Random Recursive Circle Nesting Pattern Generator

Description

Construct contained binary circle hierarchies with opposite tangent children and disjoint sibling interiors; drawing and clipping remain external.

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.

Construct contained binary circle hierarchies with opposite tangent children and disjoint sibling interiors; drawing and clipping remain external.

When to use this tool

Use this generator to create reproducible recursive circle nesting 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 radius, depth, childRatio, axisTurns 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 circles, bounds. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.

Recursively place two child circles on opposite directions of a sampled axis, each with radius parent radius times childRatio and center distance parent radius minus child radius. Return parent indexes and the original outer-circle bounds.^[1]

Input field	Meaning and units	Accepted input
radius	Radius	Required. Finite number; strictly above 0.
depth	Depth	Required. Safe integer; minimum 0; maximum 5.
childRatio	Child Ratio	Required. Finite number; maximum 0.5; strictly above 0.
axisTurns	Axis Turns	Required. Array of Finite number.; minimum items 1; maximum items 16.
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
circles	Circles	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 63.
circles[].id	Literal identifier used to match records, independently of their display labels.	Required. Safe integer; minimum 0; maximum 62.
circles[].parent	Parent	Required. Safe integer; minimum -1; maximum 62.
circles[].center	Center	Required. Array of Finite number.; minimum items 2; maximum items 2.
circles[].radius	Radius	Required. Finite number; strictly above 0.
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": {
    "radius": 4,
    "depth": 1,
    "childRatio": 0.25,
    "axisTurns": [
      0
    ],
    "seed": 42
  },
  "output": {
    "circles": [
      {
        "id": 0,
        "parent": -1,
        "center": [
          0,
          0
        ],
        "radius": 4
      },
      {
        "id": 1,
        "parent": 0,
        "center": [
          -3,
          0
        ],
        "radius": 1
      },
      {
        "id": 2,
        "parent": 0,
        "center": [
          3,
          0
        ],
        "radius": 1
      }
    ],
    "bounds": {
      "minX": -4,
      "minY": -4,
      "maxX": 4,
      "maxY": 4,
      "width": 8,
      "height": 8
    },
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

Parent index minus one identifies the root. Internal tangency to the parent does not imply siblings never overlap. Depth grows the output exponentially and tiny geometry can fail numeric resolution.
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 Recursive Circle Nesting Pattern Generator — tool.io

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