Ellipse Focal Radius Derivative Tool

Description

Return sqrt(a²-b²), mark derivativeDefined only when a>b, and return a/c or -b/c accordingly, otherwise derivative zero.

A conic curve has parameters tied to a chosen axis system and parameterization. Parametric angle, polar angle, focus distance, tangent, and normal describe different relationships and cannot be interchanged without their definitions.

Return sqrt(a²-b²), mark derivativeDefined only when a>b, and return a/c or -b/c accordingly, otherwise derivative zero.

When to use this tool

Use Ellipse Focal Radius Derivative when you need a local conic-curve parameter or derivative calculation.

How to use the tool and how it works

Enter semiMajor, semiMinor, vary. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read focalDistance, derivativeDefined, derivative. Inspect derivativeDefined before interpreting the associated values.

Require semiMajor at least positive semiMinor and vary major or minor. Return sqrt(a²-b²), mark derivativeDefined only when a>b, and return a/c or -b/c accordingly, otherwise derivative zero.^[1]

Input field	Meaning and units	Accepted input
semiMajor	Semi Major	Required. Finite number.
semiMinor	Semi Minor	Required. Finite number; strictly above 0.
vary	Vary	Required. Text; maximum UTF-16 length 8192.

Output field	Interpretation	Returned shape
focalDistance	Focal Distance	Required. Finite number.
derivativeDefined	False at the circular limit, where derivative is a zero placeholder.	Required. Boolean.
derivative	Local sensitivity with respect to the stated variable, holding other model inputs fixed.	Required. Finite number.

json

{
  "input": {
    "semiMajor": 5,
    "semiMinor": 3,
    "vary": "major"
  },
  "output": {
    "focalDistance": 4,
    "derivativeDefined": true,
    "derivative": 1.25
  }
}

Limitations and assumptions

Focal distance is center to one focus, not focus-to-focus; zero derivative is a placeholder at a circle and no complete ellipse or numerical sensitivity model is validated.
Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.

Alternative or Complementary methods

Use the full conic equation and coordinate transform for rotated or translated curves. Check semiaxis ordering, parameter units, and degeneracy before interpreting a local tangent or focus identity.

References

Calculation and input contract: geometry/ellipse-focal-radius-derivative — tool.io

Similar or alternative tools

Ellipse Focal Distance Residual
Use their ratio to compute center-to-focus distance and return suppliedFocusDistance minus the derived value.
Ellipse Eccentric Angle Focal Distance
Compute c=sqrt(a²-b²) and return a-c cos(parameter) and a+c cos(parameter).