - pts[0]
for p in pts[2:]:
r.append(_cross2(base, p - pts[0]))
elif isinstance(c, Intersection):
a, b = _get(coords, c.a), _get(coords, c.b)
cc, d = _get(coords, c.c), _get(coords, c.d)
p = _get(coords, c.point)
r.append(_cross2(b - a, p - a))
r.append(_cross2(d - cc, p - cc))
else:
raise TypeError(f"unknown constraint {type(c)}")
for circ in ir.circles:
if circ.center and circ.through and circ.center in coords:
center = _get(coords, circ.center)
if circ.radius is not None:
rad = circ.radius
else:
refs = [p for p in circ.through if p in coords]
if not refs:
continue
rad = _norm(_get(coords, refs[0]) - center)
for p in circ.through:
if p in coords:
r.append(_norm(_get(coords, p) - center) - rad)
elif circ.through and len(circ.through) >= 3:
pts = [p for p in circ.through if p in coords]
if len(pts) >= 3:
geom = _circle_center_radius_from_def(circ, coords)
if geom is not None:
center, rad = geom
for p in pts:
r.append(_norm(_get(coords, p) - center) - rad)
return r
def evaluate_residuals(
ir: GeometryIR, coords: dict[str, tuple[float, float]]
) -> list[float]:
return [abs(x) for x in constraint_residuals(ir, coords)]
def _initial_coords(ir: GeometryIR) -> dict[str, tuple[float, float]]:
coords: dict[str, tuple[float, float]] = {}
n = len(ir.points)
for i, p in enumerate(ir.points):
if p.hint is not None:
coords[p.name] = (float(p.hint[0]), float(p.hint[1]))
else:
ang = 2 * math.pi * i / max(n, 1) - math.pi / 2
coords[p.name] = (math.cos(ang), math.sin(ang))
return coords
def _gauge_point_names(ir: GeometryIR) -> tuple[str | None, str | None]:
"""Choose two points to fix: prefer a length-constrained segment as base."""
names = ir.point_names()
if not names:
return None, None
for c in ir.constraints:
if isinstance(c, Length):
return c.a, c.b
for c in ir.constraints:
if isinstance(c, Midpoint):
return c.a, c.b
if ir.segments:
return ir.segments[0].a, ir.segments[0].b
p0 = names[0]
p1 = names[1] if len(names) > 1 else None
return p0, p1
def _triangle_area_residuals(
ir: GeometryIR, coords: dict[str, tuple[float, float]]
) -> list[float]:
"""Soft non-degeneracy: penalize near-zero area for 3-cycles of segments."""
adj: dict[str, set[str]] = defaultdict(set)
for seg in ir.segments:
adj[seg.a].add(seg.b)
adj[seg.b].add(seg.a)
residuals: list[float] = []
seen: set[tuple[str, str, str]] = set()
for a, nbrs in adj.items():
for b in nbrs:
for c in nbrs:
if b >= c:
continue
if c not in adj[b]:
continue
key = tuple(sorted((a, b, c)))
if key in seen:
continue
seen.add(key)
if not all(p in coords for p in key):
continue
pa, pb, pc = (_get(coords, p) for p in key)
area2 = abs(_cross2(pb - pa, pc - pa))
residuals.append(max(0.0, 0.5 - area2))
return residuals
def _remap_init_to_gauge(
init: dict[str, tuple[float, float]],
p0: str,
p1: str,
gauged: dict[str, tuple[float, float]],
) -> dict[str, tuple[float, float]]:
"""Rigidly map hinted positions onto the gauged base segment."""
if p0 not in init or p1 not in init:
return gauged
o0 = np.asarray(init[p0], dtype=float)
o1 = np.asarray(init[p1], dtype=float)
n0 = np.asarray(gauged[p0], dtype=float)
n1 = np.asarray(gauged[p1], dtype=float)
o_len = _norm(o1 - o0) or 1.0
n_len = _norm(n1 - n0) or 1.0
scale = n_len / o_len
od = (o1 - o0) / o_len
nd = (n1 - n0) / n_len
ang = math.atan2(nd[1], nd[0]) - math.atan2(od[1], od[0])
ca, sa = math.cos(ang), math.sin(ang)
rot = np.array([[ca, -sa], [sa, ca]])
remapped = dict(gauged)
for n, xy in init.items():
if n in (p0, p1):
continue
local = (np.asarray(xy, dtype=float) - o0) * scale
remapped[n] = tuple(n0 + rot @ local)
return remapped
def _degrees_of_freedom(ir: GeometryIR) -> tuple[int, int]:
"""Rough DOF count: 2*|points| - 3 (gauge) vs number of scalar residuals."""
n_coords = 2 * len(ir.points)
gauge = min(3, n_coords)
probe = _initial_coords(ir)
n_res = len(constraint_residuals(ir, probe))
return n_coords - gauge, n_res
def solve_geometry(
ir: GeometryIR,
*,
tol: float = DEFAULT_TOL,
max_nfev: int = 2000,
) -> SolvedScene:
"""Solve for point coordinates. Returns exact mode if residuals within tol."""
names = ir.point_names()
if not names:
return SolvedScene(
ir=ir,
coordinates={},
mode=SolveMode.failed,
max_residual=0.0,
message="no points",
)
init = _initial_coords(ir)
dof, n_res = _degrees_of_freedom(ir)
if n_res == 0 or n_res < max(1, dof - 1):
res = evaluate_residuals(ir, init)
return SolvedScene(
ir=ir,
coordinates=init,
mode=SolveMode.failed,
max_residual=max(res) if res else 0.0,
residuals=res,
message=f"underconstrained (dof≈{dof}, residuals={n_res})",
)
p0, p1 = _gauge_point_names(ir)
free_names: list[str] = [n for n in names if n != p0]
def apply_gauge(partial: dict[str, tuple[float, float]]) -> dict[str, tuple[float, float]]:
full = dict(partial)
if p0:
full[p0] = (0.0, 0.0)
if p1 and p1 in full:
x, _y = full[p1]
if abs(x) < 1e-9:
x = 1.0
full[p1] = (abs(x), 0.0)
elif p1:
full[p1] = (1.0, 0.0)
return full
def pack_free(coords: dict[str, tuple[float, float]]) -> np.ndarray:
vals: list[float] = []
for n in free_names:
x, y = coords[n]
if n == p1:
vals.append(x if abs(x) > 1e-9 else 1.0)
else:
vals.extend([x, y])
return np.asarray(vals, dtype=float)
def unpack_free(v: np.ndarray) -> dict[str, tuple[float, float]]:
partial: dict[str, tuple[float, float]] = {}
idx = 0
for n in free_names:
if n == p1:
partial[n] = (float(v[idx]), 0.0)
idx += 1
else:
partial[n] = (float(v[idx]), float(v[idx + 1]))
idx += 2
return apply_gauge(partial)
def fun(v: np.ndarray) -> np.ndarray:
coords = unpack_free(v)
res = list(constraint_residuals(ir, coords))
res.extend(_triangle_area_residuals(ir, coords))
if not res:
extras = []
if p1 and p1 in coords:
extras.append(coords[p1][0] - 1.0)
return np.asarray(extras or [0.0], dtype=float)
return np.asarray(res, dtype=float)
gauged_init = apply_gauge(init)
if p0 and p1:
gauged_init = apply_gauge(_remap_init_to_gauge(init, p0, p1, gauged_init))
x0 = pack_free(gauged_init)
if x0.size == 0:
coords = gauged_init
res = evaluate_residuals(ir, coords)
max_r = max(res) if res else 0.0
mode = SolveMode.exact if max_r <= tol else SolveMode.failed
return SolvedScene(
ir=ir,
coordinates=coords,
mode=mode,
max_residual=max_r,
residuals=res,
message="only gauge points",
)
try:
result = least_squares(
fun, x0, ftol=1e-12, xtol=1e-12, gtol=1e-12, max_nfev=max_nfev
)
coords = unpack_free(result.x)
except Exception as exc: # noqa: BLE001
coords = gauged_init
res = evaluate_residuals(ir, coords)
return SolvedScene(
ir=ir,
coordinates=coords,
mode=SolveMode.failed,
max_residual=max(res) if res else float("inf"),
residuals=res,
message=f"solver error: {exc}",
)
res = evaluate_residuals(ir, coords)
max_r = max(res) if res else 0.0
if max_r <= tol:
mode = SolveMode.exact
msg = "constraints satisfied"
else:
mode = SolveMode.failed
msg = f"residuals too large ({max_r:.4g})"
return SolvedScene(
ir=ir,
coordinates={k: (float(v[0]), float(v[1])) for k, v in coords.items()},
mode=mode,
max_residual=float(max_r),
residuals=res,
message=msg,
)
def scene_from_coordinates(
ir: GeometryIR,
coords: dict[str, tuple[float, float]],
*,
mode: SolveMode,
message: str = "",
) -> SolvedScene:
res = evaluate_residuals(ir, coords)
max_r = max(res) if res else 0.0
return SolvedScene(
ir=ir,
coordinates=coords,
mode=mode,
max_residual=float(max_r),
residuals=res,
message=message,
)
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