La projection orthogonale est un concept fondamental dans algèbre linéaire and geometry, used to project vectors onto a specific subspace. In simpler terms, it refers to the process of dropping a perpendicular (orthogonal) line from a point (or vector) to a line (or plane) in a vector space, resulting in the closest point in that line or plane.
Mathématiquement, pour un vecteur v in a vector space and a subspace defined by an base orthonormale, the orthogonal projection is calculated using the formula:
projW(v) = Σ (v · wi) wi, where wi are the orthonormal basis vectors of the subspace W.
Cette projection minimise le Distance Euclidienne between the original vector v and its projection onto the subspace, ensuring that the resulting vector is as close as possible to v tout en restant dans le sous-espace.
Les projections orthogonales sont largement utilisées dans divers domaines, notamment infographie, where they help in rendering scenes by projecting 3D points onto 2D planes for display on screens. Additionally, they play a significant role in data science, particularly in dimensionality reduction techniques such as Analyse en Composantes Principales (PCA), where data is projected onto lower-dimensional subspaces to reveal patterns and structures.