Originally Posted by BigE
… I often thought of the turn as an infinite sequence of very small arcs, where the radius of each arc segment changes as you progress through the turn. Such a model implies that the center of each of the circle that defines each arc segment is moving. …
Originally Posted by BigE
… Maybe a parabolic approximation IS better, as the focus remains constant…
This is an interesting thought, and has been suggested before.
One could, of course, construct a continuous ski path by splicing together a series of alternately flipped parabolas. In fact, one could not only match the function values at the splice points, but even match their 1st derivatives where they are joined. Unfortunately, it would be impossible to match their second derivatives where they join – the 2nd derivatives would, of course, be of opposite sign and of constant, non-zero magnitude on either side of the splice points.
I suspect you may be thinking about parabolas because you may once have heard someone say that the 2nd derivative *is* the curvature of a function, so since the 2nd derivative of a parabola is constant, a skier’s path constructed out of linked parabolic arcs would lead to regions of constant radius of curvature.
Unfortunately, the above statement is not true. The full formula for the curvature (or, its reciprocal, the radius of curvature) is not quite so simple as just the 2nd derivative. It’s given by the first formula shown on this page: http://en.wikipedia.org/wiki/Curvature .
In fact, slightly further down on the same page, they work out the formula for the radius of curvature of a parabola (…just under “Example”). It most certainly does not remain constant, and, unfortunately, for a parabola, just like for the sine curve, both the radius of curvature and the position of the center of curvature both change as you progress along the curve. In fact, they change in remarkably similar manners for both types of curves. For example, the radius of curvature for both types of curves rapidly grows extremely large as you move away from the apices in both curves.
You mentioned the concept of the focus of a parabola. This is a quantity which does indeed remain at a fixed point in space, but the focus of the parabola is unrelated to its instantaneous radius or center of curvature.
While I could have used linked parabolas as the model for the path of skier, I did not because doing so would not have provided either more accuracy or ease of computation. In fact, sinusoids are considerably easier to manipulate. In addition, they are smoother than linked parabolas because all their derivatives exist, are bounded and continuous. The 2nd derivative of linked parabolas would be discontinuous.
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Tom / PM