Cubic Bezier Second Derivative Value

Description

Evaluate second derivative using De Casteljau arithmetic, not a screen-space or arc-length parameter.

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.

Evaluate second derivative using De Casteljau arithmetic, not a screen-space or arc-length parameter.

When to use this tool

Use this tool to evaluate second derivative using De Casteljau arithmetic, not a screen-space or arc-length parameter.

How to use the tool and how it works

Enter points, t using the input definitions below.
Run the operation. Resolve malformed-input errors before interpreting the result.
Read result.value. Interpret signs, units, ordering and null values using the rules below.

Require 4 ordered planar controls. Apply the degree-scaled adjacent-difference derivative construction twice, then evaluate at t by De Casteljau interpolation.^[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.
t	Dimensionless curve parameter from 0 through 1.	Finite number satisfying the stated comparisons.

Output field	Interpretation
result	Object containing the fields below.
result.value	Second derivative [d²x/dt²,d²y/dt²], in coordinate units per t².

json

{
  "input": {
    "points": [
      [
        0,
        0
      ],
      [
        1,
        0
      ],
      [
        2,
        0
      ],
      [
        3,
        0
      ]
    ],
    "t": 0.5
  },
  "output": {
    "result": {
      "value": [
        0,
        0
      ]
    }
  }
}

Limitations and assumptions

Second derivative has coordinate units per t². It is not curvature, and t is not arc length.
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: Cubic Bezier Second Derivative Value — tool.io

Similar or alternative tools

Cubic Bezier Point Value
Evaluate point using De Casteljau arithmetic, not a screen-space or arc-length parameter.
Cubic Bezier Tangent Value
Evaluate first derivative tangent using De Casteljau arithmetic, not a screen-space or arc-length parameter.