Tracing curvature paths on trimmed multipatch surfaces

Abstract

Advances in architectural geometry and computation have created strategies to rationalize complex building envelopes. This paper presents techniques to identify paths on freeform surfaces by prescribing specific curvature properties. The paths can be defined by the normal curvature or the geodesic torsion. In special cases, asymptotic and principal curvature lines can be determined. Such paths are used for the design of gridshells with lamellar elements. A brief introduction to differential geometry of freeform surfaces provides the relevant foundations for the method. The relevant quantities and relations are highlighted with illustrations. The consistent description of the paths in the parameter space of the surface avoids unnecessary and complex projection operations. This allows a computationally efficient and robust implementation. The tracing is explained for the simple case of a single surface and extended to trimmed multipatches which are used for geometric modeling in common computer-aided design (CAD) programs. The topological structure of the multipatch was used to trace paths across several surfaces. To enable interactive design, the path tracing techniques were integrated into the parametric CAD package Grasshopper for Rhino. Additional components for curvature analysis were implemented to analyze and evaluate designs. To simplify the handling of curvature lines, a geometry type for embedded curves was introduced and implemented within a CAD environment. Finally, the use of these tools for the design of architecturally sophisticated gridshells is presented along with three projects of increasing complexity.