Planar projections of spatial Pythagorean-hodograph curves

• Published in: Computer aided geometric design 2021-11, Vol.91, p.102049, Article 102049
• Main Authors: Farouki, Rida T., Knez, Marjeta, Vitrih, Vito, Žagar, Emil
• Format: Article
• Language: English
• Subjects: Helical curves; Planar orthogonal projections; Polynomial factorizations; Pythagorean–hodograph curves; Quaternions; Spatial rotations
• ISSN: 0167-8396; 1879-2332
• DOI: 10.1016/j.cagd.2021.102049

Although the orthogonal projection of a spatial Pythagorean–hodograph (PH) curve on to a plane is not (in general) a planar PH curve, it is possible to construct spatial PH curves so as to ensure that their orthogonal projections on to planes of a prescribed orientation are planar PH curves. The construction employs an analysis of the root structure of the components of the quaternion polynomials that generate spatial PH curves, and it encompasses both helical and non–helical spatial PH curves. An initial characterization for orthogonal projections of spatial PH curves on to the coordinate planes provides the basis for a generalization to projections of arbitrary direction, based on unit quaternion rotation transformations of R3. Examples of the special class of spatial PH curves (gray) that possess planar PH curves (blue) as their projections on to one of the coordinate planes. •The conditions under which spatial PH curves admit planar PH projections are characterized.•Spatial PH curves with planar PH projections include helical and non-helical polynomial curves.•The product of sums of squares of polynomials must equal the perfect square of a polynomial.•The case of projections onto a coordinate plane is extended to projections onto arbitrary planes.