Wednesday, January 1, 2014

Bearing/Distance curve

Suppose a curve C has a certain starting point in a chosen coordinate system. For example -3, 2 in Cartesian coordinates.
My idea was that you could have a Bearing/Distance plot that for a curve without discontinuities takes the form of another smooth curve Bd

The distance is an amount d that is the arclength of the curve C, and the bearing varies mod 2 between 0 and 2 and indicates the direction the tangent line to the curve C is pointing when the arclength equals d.
The bearing direction is mapped as follows.
 But any direction within the range -inf..inf is valid the direction is mod 2, but a value of 6 might indicate that you've turned a full circle 3 times. So Bd generates a curve C as shown.

See the red curve C starts heading in direction 0 (also direction 2) but rapidly turns it's bearing until it is facing in the 1 direction, stays heading in that direction for about 3 arclength and then gradually turns until it is not quite facing in the 0 or 2 direction.

I think that one nice thing about this setup is that because straight lines in the Bd plot generate circles of varying diameter, the curvature of the C plot at a certain point is the slope of the Bd plot at the corresponding point, with decreasing curvature in the C plot corresponding to a flatter line in the Bd plot.

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