Space curve

A space curve is a curve in three-dimensional Euclidean space.

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.

The displayed space curve, $s>0$, lies on a cone and is parametrized by arc length. It has a constant tangent angle with its axis, while the turn spacing shrinks geometrically toward the limiting apex. Its curvature is $1/(\sqrt{2}s)$: differentiate its unit tangent with respect to $s$. Infinitely many turns accumulate at the apex with finite arc length; the apex itself is not a regular point of the curve.

The pitch is the axial displacement per complete turn. For a circular helix with $z=a\varphi +b$, the signed pitch for increasing angle is $2\pi a$, and its magnitude is $2\pi |a|$. Pitch is independent of the cylinder radius.