Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Tangential and normal components
Summary
In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector. Similarly, a vector at a point on a surface can be broken down the same way.
More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point of N, it can be decomposed into the component tangent to N and the component normal to N.
More formally, let be a surface, and be a point on the surface. Let be a vector at . Then one can write uniquely as a sum
where the first vector in the sum is the tangential component and the second one is the normal component. It follows immediately that these two vectors are perpendicular to each other.
To calculate the tangential and normal components, consider a unit normal to the surface, that is, a unit vector perpendicular to at . Then,
and thus
where "" denotes the dot product. Another formula for the tangential component is
where "" denotes the cross product.
Note that these formulas do not depend on the particular unit normal used (there exist two unit normals to any surface at a given point, pointing in opposite directions, so one of the unit normals is the negative of the other one).
More generally, given a submanifold N of a manifold M and a point , we get a short exact sequence involving the tangent spaces:
The quotient space is a generalized space of normal vectors.
If M is a Riemannian manifold, the above sequence splits, and the tangent space of M at p decomposes as a direct sum of the component tangent to N and the component normal to N:
Thus every tangent vector splits as , where and .
Suppose N is given by non-degenerate equations.
If N is given explicitly, via parametric equations (such as a parametric curve), then the derivative gives a spanning set for the tangent bundle (it is a basis if and only if the parametrization is an immersion).
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (3)
PHYS-101(a): General physics : mechanics
Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de pr
ME-201: Continuum mechanics
Continuum conservation laws (e.g. mass, momentum and energy) will be introduced. Mathematical tools, including basic algebra and calculus of vectors and Cartesian tensors will be taught. Stress and de
MATH-213: Differential geometry
Ce cours est une introduction à la géométrie différentielle classique des courbes et des surfaces, principalement dans le plan et l'espace euclidien.
Related lectures (32)
Geodesic Curves: Minimizing Distance and Constant Speed
Covers geodesic curves parameterized at constant speed to minimize distances between points.
Motion Analysis: Trajectories and Acceleration
Explores trajectories, velocities, and accelerations in motion analysis, including tangential and normal acceleration components.
Trajectories and Motion Equations
Covers trajectories, motion equations, velocity properties, acceleration components, and integration of motion equations.
Related publications (5)
Benchmarking of the plane-to-plane metric
Touradj Ebrahimi, Evangelos Alexiou
In this report we benchmark the plane-to-plane objective quality metric. This is, a metric that measures the angular similarity of tangent planes between two point cloud models and relies on normal vectors that are carried with associated pairs of points. ...
2020MATHICSE Technical Report : Spectral based discontinuous Galerkin reduced basis element method for parametrized Stokes problems
Alfio Quarteroni, Paolo Pacciarini
In this work we extend to the Stokes problem the Discontinuous Galerkin Reduced Basis Element (DGRBE) method introduced in [1]. By this method we aim at reducing the computational cost for the approximation of a parametrized Stokes problem on a domain part ...
MATHICSE2016Spectral based Discontinuous Galerkin Reduced Basis Element method for parametrized Stokes problems
Alfio Quarteroni, Paolo Pacciarini
In this work we extend to the Stokes problem the Discontinuous Galerkin Reduced Basis Element (DGRBE) method introduced in Antonietti et al. (2015). By this method we aim at reducing the computational cost for the approximation of a parametrized Stokes pro ...
Pergamon-Elsevier Science Ltd2016Related concepts (3)
Normal (geometry)
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (in which case it is a unit normal vector) or its length may represent the curvature of the object (a ); its algebraic sign may indicate sides (interior or exterior).
Cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".