Curvature Map on Terrains

This is the last of the site analysis tools we previously used in first-year architecture education. As you may recall, we previously introduced a tool that generates topographies based on specific parameters. Afterward, we examined two tools that perform slope and flow analyses on these topographies. Today, we are presenting a tool that analyzes curvature. This topic actually constitutes one of the interesting intersection points between architectural design and differential geometry. Curvature is not something architects normally observe directly on a topography; they are mostly concerned with slope. However, in this educational experiment, we saw that reading curvature on the site as well contributes to both fields.

Curvature is a measure of how much a geometric object deviates from being flat at a point. Of course, this definition falls short. Surface curvature is divided into two: mean curvature and Gaussian curvature. We use Gaussian curvature in this studio setup. This curvature measures the amount and characteristic of bending at any point on a surface, giving valuable spatial information. It is the product of the two principal curvatures. In positive Gaussian curvature, the principal curvatures (you can think of them as two mutually perpendicular cross-sectional curves of the surface) bend in the same direction. In other words, at these points, the surface orients entirely to one side of the tangent plane, just like a dome or a pit. Shapes like spheres and ellipsoids have positive Gaussian curvature everywhere. In negative Gaussian curvature, the principal curvatures bend in opposite directions. At these points, the surface resembles a saddle; the tangent plane intersects the surface. A hyperboloid, or a Pringles, has negative Gaussian curvature. In zero Gaussian curvature, at least one of the principal curvatures is zero, meaning the cross-section taken in that direction is a straight line. Naturally, the result of the multiplication is also zero. A characteristic of such surfaces is that they are developable. Shapes like cylinders and cones can therefore be unrolled onto a plane.

One of Carl Friedrich Gauss‘s most important discoveries, this theorem reveals that Gaussian curvature is an intrinsic property. That is, if you bend a surface without stretching, tearing, or compressing it (for example, rolling up a flat piece of paper into a cylinder), the Gaussian curvature remains constant even if the surface’s shape in 3D space changes from an external perspective. This is why you can flatten a cylinder-shaped piece of paper back onto a completely flat table without tearing it, whereas you cannot flatten an orange peel in the shape of a sphere without tearing it, because the sphere’s Gaussian curvature is non-zero.

In architecture, it is a concept that must be known, especially in the production of complex geometries, surface panelization, and predicting the structural behavior of shell structures. This Grasshopper code generates a color-coded Gaussian Curvature map on a surface. This can be used as information alongside other analyses when making design decisions. Students used this code to identify and present more enclosed and open, accessible regions on their sites. From a pedagogical perspective, such concepts should not remain limited to geometry classes but should be directly usable in the studio.