Quote:
Originally Posted by

**DWFII**
So I will beg indulgence as I explain this in the terms I understand...and someone can correct me if I make a mistake.

A vector is a direction or even a measure of velocity in one direction. Any change in the acceleration/velocity, plus or minus, is going to cause a change of direction. Too abrupt a change and you get an awkward curve--one that doesn't seem coherent or doesn't "flow."

The best example of an accelerating "fair" curve might be a mathematically perfect spiral.

OK...now I'm getting myself into deeper water than I can swim in, and I think I will leave it to others.

Thank you for the interesting post DWFII, I agree with your breast v/s foot example! There are a few inaccuracies in your post, so I'll try to complement the definitions you offered.

Velocity is a good example of a vector, but there are many others. Acceleration is also a vector, and so is any quantity that has a direction associated. Changes in magnitude do not necessarily imply a change in direction.

What you're trying to explain is a concept called "smoothness". The strict definition involves derivatives (smooth means all derivatives are continuous), and can be found at http://en.wikipedia.org/wiki/Smooth_function for those mathematically-inclined.

As long as there are no "sudden" changes in acceleration, the trajectory we're tracing qualifies as smooth. As you can see, when tracing the toe of a shoe you have a very drastic change of direction, but that doesn't affect the smoothness of the curve. If you were to "bump" the pencil while you were tracing, then the resulting curve wouldn't be smooth at all, because that bump would mean a sudden change in acceleration, and therefore "break" the smoothness of the curve.

Any trajectory without "bumps" would be a good example of a "fair" curve, such as a sine, cosine, or, indeed, a spiral. If you see any "sharp corners", that generally means the curve is not smooth.

The golden mean is an interesting concept, and is intrinsically related to how we perceive proportion, but it's too deep a subject to explain in a few lines!

Hope it helps!