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Q: Is the edge of a circle with an infinite radius curved or straight?

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Posted in Strange

Physicist: Straight.

In fact, in mathematics the “curvature” of a curve is usually defined as the “reciprocal of the radius of the osculating circle”. This is fancy talk for: fit a circle into the curve as best you can, then measure the radius of that circle, and flip it over.

It makes sense; a smaller circle should have a higher curvature (it’s turning faster) but it has a smaller radius, R. So, use 1/R, which is big when R is small, and small when R is big. There are some more technical reasons to use 1/R (like that you can apply it directly to calculating the centrifugal force on a point following that path, or that it gives rise to the entirely kick-ass Descartes’ theorem), but really it’s just one of the more reasonable definitions.

So, just working with the standard definition, you say: the curvature of a circle is 1/R, if I let the radius become infinite, the curvature must go to zero. Zero curvature means no bending of any kind. Must be a line.

Old school topologists get very excited about this stuff.

Say you have two lines on a plane. They’ll always intersect at exactly one point, unless they’re parallel in which case they’ll never intersect at all. But the Greek Geometers, back in the day, didn’t like that; they wanted a more universal theorem. So they included the “line at infinity” with their plane, and created the “projective plane”. In so doing they created a new space where every pair of straight lines intersect at one point, no matter what.

To picture this, imagine the ground under your feet as the plane and the horizon as the line-at-infinity.

Parallel lines meet at two points on the horizon (in opposite directions). So the line at infinity is weirdly defined with opposite points on the horizon being the same point. Mathematicians would say “antipodal points are identified”. In the projective plane two lines always meet at one point.

Notice that east-west parallel lines meet at the east-west point on the line at infinity, and north-south parallel lines meet at the north-south point at infinity. So you do need the entire horizon, not just a single “far away point”. A single point would yield the Riemann sphere, which is also good times.

Back to the point. If you think about circles in this new-fangled projective plane, one of the first questions that comes to mind is “what happens if the circle includes a point on the line at infinity?”.

No matter how big the circle is, or where its center is, the whole thing will always be in the plane (not all the way out to the line at infinity). If the circle does have a point on the horizon, then you’ll find that the center also has to be at infinity (if it’s in the plane, then it’ll be closer to some points on the circle than others, but the center point is the same distance to every point on the circle). Specifically, the center will be on the line at infinity exactly 90° away from where the circle intersects the line.

This agrees surprisingly well with the intuition behind the more colorful circle picture above. Science and math truly are a beautiful tapestry of interconnections and nerding right the hell out.