I did a little more research on this topic & found that the answer appears to be very simple. Consider a moving particle going around the outside of a triangle as fast as it can. It obviously has to change direction in order to complete a circuit of the triangle -- the simplest way to do this is obviously the circumcircle. However, going around the circumcircle may take longer than utilizing this same potential for acceleration & deceleration to navigate closer to the triangle. For example, if the triangle is very long and thin -- approximating a line segment -- then the fastest way to circumnavigate it would be to utilize all of the acceleration to gain as much speed as possible up to the midpoint, and then to utilize all of the acceleration to slow down towards the end. The reason is to avoid the spike in acceleration that would be required to turn the 180 degree corner at the end (we have limited the maximum magnitude of acceleration for the entire round trip). It would appear that the fastest circuits around triangles utilize pieces of _parabolas_ to get around the triangle faster -- three pieces to be exact. During each of the three parabolic pieces, the entire ability to accelerate is used to create a uniform acceleration field much like that studied in high school physics by the acceleration due to gravity, except that the uniform acceleration is in the plane of the triangle, and used to slow down the particle just enough to clear each vertex before accelerating to the next. One way to think about this curve is to follow Hamilton and consider its "hodograph" -- take all of the _velocity_ vectors and bunch them at the origin. Then the hodograph for the fastest curve can be constructed by taking each side of the triangle and considering it as a vector in the direction that the particle will "trace" it (i.e., stay "close" to it); place each of these three vectors at the origin. Then the acceleration vectors for the fastest curve will trace out the perimeter of the triangle formed by these three vectors (suitably scaled). Since the _magnitude_ of the acceleration is at all times the same, the perimeter will be traced out at constant speed (we aren't charged for the change in direction of the acceleration at the corners of this derived triangle). Around the original triangle, however, the particle will speed up & slow down as it goes around the parabolic path pieces. (The intuitive idea is that the longer sides of the triangle give the particle more distance to speed up and more distance to slow down, so the particle can achieve a higher velocity "along" those sides.) If the original triangle is isosceles, then the overall shape of the 3 parabolic pieces will look egg-shaped. I haven't yet worked out what triangle shape will produce a pleasing egg shape. If someone already has a particle simulation applet, then perhaps these data could be put into it. At 08:09 AM 2/22/2007, Henry Baker wrote:
In bicycle/car races, the vehicles have a maximum acceleration potential. For simplicity's sake, let's assume that the max acceleration in any direction is capped at a certain constant. When going around corners, these vehicles "go wide", so that they don't go beyond their maximum acceleration when turning the corner. On the other hand, their ability to "go wide" is limited by the width of the road/track.
So, one way to produce an egg-shape is to consider a racing track of a certain width, whose overall shape is in the form of an isosceles triangle. If the triangle is equilateral, and the width is wide enough, then the curve is circular. However, when we start reducing the width, we get a curve which starts getting pinched in 3 places. If we then lengthen the altitude of the triangle so that it is no longer equilateral, then the curve starts looking egg-shaped.
I assume that curves such as these have been studied. Does anyone know what their equations are? Can anyone point me to any references?