Bilinear Saddle Stationary Coordinate Tool

Description

Mark insideUnitCell inclusively.

A bilinear cell combines constant, two linear, and one mixed coordinate term. Corner contrasts, stationary points, and edge intersections use normalized coordinates unless spans are supplied.

Mark insideUnitCell inclusively.

When to use this tool

Use Bilinear Saddle Stationary Coordinate when you need one explicit bilinear coefficient, sensitivity, or level-set term.

How to use the tool and how it works

Enter linearX, linearY, cross. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read x, y, insideUnitCell. Inspect insideUnitCell before interpreting the associated values.

Require nonzero cross and return x = -linearY/cross and y = -linearX/cross for the stationary point of constant + linearX·x + linearY·y + cross·xy. Mark insideUnitCell inclusively.^[1]

Input field	Meaning and units	Accepted input
linearX	Linear X	Required. Finite number.
linearY	Linear Y	Required. Finite number.
cross	Cross	Required. Finite number.

Output field	Interpretation	Returned shape
x	X	Required. Finite number.
y	Y	Required. Finite number.
insideUnitCell	Inside Unit Cell	Required. Boolean.

json

{
  "input": {
    "linearX": -2,
    "linearY": -1,
    "cross": 4
  },
  "output": {
    "x": 0.25,
    "y": 0.5,
    "insideUnitCell": true
  }
}

Limitations and assumptions

The constant does not affect the stationary location; no level set or sampled grid is constructed.
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 a complete cell model and a documented contour or interpolation policy when reconstructing a surface.

References

Calculation and input contract: interpolation/bilinear-saddle-stationary-coordinate — tool.io

Similar or alternative tools

Bilinear Level Set Saddle Sign
For nonzero crossCoefficient, return saddleX = -yCoefficient/crossCoefficient, saddleY = -xCoefficient/crossCoefficient, saddleValue = constant - xCoefficient·yCoefficient/crossCoefficient, and sign(targetLevel - saddleValue).
Bilinear Level Set Edge Intersection
Return (target - first)/(second - first) and an inclusive onEdge flag for parameters in [0, 1].