# Interpolating free-form curve networks using recursive subdivision surfaces

The purpose of this work is to find an approach to extend subdivision based surface modeling with methods that utilize free-form curves to describe a surface. Amongst those different methods lies the framework of combined subdivision schemes which introduce additional external geometric constraints to the subdivision process, not just the local properties of the polyhedron. Adi Levin defined a method which describe an initial curve network and a corresponding control-polyhedron. The method utilizes some of the properties of the given curve network to ensure that the resulting surface of the subdivision interpolates the network at the limit.

This diploma work tries to address the upcoming difficulties of applying this method. One considerable constraint is that the control-polyhedron must be initialized manually for the curve network, while the network’s topology basically describes the requested initial structure. In addition to that, there is an option to incorporate additional geometric constraints to the process, which help to control the limit surface more precisely.

The extension of the algorithm is made from two directions. The first preprocessing stage creates a connected curve network from free parametric curves. After the curve network is successfully created, an automated process builds an initial polyhedron using the topology of the network. An additional concern is the effect of the control-polyhedron’s initial topology to the resulting limit surface. For the initial faces of the polyhedron, it is advantageous to define a more natural center point which corresponds to the shape to the nearby curve segments.

After the pre-processing stage, the subdivision method can be applied to the generated polyhedron. A further extension of the algorithm focuses on the curve based local corrections on the polyhedron. The local corrections operate by the estimated metrics of the resulting curve segments while the subdivision is applied. The accuracy of the estimations can be improved, when the sampling of the curves are made on a localized parameter range.

Utilizing the tangent vectors defined by the joints of the curve network, the shape of the network can affect the resulting polyhedral surface more directly. By interpolating the cross-curve tangents on each surface edge, the corrections achieve a more natural result in respect of the shape of nearby curves. The method also enables corrections to be applied on boundary polyhedral edges, thus preserving the control of the curve network. These corrections can be further improved, when they use the corresponding curve points directly, which results in a smoother limit surface.