Laymen’s terms explanation of curvature

Question

I’m attempting to explain curvature in layman’s terms to my class before explaining the formula. I like to do this first to give my students an idea of what we are finding.

Some people explain curvature as a “measure of how fast a curve is changing direction at a given point.” But this seems misleading to me. It seems to me that when someone say “how fast” most people interpret that as a change in direction per unit time. But time has nothing to do with it correct?

For example the curvature is the same for any particular curve regardless of the speed of its parametrization. So time has nothing to do with it. The way I’ve been explaining curvature in laymen’s terms is that it is a “measure of how ‘hard’ a curve is changing direction at a given point”.

Do you think this is sufficient for a laymen’s term explanation without leading the students astray with a time component. Perhaps you have another way to explain it? Or perhaps other people’s laymen definition is perfectly fine and I’m over analyzing.

Advice?

Answer 1

The standard definition for the curvature at a particular point on a curved path is just $\frac{1}{R}$, where R is the radius of curvature back to some notional associated point (valid for that particular point on the curved path only). The definition need not involve time, but most applications involve time and motion.

e.g. if in a racing car the absolute centripetal acceleration, $a$ (measured using an 3 axis-accelerometer) and forward speed, $v$ are measured regularly as the car drives around a racing track, the curvature can be determined at all the points of measurement using $a=\frac{v^2}{r}$. From this data a map of the track can be drawn. The usefulness of allowing a ever changing curvature is that any arbitrary track can be mapped not just circular ones, with a fixed radius of curvature. The forward speed of the car can vary as this acceleration/deceleration is perpendicular to the centripetal acceleration.

Another application for the concept of curvature and radius of curvature is refraction of light by a varying density gas, such as for light from the sun or stars passing down through the earths atmosphere.

Answer 2

Time can be an ok way to talk about it. Take a particle travelling along a curve, maybe defined by $x(t)$ or maybe it happens to be the graph of some function of time. Then, curvature is measuring how the velocity is changing, encoding the sign (think concavity/convexity) of this change as well.

Answer 3

I think about it as related to the radius of a circle or ball that will juuust lie tangent to the curve at that point. Any larger radius and the circle will be too large to touch at that point, and so will necessarily touch the curve in 2 other points.

Of course this isn't exactly rigorous, but I found it to be a helpful conceptualization.