coil curve | cylindrical spiral

x(t) = r cos(t) y(t) = r sin(t) z(t) = c t

x^2 = r^2 - y^2 ∧ x/r = cos(z/c)

left bracketing bar x(t) right bracketing bar = sqrt(c^2 t^2 + r^2)

T^^(t) = (-(sin(t) r)/sqrt(c^2 + r^2), (cos(t) r)/sqrt(c^2 + r^2), c/sqrt(c^2 + r^2))

N^^(t) = (-cos(t), -sin(t), 0)

B^^(t) = ((sin(t) c)/sqrt(c^2 + r^2), -(cos(t) c)/sqrt(c^2 + r^2), r/sqrt(c^2 + r^2))

s(t) = t sqrt(c^2 + r^2)

κ(t) = r/(c^2 + r^2)

τ(t) = c/(c^2 + r^2)

parametric space curves | spiral space curves