How this instrument works
An involute of a circle is the path traced by the free end of a taut string as it unwinds from a spool of radius r. At roll angle θ the string has come free along an arc of length rθ, and because the string stays straight and tangent to the circle, that whole unrolled length gets added in the tangent direction rather than curving with the circle. The result is a curve that starts on the circle and spirals steadily outward.
The formula splits cleanly into two motions. The cosθ and sinθ terms place the point of tangency on the circle as it rotates; the θsinθ and θcosθ terms add the straight, unrolled length of string, pointing along that same tangent line. Because one part rotates like a circle and the other grows in a straight line, the involute is not a circle, an ellipse, or any other conic — it is a transcendental curve whose distance from the center grows without bound as more string comes free.
At θ = 0 no string has unrolled yet, so the curve sits exactly on the base circle at (r, 0) — the one point the involute and its own generating circle share. Run θ into negative values and the string unwinds the other way, drawing the mirror-image curve on the far side of that same starting point. The circle itself is called the evolute of the involute, since it is exactly the locus of the involute's centers of curvature at every point along the curve.
- Enter the spool's size in Base circle radius — this is r, the circle the string is wound around.
- Enter how far the string has unrolled in Roll angle, θ — use the unit selector next to it if you would rather work in degrees or turns.
- Read x and y for the coordinates of the string's free end at that roll angle.
- Raise θ in small steps and watch x and y trace the outward spiral, one arc-length rθ at a time.
Worked example — one radian off a radius-2 spool
Take a spool with base circle radius r = 2 and unroll one radian of string, so θ = 1. Then x = 2(cos 1 + 1·sin 1) = 2 × 1.38177329 = 2.76354658 and y = 2(sin 1 − 1·cos 1) = 2 × 0.30116868 = 0.60233736.
That point sits at distance √(2.76354658² + 0.60233736²) = √8 = 2√2 ≈ 2.82843 from the center — exactly r√(1+θ²) with r = 2 and θ = 1, confirming the point has moved well clear of the radius-2 circle it started on.
Questions
What does the involute of a circle actually trace?
It traces the path swept by the free end of a taut string as it is unwound from a spool — the base circle. Hold the string taut against the circle, unwrap it steadily, and the end draws the involute; that is also, exactly, the shape ground into the flank of a standard gear tooth.
Why does the formula mix trigonometric terms with a bare θ?
Because two different motions are added together. cosθ and sinθ place the point of tangency on the rotating circle; θsinθ and θcosθ add the straight length of string, rθ, that has come free along that tangent direction. One term rotates, the other grows linearly, and that combination is what makes the curve transcendental rather than a simple conic section.
Why do gear teeth use an involute profile specifically?
Because the line perpendicular to an involute at any point is always tangent to its base circle, so a pair of meshing involute teeth push against each other along a fixed line of action. That keeps the pressure angle, and so the direction of force transfer, constant through the whole mesh, which is what gives involute gears their smooth engagement even if the center distance is slightly off.
How far does the traced point sit from the circle's center?
Exactly r√(1+θ²). Squaring and adding the x and y formulas cancels every cross term, leaving x² + y² = r²(1+θ²) regardless of θ. At θ = 1 that distance is r√2 — for a radius-2 circle, 2√2 ≈ 2.828, noticeably past the circle's own edge at 2.
What is the difference between an involute and an evolute?
They describe the same pair of curves from opposite directions. The involute is the path traced by unwinding a string from a curve; the evolute is the locus of that base curve's centers of curvature. Here, the base circle is the evolute and the traced spiral is its involute — the circle's own curvature centers are, at every point, exactly where the taut string was still touching it.