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8. Security - check for the yellow padlock on the Intrinsic Coordinates site before you buy, and the s after http:/ /i.e. https:// = a secure site
9. Contact - got a question about Intrinsic Coordinates, or want to leave a comment then check out the sites contact page. Reputable companies have them and respond.
10. Payment - ready to pay for your Intrinsic Coordinates, then use your credit card or PayPal! Be aware of companies that don't accept them, there may be genuine reasons but given the huge amount of choice you have when buying online there is no reason at all not to buy via credit card or PayPal.
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;
ψ =
f(s) is the
intrinsic equation of the curve.
This is arguably not a true coordinate system, since it describes points on a curve relative to the curve itself rather than the space in which it is embedded, and points not on the curve have no intrinsic coordinates.
Inspection reveals three properties regarding the rate of change of the variables, namely:
\frac{dy}{dx} = \tan \psi
\frac{dx}{ds} = \cos \psi
\frac{dy}{ds} = \sin \psi
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 turns anticlockwise as
s increases; if negative, the curve turns clockwise. It is given by:\rho = \frac{ds}{d\psi}.
It can be proven that the following is true:
\rho = \frac {\big( 1 + (\frac{dy}{dx})^2\big)^{3/2-->{\frac {d^2y}{dx^2-->.
This allows the radius of curvature of a line to be found from only
Cartesian coordinates.
Another useful formula can relate the above to parametric equation:
\frac{ds}{d\psi} = \frac {\big({\dot{x}^2 + \dot{y}^2}\big)^{3/2-->{\dot {x}\ddot{y} - \dot{y}\ddot{x-->,
where
\dot{x} = \frac{dx}{dt}\ \mbox{and}\ \ddot{x} = \frac{d^2x}{dt^2}.
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.
The general expression for finding the intrinsic equation is:
s = \int_\,\sqrt{1 + \tan^2 \psi}\frac{dx}{d\psi}\,d\psi
This integration is only practical where \frac{dx}{d\psi} is known in terms of \psi
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;
ψ =
f(s) is the
intrinsic equation of the curve.
This is arguably not a true coordinate system, since it describes points on a curve relative to the curve itself rather than the space in which it is embedded, and points not on the curve have no intrinsic coordinates.
Inspection reveals three properties regarding the rate of change of the variables, namely:
\frac{dy}{dx} = \tan \psi
\frac{dx}{ds} = \cos \psi
\frac{dy}{ds} = \sin \psi
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 turns anticlockwise as
s increases; if negative, the curve turns clockwise. It is given by:\rho = \frac{ds}{d\psi}.
It can be proven that the following is true:
\rho = \frac {\big( 1 + (\frac{dy}{dx})^2\big)^{3/2-->{\frac {d^2y}{dx^2-->.
This allows the radius of curvature of a line to be found from only
Cartesian coordinates.
Another useful formula can relate the above to
parametric equation:
\frac{ds}{d\psi} = \frac {\big({\dot{x}^2 + \dot{y}^2}\big)^{3/2-->{\dot {x}\ddot{y} - \dot{y}\ddot{x-->,
where
\dot{x} = \frac{dx}{dt}\ \mbox{and}\ \ddot{x} = \frac{d^2x}{dt^2}.
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.
The general expression for finding the intrinsic equation is:
s = \int_\,\sqrt{1 + \tan^2 \psi}\frac{dx}{d\psi}\,d\psi
This integration is only practical where \frac{dx}{d\psi} is known in terms of \psi
