**Curvilinear lines** are smooth lines or curves, like the curves we see in trigonometric functions or parabolic functions. “Curvilinear” refers to something more complicated though, than just a simple curve or set of curves—it’s a set of curves with a *purpose*.

For example, during curvilinear regression, the procedure considers an infinite number of curves—or at the least, a very large family of curves. That leads us to a slightly more formal (and general) definition: In general, we can say that something is curvilinear if it is formed (or characterized by) a set of curved lines. That doesn’t mean you need to have a large number of lines: a single curved line still counts as a set of one.

## Curvilinear Coordinates

**Curvilinear coordinates** are a coordinate system used for Euclidean space where the coordinate lines (think: the x and y-axis, and the grid lines on your graph paper) may be curved. You can think of curvilinear coordinates as a generalized coordinate system, free of the straight line restraints that you see with a Euclidean system.

Curvilinear coordinates don’t *have *to be based on curved lines. Cartesian coordinates are one type of curvilinear coordinate systems. But so are cylindrical coordinates (an extension of two-dimensional polar coordinates to three-dimensional) and spherical polar coordinates. If there’s a one to one mapping between coordinate systems, we can convert between them. For example, the mapping between spherical polar coordinates and Cartesian coordinates uses these equivalences:

Different systems are given different names, depending on how they intersect:

- If the intersections are right angles, the system is called an
**orthogonal coordinate system**. - Otherwise, it is a
**skew coordinate system**.

The image below shows three coordinate systems in two dimensional space, as well as the mappings between them.

## Why use Curviliear Coordinates?

If you wanted to map a roller coaster, a spherical cell, or the shape of a biological cell, the usual lines in a x-y plane won’t quite do. However, if you have an **infinite number of curves** to play with, then you would be able to create a more accurate representation of just about every real-life system. These representations can help you to see aspects of the problem that might not be readily apparent in a regular Cartesian plane.

However, a Cartesian coordinate system is usually easier to work with.

## Curvilinear Regression

**Curvilinear regression** involves finding the equation of a curved line that fits a particular set of data points. You might use this type of regression if you’ve begun analyzing a data set with linear regression, and then you realize that the correlation between your x and y variables doesn’t really look like a straight line— it looks more like a curve.

**Polynomial regression** (which includes quadratic regression and cubic regression) is a frequently used method of curvilinear regression. What type of curve you decide to fit your data to should depend on a preliminary study of the data. You will want to use the simplest possible curve that fits the data well.

## References

Antoni, M. (2018). Calculus with Curvilinear Coordinates: Problems and Solutions. Springer.

1.16 Curvilinear Coordinates. Retrieved September 12, 2019 from: http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_16_Curvilinear_Coordinates.pdf

McDonald, J.H. 2014. Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland. Pages 213-219. Retrieved online from http://www.biostathandbook.com/curvilinearregression.html on June 22, 2019.