Random Parametric Knot Projection Generator

Description

Sample caller-reviewed coprime torus-curve templates in 3D and project onto a selected coordinate plane; sparse samples do not classify topology or crossings.

Procedural artwork expresses a visual idea through coordinates, repeated motifs, colors, or explicit placement constraints. A coordinate plan can be checked and reproduced before rendering, while physical material behavior and fabrication require separate information.

Sample caller-reviewed coprime torus-curve templates in 3D and project onto a selected coordinate plane; sparse samples do not classify topology or crossings.

When to use this tool

Use this generator to create reproducible parametric knot projection cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply templates, axes, intervals in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read template, axis, samples, points, bounds. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.

Select a torus-knot template with coprime positive winding counts and major radius greater than minor radius. Sample its three-dimensional parametric curve over one period, repeat the start exactly, and project onto the selected coordinate plane.^[1]

Input field	Meaning and units	Accepted input
templates	Templates	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 16.
templates[].id	Literal identifier used to match records, independently of their display labels.	Required. Text; minimum UTF-16 length 1; maximum UTF-16 length 80.
templates[].source	Source	Required. Text; minimum UTF-16 length 1; maximum UTF-16 length 80.
templates[].majorRadius	Major Radius	Required. Finite number; strictly above 0.
templates[].minorRadius	Minor Radius	Required. Finite number; strictly above 0.
templates[].p	P	Required. Safe integer; minimum 1; maximum 32.
templates[].q	Q	Required. Safe integer; minimum 1; maximum 32.
axes	Axes	Required. Array of Text; one of "xy", "xz", "yz".; minimum items 1; maximum items 3.
intervals	Intervals	Required. Safe integer; minimum 4; maximum 127.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
template	Template	Required. Object; no unknown keys.
template.id	Literal identifier used to match records, independently of their display labels.	Required. Text; minimum UTF-16 length 1; maximum UTF-16 length 80.
template.source	Source	Required. Text; minimum UTF-16 length 1; maximum UTF-16 length 80.
template.majorRadius	Major Radius	Required. Finite number; strictly above 0.
template.minorRadius	Minor Radius	Required. Finite number; strictly above 0.
template.p	P	Required. Safe integer; minimum 1; maximum 32.
template.q	Q	Required. Safe integer; minimum 1; maximum 32.
axis	Axis	Required. Text; one of "xy", "xz", "yz".
samples	Samples	Required. Array of Array of Finite number.; minimum items 3; maximum items 3.; minimum items 5; maximum items 128.
points	Points	Required. Array of Array of Finite number.; minimum items 2; maximum items 2.; minimum items 1; maximum items 128.
bounds	Bounds	Required. Object; no unknown keys.
bounds.minX	Min X	Required. Finite number.
bounds.minY	Min Y	Required. Finite number.
bounds.maxX	Max X	Required. Finite number.
bounds.maxY	Max Y	Required. Finite number.
bounds.width	Width	Required. Finite number.
bounds.height	Height	Required. Finite number.
randomness	Replay metadata: source is seed or context; seed is the normalized 32-bit seed or null for a context source.	Required. Object; no unknown keys.
randomness.seed	Optional numeric replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "templates": [
      {
        "id": "trefoil",
        "source": "caller reviewed torus template",
        "majorRadius": 3,
        "minorRadius": 1,
        "p": 2,
        "q": 3
      }
    ],
    "axes": [
      "xy"
    ],
    "intervals": 4,
    "seed": 42
  },
  "output": {
    "template": {
      "id": "trefoil",
      "source": "caller reviewed torus template",
      "majorRadius": 3,
      "minorRadius": 1,
      "p": 2,
      "q": 3
    },
    "axis": "xy",
    "samples": [
      [
        4,
        0,
        0
      ],
      [
        -3,
        0,
        -1
      ],
      [
        2,
        0,
        0
      ],
      [
        -3,
        0,
        1
      ],
      [
        4,
        0,
        0
      ]
    ],
    "points": [
      [
        4,
        0
      ],
      [
        -3,
        0
      ],
      [
        2,
        0
      ],
      [
        -3,
        0
      ],
      [
        4,
        0
      ]
    ],
    "bounds": {
      "minX": -3,
      "minY": 0,
      "maxX": 4,
      "maxY": 0,
      "width": 7,
      "height": 0
    },
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

Projected crossings do not identify over/under topology. Bounds cover sampled projected points rather than continuous extrema, and this is artwork rather than a physical knot construction.
A replay requires the same complete input and implementation. Randomized construction does not guarantee uniform coverage of every feasible output; finite samples can repeat unless a distinct-selection rule is explicitly stated.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Parametric Knot Projection Generator — tool.io

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