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.

$K$-frames are more generalized than ordinary frames, particularly in terms of their weaving properties. The study of weaving $K$-frames in Hilbert space has already been explored. Given the significance of $n$-Hilbert spaces in functional analysis, it is essential to study weaving $K$-frames in $n$-Hilbert spaces. In this paper, we introduced the notion of weaving $K$-frames in $n$-Hilbert spaces and obtained some new properties for these frames using operator theory methods. First, the concept of weaving $K$-frames in $n$-Hilbert spaces is developed, and examples are given. By virtue of auxiliary operators, such as the preframe operator, analysis operator, and frame operator, some new properties and characterizations of these frames are presented, and several new methods for their construction are given. Stability and perturbation results are discussed and new inequalities are established as applications.