{"uid":"cap_0RnhYDXDZhT8zWbNcu0jI","slug":"nurbs-surface-second-jet-principal-curvatures-88dcfa6f","name":"NURBS Surface Second-Jet & Principal Curvatures","description":"Compute local differential geometry of an explicit regular NURBS patch.","url":"https://detect.halowerk.com/v1/F-TD2000-GEO-INDUSTRY-B005/nurbs-surface-second-jet-fundamental-principal-curvatures?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"artifact":{"type":"object","required":["nurbsJet"],"properties":{"nurbsJet":{"type":"object","required":["data","claim"],"properties":{"data":{"type":"object","required":["degree_u","degree_v","knots_u","knots_v","control_net","weights","parameters"],"properties":{"knots_u":{"type":"array","items":{"type":"number","maximum":1,"minimum":0},"maxItems":10,"minItems":4},"knots_v":{"type":"array","items":{"type":"number","maximum":1,"minimum":0},"maxItems":10,"minItems":4},"weights":{"type":"array","items":{"type":"array","items":{"type":"number","maximum":1000,"minimum":0.001},"maxItems":6,"minItems":2},"maxItems":6,"minItems":2},"degree_u":{"type":"integer","maximum":3,"minimum":1},"degree_v":{"type":"integer","maximum":3,"minimum":1},"parameters":{"type":"array","items":{"type":"array","items":{"type":"number","maximum":10000,"minimum":-10000},"maxItems":2,"minItems":2},"maxItems":16,"minItems":1},"control_net":{"type":"array","items":{"type":"array","items":{"type":"array","items":{"type":"number","maximum":10000,"minimum":-10000},"maxItems":3,"minItems":3},"maxItems":6,"minItems":2},"maxItems":6,"minItems":2}},"additionalProperties":false},"claim":{}},"additionalProperties":false}},"additionalProperties":false}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm__e41c9n9sNlHHVuFe0RLp","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes local differential geometry (second fundamental form, principal curvatures, and related quantities) of an explicit regular NURBS patch at given parameter points.","exampleAgentPrompt":"Compute the principal curvatures and second fundamental form for a degree-2×2 NURBS patch with knots [0,0,0,1,1,1] in both u and v, a 3×3 control net with all weights 1.0, evaluated at parameter point (0.5, 0.5).","exampleUseCases":[{"title":"CAD surface quality inspection","prompt":"I have a bicubic NURBS surface from my CAD model — degree 3 in both u and v — and I need to check its curvature at the point (0.3, 0.7) to verify it's smooth enough for a Class-A automotive surface. Can you compute the principal curvatures and Gaussian curvature for me?"},{"title":"CNC toolpath curvature analysis","prompt":"For our CNC machining pipeline, I need the principal curvatures of a NURBS patch at 8 sample parameter points so we can adapt the tool contact radius. The patch is degree 2×3, here are my knot vectors, control net, and weights — please evaluate the second fundamental form at all those points."},{"title":"Mesh refinement curvature-driven sampling","prompt":"I'm building an adaptive mesh of a NURBS hull surface and I need to know the Gaussian and mean curvature at several candidate parameter locations to decide where to add more triangles. Can you give me the full second-jet differential geometry data at those points for a degree-2×2 rational NURBS patch?"}],"resultDescription":"Returns the computed local differential geometry of the NURBS patch at each queried parameter point, including components of the second fundamental form (L, M, N coefficients), principal curvatures (κ₁, κ₂), principal curvature directions, Gaussian curvature (K = κ₁·κ₂), and mean curvature (H = (κ₁+κ₂)/2).","failureModes":["Invalid or non-clamped knot vectors (knots not in [0,1] or wrong count for given degree and control net) → validation error","Control net dimensions inconsistent with degree or weight array dimensions → schema validation failure","Evaluation at a singular or degenerate point where the surface normal vanishes → undefined geometry error","Degree outside allowed range 1–3 → validation rejection","Parameter values outside domain implied by knot vectors → out-of-domain error","Payment not included or insufficient USDC → HTTP 402 Payment Required"],"whenToPreferThis":"Choose this endpoint when you need analytically exact local differential geometry (second-order quantities) of a NURBS surface rather than finite-difference approximations from a mesh. It is ideal for CAD/CAM pipelines, surface quality inspection, curvature-driven adaptive sampling, and any workflow where the surface is already represented as a NURBS patch with explicit knot vectors and control nets. Prefer it over mesh-based curvature estimators when precision and mathematical exactness matter.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-04T06:40:05.721Z","isFirstParty":false,"canonicalSlug":"nurbs-surface-second-jet-principal-curvatures-88dcfa6f"}