normalul

Normalul (the normal) is a geometric concept describing a line or a vector perpendicular to a curve, a surface, or a plane at a specific point. It is used to specify orientation, contact forces, and directions of projection. The term comes from Latin normalis, meaning perpendicular. At a point on a smooth surface, the normal line is unique, defined up to a sign; the accompanying normal vector can be reversed without changing the geometric line.

In the plane, the normal at a point on a curve is the line perpendicular to the tangent at that point. If a function y = f(x) has a point x0 with y0 = f(x0) and slope f′(x0), then the tangent slope is f′(x0), and the normal has slope −1/f′(x0) (when f′(x0) ≠ 0). The equation of the normal line is y − y0 = −(1/f′(x0))(x − x0). If f′(x0) = 0, the normal is a vertical line x = x0.

In three-dimensional space, the normal to a surface defined implicitly by F(x, y, z) = 0 at a point P is parallel to the gradient ∇F(P). The normal line is X(t) = P + t∇F(P), and the tangent plane satisfies ∇F(P) · (X − P) = 0. For a parametric surface r(u, v), the normal is the cross product r_u × r_v.

Unit normals can be used, such as û = ∇F/||∇F|| or T × B in curves, where T is the unit tangent and B the binormal. Normals have broad applications in physics, engineering, and computer graphics, including normal forces, shading, and surface analysis.