Find the Ideal Curvature for a Curve

Imagine you are building a roller coaster. You need to create a curved shape for the track, which will be designed on a computer before being built out of metal. You want a curve that is fun to travel along, which means you want a lot of sharp curves—but not too sharp, unless you want your amusement park guests to get sick or pass out.

A similar consideration is faced by engineers building a railroad or a highway: you want the path of the road to have curves that are not too sharp—in this case, to prevent the cars or trains from having to slow down (reducing efficiency) or even to wreck. The reason, of course, is that when you travel along a strongly curved section of track or road, you feel an acceleration. The higher the curvature, the stronger the acceleration, all other things being equal. However, the acceleration you feel depends on how fast you are going along the track (the faster you go, the greater the acceleration), while curvature is a property intrinsic to the track itself.

Curvature is essentially defined as how fast the direction of motion changes as you move along the track. The curvature is highest on the sharpest curves of track. Along straight sections of track, the curvature is zero. Also important in applications is the rate of change of curvature as you go along the track (this produces a “jerk” just as curvature produces acceleration). In both cases, we think of how fast something changes if you move along the track at a constant (unit) speed. So, technically, these are derivatives with respect to arc length.

Here is an example of a fun curve you might want to use in your roller coaster:

Besides giving the curvature and plotting the curve and the point you specify, notice that Wolfram|Alpha also shows you something called the “osculating sphere” and its center, radius, and equation. The osculating sphere and its two-dimensional cousin, the osculating circle, are great ways to visualize the curvature at a specific point on a curve.

What is the osculating circle/sphere? Suppose you are driving along a curved part of road, so you are holding the steering wheel of your car in a rotated position. If you then continued to hold the steering wheel at that exact position, what would happen? You would drive in a precise circle, of course (assuming you didn’t crash into anything first!). This “osculating” (literally, “kissing”) circle has the same curvature as the part of road you are on. The greater the curvature at that point in the road, the smaller the osculating circle will be. In fact, the curvature is equal to the inverse of the radius of the osculating circle.

You can ask for the osculating circle directly, for the “radius of curvature” (which is the radius of the osculating circle), or for the “center of curvature” (the center of the osculating circle). For example:

Besides investigating the curvature of standard functions from the real numbers to the real numbers, Wolfram|Alpha can also find curvatures and osculating circles of curves given in polar coordinates, implicit curves in two dimensions, and parametric curves in any number of dimensions:

You can even include unspecified parameters in your query, and Wolfram|Alpha will try to include them in the results. In this case, we find that the osculating circle of a circle is itself:

You don’t have to ask for the curvature at a specific point. If you don’t, Wolfram|Alpha will try and show you the curvature and osculating circle as a function of the independent variable, both algebraically and graphically:

Curvature can also be generalized to surfaces of any dimension, although there are several different ways. We hope to bring you such advanced functionality soon. In the meantime, enjoy exploring curvature on curves!