Description
Return tangent [-a sin t, b cos t], normal [cos t/a, sin t/b], and their dot product as residual.
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 tangent [-a sin t, b cos t], normal [cos t/a, sin t/b], and their dot product as residual.
When to use this tool
Use Ellipse Normal Tangent Orthogonality Residual when you need a local conic-curve parameter or derivative calculation.
How to use the tool and how it works
- Enter a, b, parameter. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read tangent, normal, residual. Use the output convention below.
Require positive a and b and use parameter in radians. Return tangent [-a sin t, b cos t], normal [cos t/a, sin t/b], and their dot product as residual.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| a | A | Required. Finite number; strictly above 0. |
| b | B | Required. Finite number; strictly above 0. |
| parameter | Parameter | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| tangent | Tangent | Required. Array of finite number; maximum items 256. |
| normal | Normal | Required. Array of finite number; maximum items 256. |
| residual | Residual | Required. Finite number. |
{
"input": {
"a": 3,
"b": 2,
"parameter": 0
},
"output": {
"tangent": [
0,
2
],
"normal": [
0.3333333333333333,
0
],
"residual": 0
}
}Limitations and assumptions
- The normal is not unit length and residual is a numerical orthogonality diagnostic; no actual ellipse, observed point, tolerance, or curve fitting is supplied.
- 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
Similar or alternative tools
- Ellipse Parametric Normal Unit Vector
Form [cos/a, sin/b] and normalize it for the outward normal.
- Geometry: Ellipse Parametric Tangent Direction
Differentiate (a cos t, b sin t), returning the tangent vector and its normalized unitTangent inside result.