Circle involutes

involutes.py

import math


def involute_point(a, t):
    c = math.cos(t)
    s = math.sin(t)
    d = t - a
    return c + d * s, s - d * c


def involute_slope(a, t):
    return math.tan(t)


def t_for_y0(a):
    assert a < 0
    t0 = 0
    t1 = math.pi / 2
    for _ in range(30):
        t = (t0 + t1) / 2
        _, y = involute_point(a, t)
        if y < 0:
            t0 = t
        else:
            t1 = t
    return (t0 + t1) / 2


def a_for_slope_at_y0(m):
    a0 = 0
    a1 = -10
    for _ in range(30):
        a = (a0 + a1) / 2
        t = t_for_y0(a)
        slope = involute_slope(a, t)
        if slope < m:
            a0 = a
        else:
            a1 = a
    return (a0 + a1) / 2


def a_for_slope_at_y0_2(m):
    return math.atan(m) - m


print(t_for_y0(-0.5))
print(involute_slope(-0.5, t_for_y0(-0.5)))
print(a_for_slope_at_y0(1.0))
print(a_for_slope_at_y0_2(math.tan(20*math.pi/180)))