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The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: [2]
An illustration of the Frenet frame for a point on a space curve. 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). It is the main tool in the differential geometric treatment of curves ...
The equation of a line is given by = +. The equation of the normal of that line which passes through the point P is given = +. The point at which these two lines intersect is the closest point on the original line to the point P. Hence:
A (parametric) curve : [,] is said to be of class C k, if exists and is continuous on [,], where derivatives at the end-points and are taken to be one sided derivatives (from the right at and from the left at ).
Each curve in this example is a locus defined as the conchoid of the point P and the line l.In this example, P is 8 cm from l. In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.
A plane algebraic curve is the set of the points of coordinates x, y such that f(x, y) = 0, where f is a polynomial in two variables defined over some field F. One says that the curve is defined over F. Algebraic geometry normally considers not only points with coordinates in F but all the points with coordinates in an algebraically closed field K.
A general straight-line thread connects the two points (0, k−t) and (t, 0), where k is an arbitrary scaling constant, and the family of lines is generated by varying the parameter t. From simple geometry, the equation of this straight line is y = −(k − t)x/t + k − t. Rearranging and casting in the form F(x,y,t) = 0 gives:
For the use in differential geometry, see Cesàro equation. For the radius of curvature of the Earth (approximated by an oblate ellipsoid); see also: arc measurement; Radius of curvature is also used in a three part equation for bending of beams. Radius of curvature (optics) Thin films technologies; Printed electronics; Minimum railway curve ...
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