Pacific Graphics 2014 J. Keyser, Y. J. Kim, and P. Wonka (Guest Editors) Volume 33 (2014), Number 7 G 2 Surface Interpolation Over General Topology Curve Networks Péter Salvi, Tamás Várady Budapest University of Technology and Economics Abstract The basic idea of curve network-based design is to construct smoothly connected surface patches, that interpolate boundaries and cross-derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G 1 ) continuity between the adjacent patches, curvature continuous connections (G 2 ) may also be required. Examples include special curve network configurations with supplemented internal edges, “master- slave” curvature constraints, and general topology surface approximations over meshes. The first step is to assign optimal surface curvatures to the nodes of the curve network; we discuss different optimization procedures for various types of nodes. Then interpolant surfaces called parabolic ribbons are created along the patch boundaries, which carry first and second derivative constraints. Our construction guarantees that the neighboring ribbons, and thus the respective transfinite patches, will be G 2 continuous. We extend Gregory’s multi-sided surface scheme in order to handle parabolic ribbons, involving the blending functions, and a new sweepline parameterization. A few simple examples conclude the paper. Categories and Subject Descriptors (according to ACM CCS): Computer Graphics [I.3.5]: Computational Geometry and Object Modeling—curvenet-based design, transfinite surfaces, Gregory patches, G 2 continuity 1. Introduction In curve network-based design, surface models are directly defined by a collection of free-form curves, arranged into a single 3D network with general topology. Curves may come from (i) sketch input, (ii) feature curves extracted from orthogonal views, (iii) curves traced on triangular meshes or (iv) direct 3D editing. Once the curves are defined, all the surfaces are generated automatically. This calls for a repre- sentation based on geometric information extracted solely from the boundaries. Transfinite surface interpolation is a natural choice, as it does not require a grid of control points to define the interior shape, and all n boundaries are han- dled uniformly, unlike in the case of trimmed quadrilateral surfaces. The ability to interactively edit prescribed bound- aries and cross-derivatives is also an advantage in contrast to recursive subdivision schemes. The first step of surface generation is to compute cross- directional data, such as tangent planes and curvatures that are shared by adjacent patches. Then interpolant surfaces, called ribbons, are generated, that carry first or second- degree cross-derivative constraints to be eventually interpo- lated by the transfinite surfaces. The majority of multi-sided transfinite surfaces are de- fined over convex domains, combining only linear ribbon surfaces and enabling G 1 continuity between the adjacent patches. At the same time, there are several practical design situations, where this approach is not sufficient, and higher degree continuity is required. (i) It often occurs that additional curves need to be in- serted into the curve network to make it suitable for apply- ing patches with convex domains. The supplemented curves must be compatible with the already defined ribbons, and it is particularly important to produce seamless transitions along these curves. Examples include handling curve con- figurations with concave angles, or connecting disjoint loops with prescribed slopes (see Figure 9 later in Section 5). (ii) Another important situation is when a designer wants to create a G 2 connection based on two existing G 1 patches. He may want to retain one surface (the master), and modify the adjacent patch (the slave) accordingly, see Figure 8, or may prefer an averaged target curvature, see Figure 7. c  2014 The Author(s) Computer Graphics Forum c  2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.