Construction of two-direction scaling functions on generalized elliptic helical curves

Based on an arc length-preserving projection, this paper investigates two-direction scaling functions on generalized elliptic helical curves. The spatial trace of the parametrization is distinguished from its parametrized carrier, so repeated spatial points caused by periodicity or self-intersection remain distinct when their parameter values differ. Under the stated parameter restrictions, the speed has a positive uniform lower bound; consequently, on the global carrier with parameter interval $I=\mathbb{R}$, the oriented arc length coordinate is a homeomorphism onto $\mathbb{R}$. This yields a nonempty global setting in which the real line two-direction multiresolution structure can be transferred to the curve without boundary truncation. Through the induced unitary identification between $L^2(\mathbb{R})$ and the carrier space, the two-direction refinement equations and the scaling and wavelet coefficient sequences are transported in a consistent arc length formulation. Riesz basis and orthogonality properties are preserved whenever the corresponding real line families possess them. The mask matrix spectral condition is treated separately from cascade convergence, $L^2$-stability, continuity, the Riesz basis property, and orthogonality, and a necessary Hermitian paraunitary mask condition is derived for orthogonal two-direction scaling functions. Under the corresponding real line approximation assumptions, approximation order estimates are likewise transported to the carrier. For numerical verification, a bounded injective spatial segment is considered independently, and the arc length map is approximated by an explicit Euler scheme. Finally, two algebraically checked symbolic mask examples, together with the numerical projection experiment, illustrate the discretization procedure and the relevant algebraic mask conditions.