## 3 Parametric Model Definition
### Surface Meshing Method  
#### 3.1.1 Generation of Geodesic Master Curves: Geometrical Fundamentals  
On any curve \(C\) in space, an intrinsic trihedron can be defined along the curve, known as the Frenet trihedron. At any point \(O\) on the curve, this trihedron is defined by the vector \(\overrightarrow{t}\) tangent to the curve at the point, the normal vector \(\overrightarrow{n}\) (unit vector of the curvature vector \(\overrightarrow{k}\)), and the binormal vector \(\overrightarrow{b}\) (cross product of the tangent and curvature vectors). The line normal to the curve at a point contains the vectors \(\overrightarrow{n}\) and \(\overrightarrow{k}\) and the centre of curvature of the curve at that point. At each point on the curve, the circumference with a centre at the centre of curvature and with the same radius of curvature as the curve is called, the osculating circumference and is contained in the plane formed by the vectors \(\overrightarrow{n}\) and \(\overrightarrow{t}\).  
At the same point \(O\) of the curve, but now contained in a surface \(S\), there is a trihedron of the curve associated with the surface called a Darboux trihedron, which is defined by the vector \(\overrightarrow{t}\) tangent to the curve and the surface, a vector \(\overrightarrow{N}\) normal to the surface, and the geodesic vector \(\overrightarrow{G}\).  
Figure 3 shows the Frenet and Darboux trihedrons at a point \(O\) on curve \(C\). It can be observed that the planes defined by the vectors \(\overrightarrow{N}\) and \(\overrightarrow{G}\), and \(\overrightarrow{n}\) and \(\overrightarrow{b}\) (planes marked in red) are coplanar since both are perpendicular to the vector \(\overrightarrow{t}\).  
Figure 4 shows the relationship between the Frenet and Darboux trihedron vectors projected onto any of the coplanar planes mentioned above and the components of the curvature vector \(\overrightarrow{k}\).  
The curvature vector \(\overrightarrow{k}\) indicates the curvature of the curve in space at point \(O\). The curvature of the curve on the surface can be evaluated by projecting the vector \(\overrightarrow{k}\) onto the vectors \(\overrightarrow{N}\) and \(\overrightarrow{G}\) of the Darboux trihedron and obtaining the vectors \(\overrightarrow{k}_{N}\) and \(\overrightarrow{k}_{G}\), which give the normal and geodesic curvature, respectively:  
\[\overrightarrow{k}=\overrightarrow{n}\ k, \tag{1}\] \[\overrightarrow{k}=\overrightarrow{k}_{N}+\overrightarrow{k}_{G}, \tag{2}\]  
Figure 3: Frenet trihedron (**a**) and Darboux trihedron (**b**).  
A geodesic curve on a surface can be defined as a curve such that at every point along its length, it is satisfied that \(\overrightarrow{k}_{G}\) is zero. Geometrically, it could be defined as the curve whose tangent vector advances over a surface rotating only around \(\overrightarrow{G}\) and never around \(\overrightarrow{N}\)[32]. In a geodesic curve, the vector \(\overrightarrow{n}\) of the curve and the vector \(\overrightarrow{N}\) of the surface are aligned, and, consequently, the rectifying plane of the curve coincides with the tangent plane of the surface or, in other words, the osculating plane of the curve is normal to the tangent plane of the surface.