Pinnakõverdused

Pinnakõverdused, also known as surface curvatures, refer to the geometric properties of a surface that describe how it bends or deviates from being flat. These concepts are fundamental in differential geometry and have applications in various fields, including computer graphics, engineering, and physics.

There are several types of surface curvatures used to quantify these properties. The most common are the Gaussian curvature and the mean curvature. Gaussian curvature (K) is the product of the two principal curvatures (k1 and k2) at a point on the surface. It is an intrinsic property, meaning it can be determined solely by measurements within the surface itself. A surface with a positive Gaussian curvature is locally shaped like a sphere or an ellipsoid, while a negative Gaussian curvature indicates a saddle shape. A zero Gaussian curvature signifies a developable surface, which can be flattened without stretching or tearing, such as a cylinder or a cone.

Mean curvature (H) is the average of the two principal curvatures: H = (k1 + k2) / 2. It is an extrinsic property, meaning it depends on how the surface is embedded in the surrounding space. Surfaces with zero mean curvature are called minimal surfaces, and they have the smallest surface area for a given boundary, often appearing in soap films. Understanding pinnakõverdused is crucial for tasks like surface reconstruction, mesh smoothing, and analyzing the behavior of physical phenomena on curved surfaces.