Pontnormál

Pontnormál is a Hungarian term used in geometry and related fields to denote the normal vector at a specific point on a curve or a surface. The normal is the direction perpendicular to the tangent of a curve or to the tangent plane of a surface at that point, and it plays a central role in tasks such as shading, lighting calculations, and surface analysis.

Definition and computation: For a smooth plane curve γ(t), the tangent vector is γ′(t) and a normal vector is any vector perpendicular to γ′(t). It is common to use a unit normal n(t) obtained by normalization. For a smooth surface X(u,v) in three-dimensional space, the normal at a point X(u0,v0) is given by the cross product n = (∂X/∂u × ∂X/∂v) divided by its length, with the orientation chosen according to a convention. In discrete geometry, such as polygonal meshes, the pontnormál at a vertex is often estimated by averaging the normals of adjacent faces, sometimes weighted by area or angle to produce smooth shading.

Applications: Pontnormál is fundamental in computer graphics for shading models (Lambertian, Phong), where the interaction of light with a surface depends on the angle between the light direction and the normal. It also informs collision detection, physics simulations, and geometric analyses of surfaces, including curvature studies.

Examples: On a sphere, the pontnormál at any point is the normalized radial vector from the center to that point, pointing outward. This aligns with the geometric intuition that the normal indicates the local orientation of the surface.

See also: normal vector, tangent, surface normal, vertex normal, face normal, shading.