Curves or surfaces with **intrinsic coordinates** do not depend on a particular coordinate system; they can be measured intrinsically — within the curve or surface itself — without referencing a larger space [1].

Intrinsic coordinates have been around since the early 19th century. However, the system isn’t very versatile and so isn’t widely used. Other names for the system include *convected*, *curvilinear*, *local*, and *natural *coordinates [2].

## Example of Intrinsic Coordinates

For example, in the Cartesian coordinate system, a point on a curve can be described by a location (x, y). In an **intrinsic coordinate system**, the point might be described by its position relative to the length of an arc *s* and an angle ψ which the tangent line at the point makes with the x-axis [3].

A curve that has the equation s = f(ψ) is said to be given

*intrinsically*[4].

## Where are Intrinsic Coordinates Used?

Intrinsic coordinates are used in **coordinate free geometry**, where geometric objects and relations can be expressed without reference to a particular coordinate system [5]. In practice, it’s usually not very useful to convert Cartesian or polar equations for curves to intrinsic coordinates as those systems are much more versatile. For example, a point (i.e., a location in space) has no intrinsic properties. Therefore, it isn’t possible to represent a general point in the plane using intrinsic coordinates; you can only represent a point on a curve [6].

## References

[1] Intrinsic Geometry. Retrieved January 29, 2022 from: https://math.etsu.edu/multicalc/prealpha/Chap3/Chap3-8/printversion.pdf

[2] Mathematical Modeling and Optimization of Complex Structures (2015). Springer International Publishing.

[3] Gaulter, B. & Gaulter, M. (2001). Further Pure Mathematics. OUP Oxford.

[4] Horadam, A. (2014). Outline Course of Pure Mathematics. Elsevier Science.

[5] Coordinate Free Geometry. Retrieved January 29, 2022 from: http://www.dgp.toronto.edu/~karan/courses/418/notes/Coordfreegeom.pdf

[6] Evans, C. (2019). Engineering Mathematics. A Programmed Approach, 3rd Edition. CRC Press.