>In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.<p>I'm surprised by that definition and always thought of a spiral as a curve with a monotonic signed-curvature function.<p>So, for example, the Euler/Cornu Spiral has a point of inflection where the curvature changes sign at the point of inflection, but the curvature increases continuously all the way from -infinity to + infinity as you travel along the length of the curve. So under my definition the whole Euler Spiral would count as a spiral, even though it stops revolving/emanating from a point just under 1/4 turn after the inflection point.<p>If you split a curve into segments at its curvature minimum and maximum points (vertices in the differential geometry sense [0]) then each segment has monotonic curvature and I'd define those as spiral segments. Vertices and monotonic curvature segments are preserved under inversion, which is mathematically useful.<p>In contrast, inflection points with zero curvature are not preserved under inversion. So the Euler spiral can be transformed by a suitable inversion to a curve like the one defined by Wikipedia, that is a curve emanating out from, for example, the origin.<p>Edit: just spotted this in the Wikipedia article on spirals 1]:<p>> Spirals which do not fit into this scheme of the first 5 examples:<p>> A Cornu spiral has two asymptotic points.<p>> The spiral of Theodorus is a polygon.<p>> The Fibonacci Spiral consists of a sequence of circle arcs.<p>> The involute of a circle looks like an Archimedean, but is not:<p>The Cornu spiral I've covered.<p>The spiral of Theodorus doesn't have a monotonic curvature function - it's a polygon approximation of the Archimedes Spiral, which does.<p>The Fibonacci Spiral's curvature function is a monotonic step-function.<p>The involute of a circle is a log-aesthetic curve, all of which have monotonic curvature functions. (The logarithmic spiral and the Euler spiral are also log-aesthetic curves.)<p>[0] <a href="https://en.wikipedia.org/wiki/Vertex_(curve)" rel="nofollow">https://en.wikipedia.org/wiki/Vertex_(curve)</a><p>[1] <a href="https://en.wikipedia.org/wiki/Spiral" rel="nofollow">https://en.wikipedia.org/wiki/Spiral</a>