11:43 AM — Straight No, not another gay marriage posting, but another math one. (Go ahead and groan anyway.)

Mathematically, what constitutes a straight line? People always say "the shortest distance between two points." I've got an alternative definition, maybe better for a line segment on a curved function. Yes, I was thinking about this after fireworks last night.

• First derivative, f', is geographically a slope, or laymen can think of it as going uphill or downhill. Positive is uphill, negative is downhill, zero is flat. But it might not be "straight," like if it's the top of a hill.


• Second derivative, f", is curvature, or concavity. Positive is concave up (like a valley), negative is concave down (like top of a hill), zero is often called an inflection point.


• Third derivative, f⁽³⁾, I'm not too sure what it means. Maybe positive is the curve is tightening (spiral in), negative is spiraling out, zero is a circle?


• Fourth derivative, f⁽⁴⁾, seriously grasping at straws here, how much the spiral is changing its tightness?

Point is, f'=0 is flat, but it's not straight. f"=0 is a point with no curvature for an instant, but the curvature is changing from one way to the other, so it's not straight. Maybe straight would be a point/segment where f⁽ⁿ⁾=0 for n>1 ? Why in the world am I thinking about this?  ¶ permalink | 0 musings