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 xaxis; 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 ndimensional space. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical onedimensional 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: The radius of curvature
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: 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:
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 and 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. ...
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). ...
