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Curve parameters, an analogy

September 28, 2013

While trying to explain curve parameters to my girlfriend (she was proofreading my previous blog post) we hit upon an apt analogy. Imagine you’re taking a car-trip from home to the nearest Hooters book-shop. It’s a good half-hour drive which takes you over two local roads and a motorway.

Road trip parameters

Road trip parameters

If the shape of the road you travel along equals the shape of a curve, then the curve parameters are analogous to the travel times. If you take this trip once every week, then you travel along the same roads every time. Thus the shape of the curve never changes. However some weeks you run into roadworks or thick fog or a traffic jam and then your travel times start to differ. Some parts of the trip may go faster than usual, others may go slower.

The point is that once you have the entire trip data (i.e. the entire curve data, not just the shape) you can then work out your location at 12:15. And you can work out in what direction you were travelling at 12:15. And you can work out your acceleration at 12:15. And you can work out the centrifugal force you were experiencing due to turning at 12:15. All of these properties can be computed as long as you have the curve (the route) and the correct curve parameter (the time).

It’s now also easy to see that parameters can have different densities in different portions of a curve, and how parameter density can be thought of as the ‘speed’ of a curve.

It even allows us to get a feeling for curve derivatives and why parameterization matters. Derivatives are used to compute the tangency and curvature and torsion of curves. A parallel can be drawn between those properties and the forces that push you into your car-seat due to acceleration. Or the forces that pull you out of your seat sideways because you’re turning too sharply. These forces don’t just depend on the shape of the road, they also depend on your travelling speed.

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