Roulette Loops

• Reads the self crossing count of one trochoid as an exact step function of the pencil's reach |p|, past the loop threshold |p| = 1, and the crossing count of two trochoids drawn from one wheel.
• The object is mrlynum::spirograph: a wheel of radius r rolls inside or outside a circle with R/r = a/b in lowest terms, a pencil sits |p| wheel radii from the wheel's centre, and trace returns the polyline every count is read back from.
• Write s = -1 inside and s = +1 outside, rho = a + s b, m = a + 2 s b. In wheel radii and in the closing parameter phi of [0, 2 pi) a pencil at the real seat |p| draws z(phi) = rho e^(i b phi) + |p| b e^(i s rho phi).
• With sigma = (phi + psi)/2 and delta = (phi - psi)/2, z(phi) = z(psi) reads rho sin(b delta) + |p| b sin(s rho delta) e^(i s a sigma) = 0, so sigma is a multiple of pi / a and the crossing equation is rho sin(b delta) = e |p| b sin(rho delta) with e = 1 or e = -1.
• The reflection of the curve in the line of angle b sigma fixes the crossing, so every self crossing lies on one of the a mirror lines k pi / a through the centre, inside and outside alike. A pencil at seat angle alpha turns the whole curve by -s b alpha / a, so its mirror lines are -s b alpha / a + k pi / a; the bare k pi / a needs a real seat.
• Not every mirror line has to carry a crossing: 4/1 outside puts all four on two of its four lines.
• (a/2)(Z_1 + Z_-1), with Z_e the roots of the crossing equation in (0, pi), counts unordered parameter pairs, so it is read off a one dimensional root count and never off a picture; it is the number of distinct double points exactly when the curve has no point of multiplicity above two and no self contact.
• The one reach with a multiple point is |p| = rho / b: there z = 0 at the a parameters phi = (2j+1) pi / a, so C(a, 2) parameter pairs collapse into the centre and the point count is C(a, 2) - 1 below the pair count. Elsewhere the two agree, because a multiple point needs every pairwise delta to be a multiple of pi / a, and such a delta solves the crossing equation only there.
• A step of the count is a double root: the crossing equation and its delta derivative cos(b delta) = e |p| cos(rho delta) together force a sin(m delta) = m sin(a delta), equivalently sinc(m delta) = sinc(a delta), with the reach |p| = |cos(b delta) / cos(rho delta)|, read as |rho sin(b delta)| / |b sin(rho delta)| where both cosines vanish, which is delta = pi/2 when a is even.
• A threshold is a tangency angle delta in [0, pi), not a reach. Several angles can share one reach, and then the step there is a multiple of a; the count |rho - b| is a count of angles with multiplicity.
• delta = 0 and delta = pi are roots of the crossing equation at every reach, and a root is born at each end as the reach passes 1, since f'(0) = rho b (1 - e |p|) and f'(pi) = rho b ((-1)^b - e |p| (-1)^rho). The single entry at delta = 0 stands for both births, which is why the jump at reach 1 is a and not a/2.
• On a tangency reach the curve touches itself: the two branches meet with equal tangents, a tacnode, and the meeting count is the transversal count below plus a/2 for each tangency angle at that reach. The step function is read on the open intervals between tangency reaches, never on one.
• Swapping the wheel frequency b and the rim frequency rho fixes every tangency angle, since {a, |m|} = {rho + b, |rho - b|} is symmetric, and inverts every reach, since |cos(b delta) / cos(rho delta)| inverts. So the thresholds depend on the ordered pair (b, rho), and the falling a < 2b staircase is the reciprocal of the rising a > 2b one.
• sin(k delta) and cos(k delta) are expanded exactly in u = sin^2(delta) by the integer recursion of Sakhnovich 2023, theorems 2.1 and 2.5, so the tangency equation becomes u Q(u) = 0, times cos(delta) when a is even, for an explicit integer polynomial Q, and |p|^2 = T(u)/B(u) for explicit integer polynomials T and B.
• The tangency equation integrates: a sin(m delta) - m sin(a delta) is 2 a m times W(delta) = int_0^delta sin(b t) sin(rho t) dt up to sign, W' vanishes on (0, pi) only at j pi / rho and k pi / b, and there W is exactly (-1)^(j+1) sin(b j pi / rho) rho / (rho^2 - b^2) and (-1)^k sin(rho k pi / b) b / (rho^2 - b^2), so the threshold count is a sign count over a merged Farey sequence with no numerics in it.
• The root scan starts at 0.4 / max(a, |m|) and hides nothing: W' > 0 on (0, pi / max(b, rho)) and max(a, |m|) = b + rho exceeds max(b, rho), so no root lies below the cut. The germ at pi makes W monotone and zero free on the last interval, so the endpoint contributes no sign change and the walk carries only the interior critical values.
• Two pencils p and q on one wheel meet at parameter pairs where rho^2 sin^2(b delta) = b^2 (P sin^2(rho delta) + E cos^2(rho delta) - s X sin(2 rho delta)) with P = |p + q|^2/4, E = |p - q|^2/4 and X = Im(q conj(p))/2, one root count in [0, 2 pi) times a/2; the equation is pi periodic in delta, so the root count is even and the halving is exact.
• The pair law needs the two curves distinct, which is the coincidence law mrlynum::spirograph::distinct already carries. It counts parameter pairs; the point count needs the further condition that no self crossing of one curve lies on the other, and the centre correction C(a, 2) - 1 per seat that sits at reach rho / b.
• Everything is checked three ways that do not share a step: the closed step function, the root count of the crossing equation, and the crossing count of the mrlynum::spirograph::trace polyline at two sample counts.
• A polyline count blurs inside a tangency, so a threshold is decided by the algebra and the trace only brackets it; the sweep's grid is offset off the rational thresholds so no sample lands on one.
• The hypotheses on the step function are |p| > 0, |p| not a tangency reach, |p| not rho / b, and m != 0; a = 2b inside, which only 2/1 reaches, has m = 0, no tangency and no crossing except at |p| = 1, where the curve is a doubled segment.
• What is taken from the literature: Wieleitner 1908 has the reach 1 trichotomy and the mirror line fact for the looped epitrochoid, qualitatively; Jaekel has both staircase endpoints a (b - 1) and a (rho - 1), the split at a = 2b, and the step sizes a and 2a, with the transition reaches solved numerically; Sakhnovich 2023 has the sin^2 recursion. What is added here: the tangency equation and the reaches in closed form, the sign count, the mirror lines with their seat hypothesis and a bound, the centre correction, the reciprocity, and the pair law.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p roulette-loops
• About three minutes, single threaded; prints only, writes nothing, holds a few megabytes.

WITNESSES

• The mirror lines: at 24001 samples the worst distance from a trace self crossing to the nearest mirror line is 4.33e-7 of the frame, over a/b in 5/1, 7/2, 8/3 inside and 5/2, 7/3 outside, reaches 1.4, 2.6, 3.9, seat angles 0 and 0.3, 30 cells. The floor is trace returning f32, not the geometry.
• The step function: a (b - 1) below the first tangency reach, a (rho - 1) above the last, and a sign(m) per tangency angle; the number of angles in [0, pi), counting delta = 0, is |rho - b|, the same integer as min(|m|, a).
• The sign count: over all 29450 coprime frequency pairs b, rho up to 220 the exact integer sign count of W returns |rho - b| every time, in integer arithmetic with no root finding.
• The census: 179 coprime fractions a/b with a at most 24, 357 cases over the two sides with m nonzero, the angle count equal to the sign count |rho - b|, the limit equal to a (rho - 1), and the closed form equal to the crossing equation's root count at the midpoint of every step, with 0 failures.
• The reciprocity: over 210 swaps of b and rho with b + rho at most 26, the worst deviation of a threshold times its partner from 1 is 1.47e-13.
• The sweep: 25578 reads of mrlynum::spirograph::trace over |p| from 0.5 to 4 in 203 steps, b in 1..6 and a in b+1..11 coprime, inside and outside, at 4001 and 12001 samples, disagree with the closed form 0 times; every jump bracket is 0.0173 wide or less and holds exactly the predicted angles.
• The contact: at |p|^2 = 27/2 the hypotrochoid 5/1 has 2 tangency angles and 2 simple roots, so 10 meetings against 5 below and 15 above, the two branches closing to 8.88e-16 at radius 2.041241; 7/2 inside gives 21 against 14 and 28, 11/4 inside 55 against 44 and 66, 3/1 outside 6 against 3 and 9, 5/2 outside 15 against 10 and 20 and 25 against 20 and 30.
• The births: 3/1 inside has no root in (0, pi) at reach 0.98 and two at reach 1.02, 0.1984 on e = 1 and 2.9432 on e = -1, count 3; 7/2 inside goes from 1.4235, 1.7181 and count 7 to those two plus 0.0750 and 3.0666 and count 14.
• The centre: at |p| = rho / b the trace's distinct point count is the step function less C(a, 2) - 1 on every odd a read, 1 at 3/1, 5/2, 5/3, 7/4 inside, 6 at 5/1 and 5/4 inside, 8 at 7/2 inside, 7 at 3/1 and 10 at 3/2 and 21 at 5/2 outside, while three percent either side the two counts agree.
• The thresholds of a/b = 5/1 inside: 1 and 3.674234614175 twice, exactly |p|^2 = 27/2, at u = 5/6 on Q(u) = 40 - 48 u; the count is 0, then 5, then 15.
• The thresholds of a/b = 7/2 inside: 1 and 2.353415666603 twice, exactly |p|^2 = (81 + 21 sqrt 21)/32, at u the smaller root of Q(u) = 192 u^2 - 336 u + 140; the count is 7, then 14, then 28.
• The thresholds of a/b = 5/1 outside: 1, 4.180967894379 twice and 5.789603394549 twice, exactly |p|^2 = (102 - 7 sqrt 21)/4 and (102 + 7 sqrt 21)/4 on Q(u) = -320 u^2 + 448 u - 140; the count is 0, then 5, then 15, then 25.
• When a is even, delta = pi/2 is always a tangency and its reach is the rational |p| = rho / b, which is also the centre reach: 3 at 4/1 inside, 5/3 at 8/3 inside, 13/5 at 8/5 outside. There the centre correction and a step fall together and the step function is not read.
• The pair law against the trace: 48 reads over 5/1, 7/2, 8/3 inside and 5/2 outside at reaches 0.6, 1.3, 2.4, 3.7 against three seat kinds, equal reach, shorter reach and the wheel's centre, agree on the parameter pair count exactly.
• The pair point count parts from it: 3/1 inside with one seat at reach sqrt 5 - 1 and one at the wheel's centre has 6 parameter pairs and 3 points, since all three self crossings of the first curve sit on the circle of radius rho the second draws; two percent either side gives 6 points.
• The centre pencil draws the circle of radius rho, the law collapses to |sin(b delta)| = b |p| / (2 rho), and the pair count is 2 a b below |p| = 2 rho / b and 0 above: 48 then 0 across 10/3 at 8/3 inside.
• The pair count 2 a b survives the loop threshold |p| = 1, which is a tangency of one curve with itself and never of two curves with each other: for two seats at one reach and half angle nu apart it first fails at 2.242763, 1.741061, 1.379486, 1.143270, 1.047854, 1.010207, 1.002194 for 5/1 inside at nu = 1, 0.5, 0.2, 0.05, 0.01, 0.001, 0.0001, excess falling 4.69 then 4.65 per decade against 10^(2/3) = 4.64. Seven samples at one fraction; the shape is a reading, not a bound.
• The pair threshold is not monotone in nu: 8/3 inside gives 1.117596 at nu = 1 against 1.523254 at nu = 0.5.
• mrlynum::spirograph::trace, frame and track are the only outside calls; the counters for one polyline and for two are carried here.