The teeth shown are derived from an analytic path of contact (see
README-derivation). Free parameters: tooth count Z₁, tooth difference d, tooth module m, centre offset e, and the
shape numbers (contact-window slip τ, tip height h, working depth w, tooth thickness, root clearance). Derived: the path of
contact (an Archimedes spiral about the pinion centre, a and b fixed by the shape numbers), the rotation φ(θ) along it from the
Willis condition, and both flanks as exact conjugates. Valid ranges come from the library's constraints().
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The teeth are laid out on reference circles of the tooth module (free: Z₁, m), so each tooth has the same pitch π·m and the rim curvature falls as Z grows. The pitch (rolling) radii R = eZ/d are derived from the offset and are much larger.
$$r_1=\tfrac{Z_1 m_t}{2},\quad r_2=\tfrac{Z_2 m_t}{2},\quad p=\pi m_t,\quad \Delta\theta_1=\tfrac{2\pi}{Z_1},\quad \kappa_1=\tfrac1{r_1}$$
d is free and fixes Z₂ = Z₁ − d. With the ring fixed and the carrier as input, the reduction ratio Z₂/d is derived, as is the reference-radius gap r₁ − r₂ = d·m/2. A larger d forces a larger eccentricity (panel 3) and fewer teeth in contact. Tested for d = 1…4.
$$Z_2=Z_1-d,\qquad i_{\text{fixed ring}}=\frac{Z_2}{d},\qquad r_1-r_2=\frac{d\,m_t}{2}$$
Path of contact (green dots: contact points measured on the model over one tooth period, drawn in the carrier frame, which is the ring frame at the drawn instant; dashed: spiral fitted about the pinion centre O₂). They lie on the contacting flanks.
e is free. The pitch radii R = e·Z/d and the contact window θ = asin(τ/e) are derived from it: the path of contact is the spiral ρ(θ) = a + bθ about the pinion centre with a and b fixed by the tip height and working depth (README-derivation), and both flanks are the exact Willis conjugates of that path. The default e = √((d/2+ρ₀)²+τ₀²) m puts a tooth's mid-contact pose (ρ₀, τ₀) = (0.256, 0.640) m, relative to its socket, on the path (0.99 m for d = 1, 1.41 m for d = 2). A larger d means a larger e, so the window spans fewer teeth and contact moves toward the centre; the centre tooth carries no load.
$$\rho=e\cos\theta-\tfrac d2,\quad \tau=e\sin\theta,\quad e^2=(d/2+\rho_0)^2+\tau_0^2,\quad R_{1,2}=\frac{e\,Z_{1,2}}{d}$$ $$\rho(\theta)=a+b\theta,\quad a=r_2+hm-b\theta_1,\quad b=\frac{-wm}{\theta_2-\theta_1},\quad \frac{d\varphi}{d\theta}=-\left[1+b\left(\frac{1}{R_2\sin\theta}-\frac{\cos\theta}{\rho\sin\theta}\right)\right]$$
All three are free, in modules. The exact conjugate flanks are continued past the end of the
contact window by tipMargin + tipRadius (pinion past θ₁, ring past θ₂), then each tip corner is rounded with a circle of radius
tipRadius tangent to the flank and the tip circle, so the fillet starts beyond the last contact point and the working flank is untouched.
Each root circle is socketRelief deeper than the deepest reach of the mating tip at top dead centre. Below each tooth's lowest contact point the flank turns, through a short tangent blend (radius 0.1 m), into a straight radial wall (no undercut, no lean) that meets the root circle with a tangent fillet whose radius equals the mating tip radius (tipRadius drives both). The mating tip, past its last contact, continues on its own flank and then rounds, so it clears the radial wall. The wall starts where the mating tip's swept envelope is at least wallClearance from the wall line (all the way down to the root); if the short blend ends higher than that, a gentle tangent-continuous cubic carries the flank down to the wall start instead. Limits: the two fillets must fit on
the tip flat (tipRadius ≤ max from constraints()), relief ≤ 0.3 m for root strength.
$$r_{f1}=r_{t2}+e+s,\quad r_{f2}=r_{t1}-e-s$$ $$\rho(\theta_1-\Delta_1)=\rho(\theta_1)+(\mu+r_c)m,\quad |C(\theta_2+\Delta_2)|=|C(\theta_2)|-(\mu+r_c)m$$
ringpinionhighlighted by the panelpath of contactreference circles