Shape-adjustable developable generalized blended trigonometric Bezier surfaces and their applications

Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bezier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bezier surfaces (for short, developable GBT-Bezier surfaces) with shape parameters. A developable GBT-Bezier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bezier surfaces that preserves the features of the developable GBT-Bezier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bezier surfaces, the necessary and sufficient G(1) and G(2) (Farin-Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bezier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bezier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.