The Trace

The combustion chamber of a Wankel rotary engine is not drawn. It is traced. A triangular rotor orbits eccentrically inside a housing. The tips of the rotor follow a path — a peritrochoid, a specific curve from the family of epitrochoids generated by a point on a circle rolling around the outside of another circle. The housing wall is the envelope of that path. It is the surface the rotor tips sweep.

Felix Wankel patented the principle in 1929, and the first working engine ran at NSU in 1957. The engine has no pistons, no connecting rods, no reciprocating motion. The rotor spins continuously. The chamber shape, which looks organic and almost arbitrary in cross-section — two shallow lobes pinched at the waist — is in fact the only shape that allows continuous contact between the three apex seals and the housing wall throughout the rotor's eccentric orbit. There is no design freedom in the chamber profile. Given the rotor geometry and the eccentricity ratio, the housing shape is determined. It is not selected from alternatives. It is derived from the motion.

The property that matters is contact. The rotor must seal three separate volumes — intake, compression-combustion, and exhaust — simultaneously. If the apex seals lose contact with the housing at any point in the rotation, the engine fails. The peritrochoid guarantees contact because the housing IS the path the seals follow. The shape cannot lose them because it was made by them.

The involute of a circle is the path traced by the end of a taut string as it unwinds from a cylinder. Leonhard Euler proposed it for gear tooth profiles in 1765. Philippe de La Hire arrived at the same curve independently. The shape replaced the cycloidal profiles that had been standard since Desargues and de La Hire's earlier work, and it replaced them for a single reason: tolerance.

Two gears with cycloidal teeth transmit rotation at a constant velocity ratio only if the center-to-center distance is exact. Any deviation — thermal expansion, bearing wear, manufacturing imprecision — causes the velocity ratio to fluctuate, producing vibration and accelerated wear. Two gears with involute teeth maintain a constant velocity ratio regardless of small variations in center distance. The contact point between meshing involute teeth always lies on a straight line — the line of action, tangent to both base circles. Moving the gears slightly closer or farther apart shifts where on the teeth contact occurs but does not change the geometry of the contact. The ratio holds.

This property is not added to the involute. It is the involute. The curve is defined as the path of a point maintaining a taut tangent to a circle. That tangent IS the line of action. The tolerance for error does not come from careful engineering of the tooth shape. It comes from the fact that the tooth shape is the trace of a string, and a string's geometry does not depend on where the cylinder is.

Before the involute became standard, gears were fitted in matched pairs. After, gears could be manufactured to a profile specification and meshed with any other gear of the same pitch and pressure angle. The industrial implication was interchangeability. The geometric reason was that the shape, defined by unwinding, carried its tolerance with it.

In 1857, Jules Antoine Lissajous mounted small mirrors on the prongs of two tuning forks oriented at right angles. A beam of light reflected off one mirror, then the other, and struck a screen. Each fork vibrated sinusoidally. The light traced the combination of the two perpendicular oscillations.

When the forks vibrated at the same frequency, the trace was an ellipse — collapsing to a straight line when the oscillations were in phase, opening to a circle at ninety degrees. When one fork vibrated at twice the frequency of the other, the trace was a figure-eight. At a ratio of two to three, a more complex closed curve. At any simple integer ratio, the figure was stable — the light retraced the same path on every cycle. At any ratio that was not a simple fraction, the figure drifted, rotating slowly, never closing.

Lissajous used this for acoustic calibration. To check whether an unknown tuning fork matched a reference, you set them perpendicular and watched the figure. A stable ellipse meant the frequencies were identical. A slowly rotating ellipse meant they were close but not equal, and the rotation rate told you by how much. The shape was not a picture of the sound. It was a measurement of the relationship between two sounds, performed by the light that passed through both.

Before electronic frequency counters existed, oscilloscope Lissajous patterns were the standard method for calibrating signal generators against reference oscillators. The engineer adjusted the unknown frequency until the figure on the screen stopped drifting. The ratio was read directly from the shape — the number of tangencies along each axis. A figure touching the top three times and the side twice was a three-to-two ratio. The figure did not represent the ratio. It was the ratio, inscribed in phosphor by the oscillations themselves.

The Wankel chamber is the path of the rotor. The involute tooth is the path of a string. The Lissajous figure is the path of combined oscillations. None of these shapes was designed by specifying coordinates. Each was generated by a motion, and the motion deposited a property in the shape: contact in the first case, tolerance in the second, measurement in the third. The property was not computed and then built. It was traced, and it arrived already present.