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 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 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.
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.
Stephanie Glen. "Curvilinear: Coordinates, Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/curvilinear/
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