Description
Sample an explicit planar quadratic/cubic Bezier at caller unit parameters in caller order, preserving repeats; not equal arc-length sampling.
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.
Sample an explicit planar quadratic/cubic Bezier at caller unit parameters in caller order, preserving repeats; not equal arc-length sampling.
When to use this tool
Use this tool to sample an explicit planar quadratic/cubic Bezier at caller unit parameters in caller order, preserving repeats; not equal arc-length sampling.
How to use the tool and how it works
- Enter points, parameters using the input definitions below.
- Run the operation. Resolve malformed-input errors before interpreting the result.
- Read result.samples. Interpret signs, units, ordering and null values using the rules below.
Validate a quadratic or cubic control polygon and evaluate De Casteljau at every supplied parameter in [0,1]. Return {t,point} records in caller order, retaining repeated parameters.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| points | Ordered planar control points: three for a quadratic, four for a cubic. Basis-conversion tools use the representation stated above. | Dense list of at most 128 rows of at most 128 finite numbers; the operation’s shape constraints also apply. |
| parameters | Caller-ordered dimensionless curve parameters in [0,1]; repeats are retained. | Dense list of at most 128 finite numbers, subject to the stated integer/shape rules. |
| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.samples | Caller-ordered {t,point} evaluations, retaining repeated parameters. |
{
"input": {
"points": [
[
0,
0
],
[
1,
1
],
[
2,
0
]
],
"parameters": [
0,
0.5,
1
]
},
"output": {
"result": {
"samples": [
{
"t": 0,
"point": [
0,
0
]
},
{
"t": 0.5,
"point": [
1,
0.5
]
},
{
"t": 1,
"point": [
2,
0
]
}
]
}
}
}Limitations and assumptions
- The operation does not sort parameters or distribute samples by physical distance or curvature.
- 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
Similar or alternative tools
- 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.
- Cubic Bezier from Power Coefficients
Convert planar ascending power coefficients to Bezier control points for explicit degree 3; exact linear basis change under finite arithmetic.