## 1 Introduction  
Lattice structures are cellular structures consisting of interconnected struts [1]. Each lattice structure is based on a lattice frame, also denoted as a wireframe, which describes with nodes and edges the inherent topological skeleton. The latter can either be irregular or consists of periodically repeated unit cells. Besides the interconnections and coordinates described by the lattice frame, the cross-section shapes and radii of the struts are local shape parameters of lattice structures. Lattice structures find use in various disciplines, often in connection with additive manufacturing, reaching from architecture and mechanical engineering to healthcare [2, 3]. Applications include medical implants [4], heat sinks [5], and crash boxes [6], exploiting the porous structure of lattices for the ingrowth of bones, the more efficient heat dissipation due to very high surface-to-volume ratio, and the high specific energy absorption capability by allowing for controlled deformation with reduced peak loads and good load uniformity. Lattice structures are particularly suitable for lightweight designs due to their high stiffness-weight ratio [3] and therefore find use in components that are designed for aeronautics and space applications [7, 8]. In [8], additively manufactured lattice structures are used as cellular infills in areas where the topology optimisation of the discussed component recommends material densities between void and solid. Even entire components consisting of lattice structures are derived in [9] from topology optimisation results. To solidify the proposed (often irregular) wireframes in CAD software, an automated algorithm is required. Many commercial CAD software tools rely on spline-based surface patches that are exactly parametrisable by their control points. To integrate the design of lattice structure infills in the design process of the surrounding component, a method based on surface patches is favourable. Having a component design based on splines is also advantageous for additive manufacturing, because it enables local manipulations, i.e., to remove large overhangings and to create support-free printable structures. Besides the wireframe topology, smooth node transitions, also called filleted nodes, are an important aspect of lattice structure design as they can improve the structural response to loadings by reducing local stress concentrations and increasing the stiffness [10; 11]. Additionally, the realisability of strut radii gradients, useful for further tailoring the lattice structure to an intended use, is improved in smooth lattice structures, according to [12]. Several methods to construct smooth lattice structures can be found in the literature, including polygon mesh methods [11; 13; 14], iso-surface methods [10; 15; 16], and surface patch methods [12; 17]. Their key properties are summarised in Table 1. Their commonality is the construction of the lattice structures surface instead of a solid object. In [11; 13; 14], lattice structures are described with polygon meshes. Starting with simple mesh models, subdivision schemes are applied to refine the mesh, leading to smooth strut intersections. The main disadvantage of polygon mesh-based approaches is the inexact parametrisability of local shape parameters such as strut radii, as discussed in [11]. In [10; 15; 16], the definition of lattice structures with iso-surfaces is based on surface equations that map the three-dimensional space to the set of real numbers. The lattice structure is defined by the set of points that is mapped to values smaller or equal to zero. Thus, the zero-value iso-surface describes the lattice structures' surface. Thereby, the lattice structures' smoothness is inherited from the surface equations. Construction methods assembling the lattice structures' surface with surface patches are presented in [12; 17]. Boundary conditions ensure smooth transition between patches.