Intrinsic coordinates

Intrinsic coordinates is a coordinate system Coordinate system

In mathematics [i] and applications, a coordinate system is a system for assigning a tuple [i] of number [i] ...

which defines points upon a curve partly by the nature of the tangent Tangent

In mathematics [i], the word tangent has two distinct but etymologically [i]-related meanings: ...

s to the curve at that point. A point is given as where s is the length of the curve from a set point and ψ is the angle which the tangent to the curve at that point makes with the x-axis; s = f is the intrinsic equation of the curve. 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:

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Encyclopedia

Intrinsic coordinates is a coordinate system Coordinate system

In mathematics [i] and applications, a coordinate system is a system for assigning a tuple [i] of number [i]...

which defines points upon a curve partly by the nature of the tangent Tangent

In mathematics [i], the word tangent has two distinct but etymologically [i]-related meanings: ...

s to the curve at that point. A point is given as where s is the length of the curve from a set point and ψ is the angle which the tangent to the curve at that point makes with the x-axis; s = f is the intrinsic equation of the curve.

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:

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:

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 coordinate system

In mathematics [i], the Cartesian coordinate system is used to uniquely determine each point [i]...

.

Another useful formula can relate the above to parametric form Parametric equation

[i]
...

:

where

Conversion

To convert a cartesian equation Cartesian coordinate system

In mathematics [i], the Cartesian coordinate system is used to uniquely determine each point [i]...

y = f to an intrinsic equation, differentiate it to get dy/dx. Then find the arc length Arc length

Determining the length of an irregular arc segment—also called rectification [i] of a curve [i]&md ...

, integrating from 0 to x. Then convert x to ψ using the dy/dx relationship above by expressing s in terms of dy/dx.