Involute Function Calculator - inv(a) = tan a - a, Inverse Involute and Involute Gears

What is the involute function?

The involute function is one of the shortest formulas in engineering:

inv(α) = tan(α) − α

The angle α must be in radians, and the result is an angle in radians too — even though nobody ever writes a unit after it. That single subtraction connects a piece of pure geometry (a curve drawn by an unwinding string) to the gearbox in your car.

Roulettes: rolling one curve on another

When mathematicians say roulette, they don't mean the casino game. A roulette is the path traced by a point attached to one curve as that curve rolls along another, fixed curve — without slipping. Two conditions apply: both curves must be differentiable (smooth everywhere, no cusps or breaks where the slope would be undefined), and they must stay tangent at every instant.

A spirograph is a roulette machine: rolling toothed circles and ellipses around each other traces those hypnotic looping curves. Change the rolling curve, the fixed curve, or the pen position, and you get a different member of the family — cycloids, epicycloids, hypocycloids, and more.

One special class comes from a straight line wrapped around a curve as taut string, traced as it unwraps. That roulette is called an involute.

The involute of a circle

Wrap a string around a circle of radius a, hold it taut, and unwind it. The tip of the string draws the involute of a circle — a spiral that never closes on itself:

x = a(cos θ + θ · sin θ)
y = a(sin θ − θ · cos θ)

Here θ is the angle at which the string separates from the circle — the point where circle and string are tangent. A few consequences fall straight out of those two equations:

• Unwound string length = a·θ. That's just the arc of circle you've released.
• Distance from the centre = a·√(1 + θ²). The string and the radius to the tangency point meet at a right angle, so Pythagoras does the rest.
• Curve length from the start = a·θ²/2. Not a·θ — the tip travels further than the string it has released, because the tangency point keeps moving too.
• Radius of curvature = a·θ — exactly the unwound string. The tangency point on the circle is the centre of curvature at that instant.

Where the two ideas meet

The curve and the function look like separate topics. They aren't — they're the same equation:

• The roll parameter is the tangent of the pressure angle: θ = tan α.
• The polar angle swept from the start of the involute to the traced point is θ − α = tan α − α = inv(α).

So the involute function isn't bolted onto the involute curve — it is the curve's polar angle. That's precisely why it shows up everywhere in gear design: it converts "how far around the gear" into "at what radius", and back.

Involute gears

Nearly every gear you have ever touched has involute flanks. The reason is a property no other simple profile matches: two involute teeth mesh with a constant velocity ratio, and they keep it even if the centre distance is slightly off. Bearings wear, housings flex, tolerances stack — an involute gear pair just keeps turning smoothly. A gear cut from any other curve would speed up and slow down within every single tooth engagement.

The involute also makes manufacturing cheap: the flanks can be generated by a straight-sided rack cutter, so one tool cuts every gear of a given module and pressure angle, whatever the tooth count.

Gear design and the involute function

The pressure angle α is the standard input. 20° is the modern default; 14.5° is the older standard still found in legacy drawings, and 25° trades smoothness for strength. Around the pressure angle the geometry unfolds:

• Pitch diameter d = m · z — module × tooth count.
• Base diameter d_b = d · cos α — the circle the string unwraps from.
• Tip diameter d_a = d + 2m, root diameter d_f = d − 2.5m.
• Base pitch p_b = π · m · cos α — the spacing that actually has to match for two gears to mesh.

The classic industrial use of inv() is tooth thickness at any radius. At the tip:

cos α_a = d_b / d_a
s_a = d_a · ( s/d + inv(α) − inv(α_a) )

If s_a comes out at zero or below, the tooth has become pointed — there is no material left at the tip and the design is not manufacturable.

Metric and US gears

The two systems name the same quantity differently. Metric gears use module m in millimetres; US gears use diametral pitch P in teeth per inch. They are reciprocals across the inch:

m [mm] = 25.4 / P

A bigger module means bigger teeth; a bigger diametral pitch means smaller teeth. This calculator accepts either and reports both.

Going backwards: the inverse involute

Gear design constantly needs the opposite direction — you know inv(α) and want α. This is where it gets awkward: tan(α) − α = y cannot be solved for α in closed form. It is a transcendental equation, so it has to be inverted numerically.

Newton–Raphson does it quickly, because the derivative is unusually tidy:

f(α) = tan α − α − y    f′(α) = sec²α − 1 = tan²α

One catch: f′(0) = 0, so the iteration crawls near the origin. The series seed α₀ = ∛(3y) − 2y/5 (from inverting inv α ≈ α³/3) starts close enough that a handful of steps reach machine precision. This calculator shows every iteration.

Reference values

FAQs

Why must α be in radians?

Because tan(α) returns a pure number while α is an angle, and the subtraction only makes sense if both are measured the same way. Radians are the natural choice: an angle in radians is an arc length divided by a radius, so tan α − α is a genuine geometric difference. Feed degrees into the formula and you get nonsense — inv(20) computed in degrees is off by a factor of about 57.

Is inv(α) an angle?

Yes — it is the polar angle in radians between the start of the involute and the point at pressure angle α. It is almost always quoted as a bare number, which hides that fact.

Why is inv(α) always positive and always increasing?

Because tan α > α for every α in (0, 90°). The derivative is tan²α ≥ 0, so inv is strictly increasing on the whole domain — which is exactly why a single inv value maps back to a single pressure angle, and the inverse is well defined.

Why is inv(α) ≈ α³/3?

The Taylor series of tan α starts α + α³/3 + 2α⁵/15 + …, so the α terms cancel and the cubic leads. That cubic growth is why inv(20°) ≈ 0.0149 is so small, and why gear tables need so many decimal places.

Can the string unwrap forever?

Mathematically yes — the involute of a circle spirals outward without limit. On a real gear only a short stretch is used, from the base circle out to the tip circle. Anything below the base circle is not an involute at all: the flank there is whatever fillet the cutter happens to leave.