normalretningen

Normalretningen, in English often called the normal direction or normal vector, is the direction that is perpendicular to a surface at a given point, or to a curve in a plane. It is a fundamental concept in geometry, physics, and applied disciplines because it encodes how a surface is oriented in space.

For a smooth surface described parametrically by r(u, v), the normal direction at a point is typically found from the cross product of the partial derivatives: n = ∂r/∂u × ∂r/∂v. This vector is perpendicular to the tangent plane of the surface at that point. For an implicit surface defined by F(x, y, z) = 0, the gradient ∇F is perpendicular to the surface, giving the normal direction n = ∇F.

In two dimensions, the normal direction to a curve can be derived from the tangent. If a curve is given by y = f(x) with tangent vector (1, f′(x)), a normal vector is perpendicular to it, for example (−f′(x), 1). For a parametric curve r(t), the tangent is r′(t) and a normal direction is any vector perpendicular to r′(t).

Unit normals are obtained by normalizing the vector: û = n/||n||. The choice of direction matters in some contexts (outward normal for closed surfaces, inward in others), and the sign of the normal can be reversed without changing its geometric meaning.

Applications of normalretningen include computer graphics shading, where normals influence how light reflects from a surface; in physics, where normal forces act along the normal direction; and in differential geometry, where normals are central to the study of surface orientation and curvature.