Characterization and construction of helical polynomial space curves

Helical space curves are characterized by the property that their unit tangents maintain a constant inclination with respect to a fixed line, the axis of the helix. Equivalently, a helix exhibits a circular tangent indicatrix, and constant curvature/torsion ratio. If a polynomial space curve is helical, it must be a Pythagorean-hodograph (PH) curve. The quaternion representation of spatial PH curves is used to characterize and construct helical curves. Whereas all spatial PH cubics are helical, the helical PH quintics form a proper subset of all PH quintics. Two types of PH quintic helix are identified: (i) the “monotone-helical” PH quintics, in which a scalar quadratic factors out of the hodograph, and the tangent exhibits a consistent sense of rotation about the axis; and (ii) general helical PH quintics, which possess irreducible hodographs, and may suffer reversals in the sense of tangent rotation. First-order Hermite interpolation is considered for both helical PH quintic types. The helicity property offers a means of fixing the residual degrees of freedom in the general PH quintic Hermite interpolation problem, and yields interpolants with desirable shape features.