Smooth joining of spline curves ensuring a high order of smoothness

Smoothly blending two curves is one of the fundamental and complex problems of geometric modeling in computer-aided design systems. This article proposes a method for constructing a smoothly blended curve between two spline curves of arbitrary format and arbitrary degrees, ensuring an arbitrary degree of smoothness. The formula of the method is as follows: the conjugation points on the original curves are fixed, then the spline segment of the first curve is fixed before the conjugation point, which will become the initial segment of the conjugation spline curve, as well as the spline segment of the second curve after the conjugation point, which will be the final segment of this curve. These fixed segments of the fillet areas are converted to the format of a rational Bezier curve, after which the degrees of these curves are converted to a single specified value by increasing the degree, while the geometry of the curves does not change. Next, the B-polygons of the Bezier curves are transformed into an open S-polygon without changing the geometry. In the next step, an S-polygon of the blend curve is constructed, the end sections of which coincide with the S-polygons of the original sections. Ultimately, an integral rational B-spline blend curve is defined on the open S-polygon. The end segments of the integral spline blend curve will geometrically precisely match the blend segments of the original curves. The smoothness order of the blend segment with the blend segments of the original curves may be higher than that of the original curves (in the C3D FairCurveModeler application, up to G9 for spline degree 10). In general, the order of smoothness of the integral curve is the same as the lowest order of smoothness of the original curves. The proposed method for constructing a conjugation is adapted for editing a section of a geometric rational Bezier spline curve.