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A general helix is a regular space curve whose unit tangent vector makes a constant angle with a fixed axis direction $u$. In arc length parameter $s$, this means ${\gamma }^{\prime }(s)\cdot u$ is constant. Equivalently, the axial coordinate grows at a constant rate per unit arc length.

The familiar circular helix has constant distance from its axis:Its speed is $\sqrt{R^2+a^2}$, and its tangent's axial component is $a/\sqrt{R^2+a^2}$. Its signed pitch of a helix is $2\pi a$. A general helix need not have constant radius or constant axial displacement per turn: the logarithmic conical helix has a varying radius while retaining a constant tangent-axis angle.

For a regular space curve parametrized by arc length $s$, its curvature is $\kappa =\Vert dT/ds\Vert$, where $T=dx/ds$ is the unit tangent. Differentiating the given parametrization givesThe curve approaches the origin as $t\to -\infty$; the origin is a limiting endpoint. Its arc length measured from that endpoint is $s={\int }_{-\infty }^t{e}^{u}du=e^t$, soThis is a logarithmic conical helix. Its unit tangent and derivative areConsequently,The infinitely many turns near the limiting origin have finite total arc length, while the curvature diverges there.