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In calculus, the method of normals was a technique invented by Descartes for finding normal and tangent lines to curves. It represented one of the earliest methods for constructing tangents to curves. 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 ...
More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet , in his thesis of 1847, and Joseph Alfred Serret , in 1851.
The normal equations can be derived directly from a matrix representation of the problem as follows. The objective is to minimize = ‖ ‖ = () = +.Here () = has the dimension 1x1 (the number of columns of ), so it is a scalar and equal to its own transpose, hence = and the quantity to minimize becomes
Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself. [6] These methods led to the development of differential calculus in the 17th century. Many people contributed.
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.
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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
Victor Katz suggests this is because Descartes had independently discovered the same new mathematics, known as his method of normals, and Descartes was quite proud of his discovery. Katz also notes that while Fermat's methods were closer to the future developments in calculus, Descartes' methods had a more immediate impact on the development. [5]
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