# NURBS Surface Second-Jet & Principal Curvatures

> NURBS Surface Second-Jet & Principal Curvatures is a paid API for AI agents from detect.halowerk.com, paid per call via x402, $0.003/call, status unknown (last checked 2026-11-04).

Computes local differential geometry (second fundamental form, principal curvatures, and related quantities) of an explicit regular NURBS patch at given parameter points.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-TD2000-GEO-INDUSTRY-B005/nurbs-surface-second-jet-fundamental-principal-curvatures?utm_source=zero.xyz
- Price: $0.003/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-04
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/nurbs-surface-second-jet-principal-curvatures-88dcfa6f
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_0RnhYDXDZhT8zWbNcu0jI

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability nurbs-surface-second-jet-principal-curvatures-88dcfa6f -d '<json body>'
```

Example prompt: 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).

## When to prefer this

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.

## Known failure modes

- 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

## How this service works

Compute local differential geometry of an explicit regular NURBS patch.

## Output

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).

## Request schema (JSON Schema)

```json
{
 "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
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/nurbs-surface-second-jet-principal-curvatures-88dcfa6f/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from detect.halowerk.com](https://www.zero.xyz/host/detect.halowerk.com/llms.txt)