Abstract. This paper describes novel methods we developed to lay out graphs using Sugiyama’s scheme [16] in a tool named GLEE. The main contributions are: a heuristic for creating a graph layout with a given aspect ratio, an efficient method of edge-crossings counting while performing adjacent vertex swaps, and a simple and fast spline routing algorithm. 1

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When designing curves on surfaces the need arises to approximate a given noisy target shape by a smooth fitting shape. We discuss the problem of fitting a B-spline curve to a point cloud by squared distance minimization in the case that both, the point cloud and the fitting curve, are constrained to lie on a smooth manifold. The on-manifold constraint is included by using the first fundamental form of the surface for squared distance computations between the point cloud and the fitting curve. For the solution we employ a constrained optimization algorithm that allows us to include further constraints such as one-sided fitting or surface regions that have to be avoided by the fitting curve. We illustrate the effectiveness of our algorithm at hand of several examples showing different applications. Key words: B-spline curve, curve fitting, constrained optimization, squared distance minimization, geometric constraints, damped Gauss-Newton method, shape approximation, free-form curves, splines on manifolds, constrained curve design. 1.

We consider the computation of an approximating curve to fit a given set of points. Such a curve fitting is a common problem in CAGD and we review three different approaches based upon minimization of the squared distance function: the point distance, tangent distance and squared distance error term. We enhance the classic setup comprising a point cloud and an approximating B-Spline curve by obstacles a final solution must not penetrate. Two algorithms for solving the emerging constrained optimization problems are presented and used to enclose point clouds from inside as well as from outside and to avoid arbitrary smooth bounded obstacles. Moreover, approximations of guaranteed quality and shaking objects are examined within this context. In a next step, we extend our work and study the curve fitting problem on parametrized surfaces. The approximation is still performed in the two dimensional parameter space while geometric properties of the manifold enter the computation at the same time. Additionally, we use this new error term to approximate the borders of point clouds on manifolds. Finally, as a minimization of the squared distance function