Consider a car driving along a curvy road. The tighter the curve, the more difficult the driving is. In math we have a number, the curvature, that describes this “tightness”. If the curvature is zero then the curve looks like a line near this point. While if the curvature is a large number, then the curve has a sharp bend.

Let assume be a curve. We also adopt the following notation

.

More formally, if is the unit tangent vector function then the curvature is defined at the rate at which the unit tangent vector changes with respect to arc length .

Definition. Curvature is given by

*Example*. Let consider an example of a circle of radius given by

.

We then see that

which implies . Thus, . Our circle is now parameterized by arc length as follows

.

The curvature vector at a given length is then

and that

.

As we stated previously, this is not a practical definition, since parameterizing by arc length is typically impossible. Instead we use the chain rule to get

.

This formula is more practical to use, but still cumbersome. is typically a mess. Instead we can borrow from the formula for the normal vector

to get the curvature

.

**Curvature of a plane curve**. If a curve in the -plane is defined by the function then there is an easier formula for the curvature. We can parameterize the curve by

We have

and

.

Their cross product is just

which has magnitude

Thus we have

Definition. Curvature is given by.

*Example*. The curve defined by has

as its curvature.

See also: http://www.ltcconline.net/greenl/courses/202/vectorFunctions/curvat.htm

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