Abstract: A class of spacial curves are defined over the algebraic-trigonometric space Ω=span{1,θ ···,
θm+1, sinθ, cosθ, θsinθ, ···, θn cosθ}. By choosing proper integral kernels, the projection of the spacial
integral curve on the xy-plane has intrinsic definition, or the whole spacial curve is a PH curve. The
Cartesian coordinates of the curve can be explicitly evaluated by integrals of the predefined kernels.
Besides, techniques of interpolation of integral curves with different integral kernels have been
studied. Given the boundary data, the coefficients within the kernel functions are obtained by solving
a system. Finally, we use PH curves to design a family of frames, which can be used to construct a
rational swept surface. Experimental results show that the piecewise swept surface is G1 continuous at
the ridge line and is nearly G1 continuous at the remaining junctions.

Key words: geometric Hermite interpolation, mixed space of functions, PH curves, frame