Placidus intermediate cusps using iterative semi-arc trisection.
Returns null inside the polar circle, |latitude| ≥ 90° − ε, where part of
the ecliptic never rises or sets and the semi-arcs are undefined.
Each cusp's right ascension α solves α = RAMC + offset + m·AD(α), with AD
the ascensional difference asin(tan φ tan δ(α)) and m = 1/3 or 2/3. With
tan δ = tan ε sin α, |dAD/dα| <= q = |tan φ| tan ε < 1 outside the polar
circle, so the map is a contraction with slope at most 2q/3. The iteration
settles to 1e-9° within 64 steps except within about 1e-8° of the polar
limit, near the sidereal times at which a cusp's right ascension reaches 90°
or 270°: it needs up to 63 steps there, and the rounding of asin near ±1
moves each step by about the tolerance. Where it does not settle, the cusp
is found by bisection on RAMC + offset ± (90° m + 1°), across which
α − RAMC − offset − m·AD(α) increases strictly, so below the limit Placidus
never falls back. Near the limit, either way, a cusp carries the rounding of
asin near ±1: up to 6.5e-7° one unit in the last place below it.
Placidus intermediate cusps using iterative semi-arc trisection. Returns
nullinside the polar circle, |latitude| ≥ 90° − ε, where part of the ecliptic never rises or sets and the semi-arcs are undefined.Each cusp's right ascension α solves α = RAMC + offset + m·AD(α), with AD the ascensional difference asin(tan φ tan δ(α)) and m = 1/3 or 2/3. With tan δ = tan ε sin α, |dAD/dα| <= q = |tan φ| tan ε < 1 outside the polar circle, so the map is a contraction with slope at most 2q/3. The iteration settles to 1e-9° within 64 steps except within about 1e-8° of the polar limit, near the sidereal times at which a cusp's right ascension reaches 90° or 270°: it needs up to 63 steps there, and the rounding of asin near ±1 moves each step by about the tolerance. Where it does not settle, the cusp is found by bisection on RAMC + offset ± (90° m + 1°), across which α − RAMC − offset − m·AD(α) increases strictly, so below the limit Placidus never falls back. Near the limit, either way, a cusp carries the rounding of asin near ±1: up to 6.5e-7° one unit in the last place below it.