Bezier Endpoint Join Position Audit

Description

Audit endpoint positional continuity of two explicit planar Bezier segments against an absolute supplied tolerance; no tangent or curvature continuity inferred.

A planar Bézier curve is defined by an ordered control polygon: three points for a quadratic or four for a cubic. Its parameter t runs from 0 to 1 and generally does not measure arc length. Derivative vectors, curve positions, control points and polynomial coefficients are different representations of the same parameterized geometry.

Audit endpoint positional continuity of two explicit planar Bezier segments against an absolute supplied tolerance; no tangent or curvature continuity inferred.

When to use this tool

Use this tool to audit endpoint positional continuity of two explicit planar Bezier segments against an absolute supplied tolerance; no tangent or curvature continuity inferred.

How to use the tool and how it works

Enter left, right, tolerance using the input definitions below.
Run the operation. Resolve malformed-input errors before interpreting the result.
Read result.distance, result.joined. Interpret signs, units, ordering and null values using the rules below.

Measure Euclidean distance between the left curve’s final control point and the right curve’s first. Return joined = distance≤tolerance.^[1]

Input field	Meaning and units	Accepted input
left	Left quadratic/cubic curve’s ordered control polygon.	Dense list of at most 128 rows of at most 128 finite numbers; the operation’s shape constraints also apply.
right	Right quadratic/cubic curve’s ordered control polygon.	Dense list of at most 128 rows of at most 128 finite numbers; the operation’s shape constraints also apply.
tolerance	Nonnegative tolerance interpreted by the specific position/tangent comparisons stated above.	Finite number satisfying the stated comparisons.

Output field	Interpretation
result	Object containing the fields below.
result.distance	Endpoint separation in coordinate units.
result.joined	Whether endpoint separation is at most tolerance.

json

{
  "input": {
    "left": [
      [
        0,
        0
      ],
      [
        1,
        1
      ],
      [
        2,
        0
      ]
    ],
    "right": [
      [
        2,
        0
      ],
      [
        3,
        1
      ],
      [
        4,
        0
      ]
    ],
    "tolerance": 0
  },
  "output": {
    "result": {
      "distance": 0,
      "joined": true
    }
  }
}

Limitations and assumptions

The tolerance uses coordinate units; a positional join does not establish tangent or curvature continuity.
Each geometric control point has two finite coordinates within ±1,000,000,000. t and interval endpoints are dimensionless in [0,1]. Calculations use floating arithmetic and do not perform curve fitting or robust topology analysis.
Every declared top-level input is required and unknown fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
Numeric outputs must be finite and within the absolute safe-integer magnitude; exact larger integers use decimal text. Source positions are zero-based UTF-16 offsets and span ends are exclusive unless a different convention is explicitly stated.

Alternative or Complementary methods

Keep control polygons and constant-first power coefficients clearly labeled. Use separate arc-length integration or adaptive sampling when physical spacing or geometric approximation error matters.

References

Tool behavior and example: Bezier Endpoint Join Position Audit — tool.io

Similar or alternative tools

Bezier Endpoint Join Tangent Audit
Audit actual degree-scaled endpoint tangents for C1 equality and same-direction alignment separately; zero tangents yield unknown alignment, position continuity is not assumed.
Bezier Control Point Order Reversal
Reverse a planar quadratic/cubic control polygon to represent the same Bezier curve under t -> 1-t, not a spatial mirror.