x(t) = (r cos(t))/sqrt(a^2 t^2 + 1) y(t) = (r sin(t))/sqrt(a^2 t^2 + 1) z(t) = -(a r t)/sqrt(a^2 t^2 + 1)

x^2 + y^2 + z^2 = r^2 ∧ y/x = -tan(z/(a sqrt(r^2 - z^2)))

left bracketing bar x(t) right bracketing bar = r

T^^(t) = ((-sin(t) - t (cos(t) + t sin(t)) a^2)/sqrt((1 + t^2 a^2) (1 + (1 + t^2) a^2)), (cos(t) + t^2 cos(t) a^2 - t sin(t) a^2)/sqrt((1 + t^2 a^2) (1 + (1 + t^2) a^2)), -a/sqrt((1 + t^2 a^2) (1 + (1 + t^2) a^2)))

N^^(t) = ((t sin(t) (a + t^2 a^3)^2 + cos(t) (-1 - (2 + 3 t^2) a^2 - (1 + 2 t^2 + 3 t^4) a^4 - t^6 a^6))/sqrt((1 + t^2 a^2) (1 + (1 + t^2) a^2) (4 t^6 a^6 + t^8 a^8 + (1 + a^2)^3 + 3 t^4 a^4 (2 + a^2) + t^2 a^2 (4 + 6 a^2 + 3 a^4))), (-t cos(t) (a + t^2 a^3)^2 - sin(t) (1 + (2 + 3 t^2) a^2 + (1 + 2 t^2 + 3 t^4) a^4 + t^6 a^6))/sqrt((1 + t^2 a^2) (1 + (1 + t^2) a^2) (4 t^6 a^6 + t^8 a^8 + (1 + a^2)^3 + 3 t^4 a^4 (2 + a^2) + t^2 a^2 (4 + 6 a^2 + 3 a^4))), (t a^3 (2 + (1 + 2 t^2) a^2))/sqrt((1 + t^2 a^2) (1 + (1 + t^2) a^2) (4 t^6 a^6 + t^8 a^8 + (1 + a^2)^3 + 3 t^4 a^4 (2 + a^2) + t^2 a^2 (4 + 6 a^2 + 3 a^4))))

B^^(t) = ((t cos(t) a^3 - sin(t) a (1 + (1 + t^2) a^2))/sqrt(4 t^6 a^6 + t^8 a^8 + (1 + a^2)^3 + 3 t^4 a^4 (2 + a^2) + t^2 a^2 (4 + 6 a^2 + 3 a^4)), (t sin(t) a^3 + cos(t) (a + (1 + t^2) a^3))/sqrt(4 t^6 a^6 + t^8 a^8 + (1 + a^2)^3 + 3 t^4 a^4 (2 + a^2) + t^2 a^2 (4 + 6 a^2 + 3 a^4)), (1 + (1 + 2 t^2) a^2 + t^4 a^4)/sqrt(4 t^6 a^6 + t^8 a^8 + (1 + a^2)^3 + 3 t^4 a^4 (2 + a^2) + t^2 a^2 (4 + 6 a^2 + 3 a^4)))

s(t) = r (log((sqrt(a^2 (t^2 + 1) + 1) + a t)/sqrt(a^2 + 1))/a + tan^(-1)((a^2 t)/sqrt(a^2 (t^2 + 1) + 1)))

κ(t) = sqrt(a^8 t^8 + 4 a^6 t^6 + (a^2 + 1)^3 + 3 (a^2 + 2) a^4 t^4 + (3 a^4 + 6 a^2 + 4) a^2 t^2)/(r (a^2 (t^2 + 1) + 1)^(3/2))

τ(t) = -(a (a^2 t^2 + 1)^(3/2) (a^4 t^4 + 4 a^4 t^2 + 2 a^2 t^2 + a^2 + 1))/(r (a^8 t^8 + 4 a^6 t^6 + 3 a^6 t^4 + 3 a^6 t^2 + a^6 + 6 a^4 t^4 + 6 a^4 t^2 + 3 a^4 + 4 a^2 t^2 + 3 a^2 + 1))

parametric space curves | spiral space curves