The Curvature of Characteristic Curves on Surfaces

nalyzing and interrogating designed surfaces remains an important and widely researched issue in computer-aided geometric design (CAGD). Treating families of characteristic curves— such as contour lines, lines of curvature, asymptotic lines, isophotes, and reflection lines—on the surface proves a popular method of doing this. All these curves have something in common: s They reflect the surface's geometric properties. Analyzing these curves gives information about shape and continuity of the surface. The curves do not depend on the parametrization of the surface. s For sufficiently complicated surfaces (such as bicubic polynomial surfaces), these curves can only be described as the solutions of differential equations. Working with the curves themselves requires solving those equations numerically. In this article we want to use the power of the characteristic curves, but avoid the latter problem described above. We achieve this by considering not the curves directly, but one of their most important properties: their curvatures. Although we do not have explicit formulas for these curves, we can express their curvatures and geo-desic curvatures. For particular curves, we introduce their thickness as another characteristic property. Finally, we discuss how to use the visualization of the characteristic curves' curvatures as a surface interrogation tool. We will be analyzing a family of curves on a para-metric surface by interpreting it as tangent curves of vector fields. Before we discuss the surface case, we briefly describe the case of 2D vector fields. Given is a 2D vector field V: IE 2 → IR 2. V assigns a vector (vx (P), vy (P)) T to any point P ∼ (u, v). We use the notation V(P) = V(u, v) = (vx(u, v), vy(u, v)) T. A point P ∈ IE 2 is called a critical point of V if V(P) = 0 is the zero vector. A curve L ⊆ IE 2 is called a tangent curve (for example, stream line, flow line, or characteristic curve) of the vector field V if the following condition is satisfied: For all points P ∈ L, the tangent vector of the curve at the point P has the same direction as the vector V (P). For every point P ∈ IE 2 there is one and only one tangent curve through it (except for critical points of V). Tangent curves do not intersect each other (except for critical points of V). The vector field may be interpreted as representing a 2D flow. …