Planar projections of spatial Pythagorean-hodograph curves

Development and Definition of Biquadratic Transformations Using Spheres and Two-Way Hyperboloids

“…The author reviews previously published materials on Cremona transformations' construction and use. Farouki et al [6] proposed a method for obtaining special Cremona transformations of three-dimensional space. The mapping described in this study is constructive in nature, and it generates not only a model of projective spatial geometry on a plane, but allows us to practically attribute their images to specific spatial objects in the sense of descriptive geometry.…”

“…This essential feature of the mapping, which has an independent theoretical significance, makes it valuable from the point of view of applications to non-elementary problems of descriptive geometry in Euclidean space. Research by Farouki et al [6] is devoted to the study of two -two-valued correspondence Т 2−2 . This correspondence consists in constructing points А 2 ′ and А 2 ′′ from point А 1 , which are projections on the frontal plane of projection П 2 of intersection points perpendicular to the horizontal plane of projection П 1 .…”

“…The curve of the second order 𝑏 1 2 in the horizontal plane of the projections П 1 is given by the traces of the projection plane Г, perpendicular to the plane of the projections П 1 . It is also given by an intersecting curve of the second order along a curve and a point belonging to the section of the curve of the second order, conjugate to the direction perpendicular to the plane of the projection П 1 [6,7].…”