BIZOMAT MAKEOS · Geometry
Linear Interpolation
What value lies between these two table entries?
Straight-line interpolation (or extrapolation) between two known points, in either direction, for reading engineering tables and charts.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Find | choice | Interpolate forwards (y from x) or backwards (x from y). |
| First point x₁ | x value of the first table row. | |
| First point y₁ | Value at x₁. | |
| Second point x₂ | x value of the second table row. | |
| Second point y₂ | Value at x₂. | |
| Known x | The x value you need (forward mode). | |
| Known y | The y value you need (backward mode). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Find
- What it is
- Whether to read a y value for an x value you know, or to work backwards and find the x that gives a y value you know.
- Units and format
- Choose one of 2 options from the list: y for a given x; x for a given y.
- Example
- The form opens with “y for a given x” selected as a worked example. Choose the option that matches your case.
- Get it right
- Only the known value for the chosen direction is used.
What this means: First point x₁
- What it is
- The x value of the first known point, such as the first table row or chart reading on either side of the value you need.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Choose the two table rows that bracket your value; points far apart make straight-line reading less accurate.
What this means: First point y₁
- What it is
- The tabulated or charted y value that goes with the first point’s x value.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4.18 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Copy it from the same row as x₁, in the same units as y₂.
What this means: Second point x₂
- What it is
- The x value of the second known point, the other table row bracketing the value you need.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must differ from x₁.
What this means: Second point y₂
- What it is
- The tabulated or charted y value that goes with the second point’s x value.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4.179 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Copy it from the same row as x₂, in the same units as y₁.
What this means: Known x
- What it is
- The x value for which you want to read y. Used only when finding y.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Outside the two points the tool extrapolates, which is less reliable.
What this means: Known y
- What it is
- The y value for which you want to find x. Used only when finding x.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4.1795 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Finding x needs the two y values to differ.
Method and formulas
- y = y₁ + (x − x₁)(y₂ − y₁) ÷ (x₂ − x₁).
- x = x₁ + (y − y₁)(x₂ − x₁) ÷ (y₂ − y₁).
- Example: between (20, 4.18) and (40, 4.179), x = 25 gives y = 4.17975.
Limitations
- Assumes a straight line between the points; strongly curved data needs closer points or another method.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Read a value between two rows of a data table
- Find the input that gives a target output
- Check a chart reading
Who it is for
- Engineers
- Technicians
- Estimators
What you get
- Interpolated value
- Slope
- Whether the point lies inside the table range
Interpreting the result
- Avoid extrapolating far beyond the table; the relation may change.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 121. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Measuring case depth
In Induction Hardening by the Numbers
What it covers: How case depth is measured and judged: sectioning, a Vickers hardness traverse, the validity of each indentation, effective case depth by interpolation, and an indentation too close to the surface.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.