Formula and Method for the Involute Function
The involute function, written inv(θ), is defined as inv(θ) = tan(θ) − θ, where θ is the angle measured in radians. It is the single formula behind the involute curve — the tooth-profile shape used in essentially every modern spur, helical, and worm gear — and it also shows up in cam design, spline geometry, and any problem that traces the path a taut string unwinds along a circle. This calculator evaluates inv(θ) directly from an angle you supply, or runs the problem backward: given a known involute value, it numerically solves for the angle θ that produces it, since that inverse has no closed algebraic form.
How the calculation works
In forward mode, enter an angle θ and choose whether it is in degrees or radians. The calculator first converts θ to radians (multiplying degrees by π/180), computes tan(θ), and subtracts θ (in radians) from it: inv(θ) = tan(θ) − θ. The result is dimensionless but expressed in radians, since it is literally the difference between two radian quantities — this is why θ must be converted to radians even when you typed it in degrees; subtracting a degree value from a tangent value would silently produce a meaningless number. In inverse mode, you supply a target involute value instead. Because inv(θ) has no algebraic inverse, the calculator searches for θ numerically: it uses a bisection search over the valid range (0° up to just under 90°, mirrored for negative targets by symmetry, since inv is an odd function) that repeatedly halves the interval until tan(θ) − θ matches your target to a very tight tolerance. This is the same general approach gear designers use when working backward from a measured tooth thickness to the pressure angle that produced it.
Where the involute function comes from
Picture a string wound tightly around a circle of radius r_b (the "base circle"). As you unwind the string while keeping it taut, its free end traces an involute curve. At any point on that curve, draw the line from the center of the circle to the point (length r), and the tangent line from the center to where the string currently leaves the circle. The angle θ between the radius to the tangent point and the radius to the involute point is exactly the pressure angle at that point, and it satisfies cos(θ) = r_b / r. The angle the whole radius vector has swept through, relative to where the involute started, works out to tan(θ) − θ — precisely the involute function. This is why gear catalogs and standards (AGMA, ISO) build entire tables around inv(θ): it converts a pressure angle at one gear radius directly into the angular tooth-thickness offset at another radius.
Common mistakes and practical notes
- Forgetting the radians conversion: tan(θ) is computed the same regardless of input unit, but the "− θ" term must use radians. Mixing degrees and radians in that subtraction is the most common error when computing inv(θ) by hand or in a spreadsheet.
- Confusing the pressure angle with the roll angle: tan(θ) alone is sometimes called the roll angle; the involute function is the difference between the roll angle and the pressure angle, not either one by itself.
- Expecting a value near ±90°: tan(θ) grows without bound as θ approaches 90°, so inv(θ) also grows without bound there — the function is only meaningful strictly between −90° and 90°.
- Assuming an algebraic inverse exists: there is no formula that isolates θ from tan(θ) − θ = k; every inverse-involute calculation, including this one, relies on iterative numerical search rather than direct algebra.