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Encyclopedia > Intrinsic coordinates
Important points in intrinsic coordinates

Intrinsic coordinates is a coordinate system which defines points upon a curve partly by the nature of the tangents to the curve at that point. A point is given as (s, ψ) where s is the length of the curve from a set point (often the origin, in the case of the diagram on the right, point A) and ψ is the angle which the tangent to the curve at that point makes with the x-axis; s = f(ψ) is the intrinsic equation of the curve. Image File history File links Illustration of intrinsic coordinates. ... Image File history File links Illustration of intrinsic coordinates. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ...

This coordinate system has limited use, it may break down entirely when straight lines are considered, but inspection reveals three properties regarding the rate of change of its variables, namely:

$frac{dy}{dx} = tan psi$
$frac{dx}{ds} = cos psi$
$frac{dy}{ds} = sin psi$

The radius of curvature, ρ, at a point is a measure of the radius of the arc which can be created by the extrapolation of that point. If this value is positive then the curve bends upwards, and if the value is negative, the curve bends downward. It is given by: $rho = frac{ds}{dpsi}.$ Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, extrapolation is a type of interpolation. ...

It can be proved that the following is true:

$rho = frac {big( 1 + (frac{dy}{dx})^2big)^{3/2}}{frac {d^2y}{dx^2}}.$

This allows the radius of curvature of a line to be found from only Cartesian coordinates. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Another useful formula can relate the above to parametric form; note that $dot{x} = frac{dx}{dt}$ and $ddot{x} = frac{d^2x}{dt^2}$ Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

$frac{ds}{dpsi} = frac {big({dot{x}^2 + dot{y}^2}big)^{3/2}}{dot {x}ddot{y} - dot{y}ddot{x}}.$

## Conversion

To convert a cartesian equation y = f(x) to an intrinsic equation, differentiate it to get dy/dx. Then find the arc length (see formula - requires the derivative), integrating from 0 to x. Then convert x to ψ using the dy/dx relationship above by expressing s in terms of dy/dx. Fig. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ... For other uses, see Curve (disambiguation). ...

Results from FactBites:

 Pellionisz "Dusseldorf-1" 1987 (4571 words) Given the existence of such intrinsic coordinates, the axiom of generalized coordinates appears inevitable if identification of the internal mathematical language, actually used by neuronal networks, is intended, especially since the mathematical fundamentals of transforming such covariant- contravariant and mixed tensorial expressions in non-orthogonal general frames are well established (30). Transformation from coordinates intrinsic to a sensory system to an intrinsic motor frame (where the latter may be of higher dimensions) can be accomplished by a three-step tensorial scheme (53). Schematic illustration of (joint-angle) coordinates intrinsic to the structure of a motor apparatus (A) and the required metric tensor transformation from a unique set of projection-type (covariant, B) to overcomplete parallelogram-type (contravariant, C) vectorial expression.
 Coordinate system - Wikipedia, the free encyclopedia (743 words) Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves. Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates. Intrinsic coordinates describe a point upon a curve by the length of the curve to that point and the angle the tangent to that point makes with the x-axis.
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