For a space curve X(t), the osculating helix at a point X(t0) is the cylindrical helix that matches the starting point X(t0) with the same unit tangent vector T(X)(t0), the same principal normal P(X)(t0), the same binormal vector B(X)(t0) and the same curvature k(t0) and the same torsion q(t0). Changing the value of t0 moves the osculating helix along the curve X(t).

The osculating helix at X(t0) is Z(u) defined over the interval -2*π ≤ u ≤ 2*π, with the equation Z(u) = a(t0)*sin(u)*E1(t0) + a(t0)*(1 - cos(u))*E2(t0) + b(t)*t*E3(t0). Here E1(t0), E2(t0), E3(t0) is an orthonormal frame where E1(t) = unit(a(t)*T(X)(t0) + b(t)B(X)(t0)), E2(t0) = P(X)(t0), and E3(t) = unit(b(t)*T(X)(t0) - a(t)*B(X)(t0)). Here a(t0) = k(t0)/(k(t0)^2 + q(t0)^2)^(1/2) and b(t0) = q(t0)/(k(t0)^2 + q(t0)^2)^(1/2).

The default curve is the space cardioid X(t) = ((1+cos(t))*cos(t), (1+cos(t))*sin(t), sin(t)) over the domain interval -π ≤ t ≤ π. Other significant examples are a cylindrical helix X(t) = (cos(t), sin(t), t) and a saddle curve X(t) = (cos(t), sin(t), sin(2*t)). Note that when the torsion q(t0) of the curve is zero, the osculating helix specializes to a multiply covered circle osculating circle.