normalens

Normalen are a class of geometric entities used to describe surface orientation in differential geometry and computer graphics. They generalize the notion of a normal vector to higher-order structures and provide a consistent way to reason about how a surface is facing in space.

In a smooth surface, the normalen at a point is a unit vector perpendicular to the tangent plane at that point. On discrete meshes, normalen are approximate and typically computed from neighboring geometry. A common approach is to compute the face normal for each polygon by taking the cross product of edge vectors, then derive vertex normals by averaging adjacent face normals and normalizing the result.

Normalen are essential for shading and lighting calculations, where the angle between normalen and light direction determines the reflected intensity. They also support geometric operations such as collision response, texture mapping with normal maps, and surface reconstruction from point clouds.

Variants include face normals, vertex normals, and weighted normals, where weights may reflect face area or angle between faces. Challenges arise with noisy data, smooth shading between disparate surface patches, and ensuring consistent orientation across a model. Techniques such as smoothing groups, angle-based weighting, and explicit normal maps help manage these issues.

Origin and terminology: the term normalen derives from the German word for normal. In English-language literature the term normals is more common, but normalen remains used in some contexts to emphasize the orientation concept across different dimensional settings. See also normal vector, unit normal, and normal mapping.