Splines for engineers: with selected applications in numerical methods and computer graphics

Abstract

Spiro splines [Lev09] are an interesting alternative for font design and are available in FontFoRge and InKscape. ‡ Two of these attributes are kerning and ligatures, which are both used extensively by X Ǝ T E X. Spotting ligatures in a text is an enjoyable game and often provides a reasonable indication of what a font has to offer! 13 2.1 The above might raise the question whether composite curves connecting with e.g. C 1 continuity can be represented more efficiently, as the middle of these three co-linear points is merely a convex combination of the other two. The answer is positive and can be obtained by studying B-spline curves. 2.1.2 B-spline curves B-splines † come in many different flavours, can be defined using a variety of ways, and form an extensive area of research. Merely interpreting them as an efficient way to represent composite Béziers does perhaps not do them justice, but as this is the direction we come from, it is our starting point. An important aspect to consider when connecting Bézier curves is whether all segments should be defined on parameter intervals of the same length. If we choose to do so, and set that interval to be of unit length, we obtain a vector of parameter values Ξ = [0, 1, 2,. .. , m], where m indicates the number of segments. These parameter values indicate where our segments are connected, or tied together, in parameter space. For this reason, they are often referred to as knots, and Ξ as the knot-vector. Let us consider a composite quadratic Bézier curve of two segments that connect with C 1 continuity. We thus have control points P 0 ,. .. , P 4 , with as corresponding basis functions M 0 (t),. .. , M 4 (t) the Bernsteins, which for the second curve segment are shifted by one unit: M 0 (t) = (1 − t) 2 for t ∈ [0, 1], M 1 (t) = 2t(1 − t) for t ∈ [0, 1], M 2 (t) = t 2 (2 − t) 2 for t ∈ [0, 1], for t ∈ [1, 2], M 3 (t) = 2(t − 1)(2 − t) for t ∈ [1, 2], M 4 (t) = (t − 1) 2 for t ∈ [1, 2].

G1 – Smooth biquintic approximation of Catmull-Clark subdivision surfaces

“…In particular, for Bézier masks of order zero, i.e. vertex points, we take M 0,0 = M 0,0 as in (1). For Bézier points of order one we will solve eq.…”

“…For the case of subdivision surfaces, analysis methodologies have been explored eg. in [1,2,12,28,34], as well as related issues, notably efficient quadrature [2,18].…”

Selected applications of subdivision surfaces and numerical quadratures for Gregory patches

“…To date, only a few recent advances in this direction have been made. In the case of Catmull-Clark subdivision, an approach to the quadrature for subdivision surfaces based on Catmull-Clark quad meshes was introduced in Barendrecht et al (2018a); Barendrecht (2019). This research discussed grouping strategies of quads around extraordinary vertices (EVs), as the surface around EVs has non-tensor product structure.…”

“…Figure 5.5: Density scatter plots for 1000 successfully converged runs (see also Barendrecht (2019)) for G 𝑢 and 𝑣 . An alpha value of 1 100 was used.…”