I’ve heard that teachers of mathematics tell their students that a line is always straight, and so a line of best fit is always straight. I’ve heard that teachers of mathematics say that there are also curves of best fit, but that these are not lines, because they are not straight. Consider a graph showing inverse proportionality; I’ve heard that teachers of mathematics say you shouldn’t draw a line of best fit for such a relationship, it should be a curve.
Euclid defined a straight line as a ‘breadthless length‘ that ‘lies evenly with respect to the points on itself‘. Huh, but what if those points form an obvious curve? Then the straight line would be the line that lies evenly amongst those points; a curve. My interpretation is that Euclid is saying a curve is, in fact, not just a line, but a straight line!
In geometry a straight line (in the colloquial sense) is an example of the more general geometric element called a geodesic. A geodesic is a line (or curve, if there’s a difference, which there isn’t) of the shortest possible length connecting two points. An example of a curved straight line (whatever that means) is a direct flight path called a great circle. Consider the great circle connecting Tahiti to Paris (ostensibly the longest domestic flight, but I think that’s stretching the definition of domestic). The shortest line (geodesic) connecting Tahiti to Paris passes through Greenland.
That might come as a surprise, but it’s because we are considering the shortest length over a curved surface. By some (most?) definitions, you cannot draw a geodesic on a curved surface and say it’s a straight line, but a straight line is just a particular type of geodesic for Euclidean geometry (flat geometry) and Euclid himself didn’t define the line in such a restrictive way.
I would argue that a curve is just an example of the shortest line that passes through points as closely as possible, which makes it straight.
But I’m not a mathematician, and someone is bound to read this and become enraged that I had the audacity to stay a curve is a straight line, but by some definitions it is, and if you’re going to point at my definitions and say they are wrong because they disagree with the mathematicians’ definitions when mathematicians cannot make their mind up what the notation of a superscript negative one means, I cannot take you seriously. Does that notation mean a reciprocal or an inverse function? I guess we’ll never know.
Reminder: Weekly Live Tuition Sessions!
SUNDAY 11th January
GCSE Physics 9:30am – 10:20am Infrared radiation A-Level Physics 10:30am – 11:20am Measurement of specific charge GCSE Astronomy 11:30am – 12:20pm Exploring the moon If you wish to enrol on the tuition sessions and haven’t yet, then click enrol below
Physics with Keith 
