Description
Compare finite tangent and normal slopes for orthogonality.
Differential-calculus formulas relate supplied local values and derivatives under explicit coordinate and unit conventions. A tangent term, curvature expression or directional derivative assumes the required differentiability and regularity; substituting numerical derivatives does not verify those hypotheses for an actual function.
Compare finite tangent and normal slopes for orthogonality.
When to use this tool
Use Calculus: Tangent Normal Slope Orthogonality when evaluating the stated local differential-calculus identity from supplied values.
How to use the tool and how it works
- Enter tangentSlope, normalSlope, tolerance. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Compare finite tangent and normal slopes for orthogonality. Require tolerance >= 0; orthogonalityResidual = tangentSlope*normalSlope+1, consistent means absolute residual <= tolerance.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| tangentSlope | Tangent Slope | Required. Finite number. |
| normalSlope | Normal Slope | Required. Finite number. |
| tolerance | Nonnegative caller-selected comparison tolerance, in matching units. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.orthogonalityResidual | Orthogonality Residual | Calculated result member: number. |
| result.consistent | Consistent | Calculated result member: boolean. |
{
"input": {
"tangentSlope": 2,
"normalSlope": -0.5,
"tolerance": 1e-9
},
"output": {
"result": {
"orthogonalityResidual": 0,
"consistent": true
}
}
}Limitations and assumptions
- This finite-slope test cannot represent a vertical/horizontal normal pair or establish that either slope belongs to the claimed curve.
- Outputs are nested inside result. 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
Differentiate or validate the complete function and its domain first; use suitable symbolic or numerical methods and check regularity and coordinate conventions for the intended interpretation.
References
Similar or alternative tools
- Calculus: Tangent Line From Point Derivative
Evaluate a supplied tangent-line approximation.
- Tangent Axis Intercept
Return axis intercepts of a supplied nonhorizontal tangent line.