Lingshu Solver

Deterministic 1D/2D/3D geometry: distances, intersections, areas, volumes, angles, convex hull, point-in-polygon, rotations. Closed-form (not iterative), exact up to floating-point rounding, identical output for identical input. Use it instead of doing geometry yourself. Pass {op, ...args} with op one of: 1D: dist1,midpoint1,divide_point,divide_ratio,harmonic_conjugate,interval_length,interval_midpoint,interval_intersect,interval_union,interval_hull,interval_overlap_length,interval_contains,point_between | 2D: dist2,midpoint2,lerp2,line_from_points,point_line_distance,point_segment_distance,project_point_line,line_intersect,line_intersect_points,segment_intersect,segment_length,angle_3points,angle_vectors,line_angle,triangle_area,triangle_centroid,triangle_incenter,polygon_area,polygon_perimeter,polygon_centroid,point_in_polygon,convex_hull,polygon_is_convex,circle_from_3points,point_circle_position,circle_line_intersect,circle_circle_intersect,circle_measure,sector_area,rotate_point,reflect_point_line,scale_point,parallel_through,perpendicular_through,triangle_circumcenter,triangle_orthocenter,triangle_excenters,triangle_five_centers,euler_line,triangle_nine_point_circle,triangle_euler_inequality,triangle_medians,triangle_altitudes,triangle_bisector_lengths,cevian_concurrent,menelaus_collinear,power_of_point,radical_axis,polar_of_point,pole_of_line,ptolemy,miquel_point,napoleon_triangles,pick_theorem,cocircular_check,minimum_enclosing_circle,circumcircle | 3D: dist3,midpoint3,lerp3,plane_from_3points,point_plane_distance,project_point_plane,reflect_point_plane,coplanar,point_line_distance3,project_point_line3,line_plane_intersect,line_line_relation,plane_plane_intersect,tetra_volume,polygon_area3,face_normal,mesh_volume,sphere_from_4points,sphere_line_intersect,sphere_plane_intersect,sphere_measure,angle_vectors3,angle_line_plane,angle_plane_plane,rotate_axis,cross3,triple_product,tetrahedron_geometry,monge_point | any: distance,midpoint,lerp,norm,normalize,dot,cross,project,reject,translate,centroid,bbox,diameter,stewart_theorem,law_of_cosines,law_of_cosines_sas,law_of_sines,solve_triangle,heron_area,brahmagupta,golden_ratio,regular_polygon,regular_tetrahedron,newton_polytope,tropical_degree_bound,tropical_system_bound,regular_subdivision,bernstein_bound,tropical_bezout_bound,sphere_equation,polynomial_equation. The remaining args are passed flat and depend on op (p, q, a, b, c, d, r, poly, pts, angle, lambda, center, axis, segment, a1,b1,c1,a2,b2,c2, ...). Send an unknown op to receive the full catalog with every signature. READ trust.trustLevel: exact = safe to use directly; definitely_none = computed proof that there is NO intersection, safe to state as a conclusion; degenerate = the INPUT is degenerate (parallel / collinear / coplanar / coincident), do NOT invent a value — report the degeneracy instead.

Report a problem to the operator when solve hits a dead end, returns an error, or produces something you believe is wrong. The text is written to a local log only and is never transmitted anywhere else.

Optional voluntary-support only. The hosted endpoint is no per-call charge today (metering off) (metering off: no credential required, no call limit, no per-call charge), so no payment is needed at all. This tool records a support contribution of any amount: the channel is a corporate static collection code (UnionPay aggregate QR) settling into a corporate bank account, and no payment-platform merchant API is used. AFTER PAYING: nothing to do — there is no self-crediting step and no waiting for reconciliation; the operator reconciles against the corporate bank statement, and any positive amount counts as support, not a bill. If the server has no payment method configured the order is still created but payIntent.payTo is null; in that case just keep calling the endpoint with no credential at all.

All real roots of a polynomial, each individually Krawczyk-certified with a strict error box. Coefficients are highest degree first: [1,-2,-5,6] means x^3-2x^2-5x+6. Use it instead of letting a language model estimate roots. Complex roots are not returned (real only).

Deterministic (non-LLM) solver for systems of real equations. Use it instead of computing the math yourself whenever a wrong number has a cost. Same input always returns the identical result — no hallucination, no randomness, so caching and retries are safe. Supports algebraic equations and common transcendentals (sin/cos/tan/log/exp/sqrt/abs), max 6 variables, no initial guess needed. READ THE trust BLOCK: it tells you, per result, whether the value is safe to use directly or must be verified first. Pass domain explicitly for exp/sin or large ranges, else truncated may be true. Not for symbolic derivation, initial-value ODEs, or integer constraints. Optional: pass honorPaid:true to declare personal or evaluation use — then it is served free with no key, no balance and no deduction; payment is voluntary and never enforced.

Check whether a claimed answer is actually correct — call this on any number you computed yourself before passing it downstream. verdict=verified means it is a certified real root; verdict=refuted_or_unverified means it is NOT, and nearestCertifiedRoot then carries the correct value so you can self-correct in the same turn. Deterministic and reproducible.