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Illustration of tangential and normal components of a vector to a surface. 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.
The method hinges on the observation that the radius of a circle is always normal to the circle itself. With this in mind Descartes would construct a circle that was tangent to a given curve. He could then use the radius at the point of intersection to find the slope of a normal line, and from this one can easily find the slope of a tangent line.
T is the unit tangent, P the unit normal, and B the unit binormal. A Frenet frame is a moving reference frame of n orthonormal vectors e i ( t ) which are used to describe a curve locally at each point γ ( t ) .
The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame (TNB frame or TNB basis), together form an orthonormal basis that spans, and are defined as follows: T is the unit vector tangent to the curve, pointing in the direction of motion.
A line is normal to γ at γ(t) if it passes through γ(t) and is perpendicular to the tangent vector to γ at γ(t). Let T denote the unit tangent vector to γ and let N denote the unit normal vector. Using a dot to denote the dot product, the generating family for the one-parameter family of normal lines is given by F : I × R 2 → R where
There is in general no natural choice of embedding, but for a given manifold , any two embeddings in for sufficiently large are regular homotopic, and hence induce the same normal bundle. The resulting class of normal bundles (it is a class of bundles and not a specific bundle because the integer could vary) is called the stable normal bundle.
The circle S and the curve C have the common tangent line at P, and therefore the common normal line. Close to P, the distance between the points of the curve C and the circle S in the normal direction decays as the cube or a higher power of the distance to P in the tangential direction.
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 line perpendicular to the tangent line to the curve at the point. A normal vector of length one is called a unit normal vector.