Singularity (mathematics) in the context of Differentiable function

The initial singularity or the Big Bang singularity is a simplified model for the origin of the universe, obtained by extrapolating the Big Bang model of cosmology backward to a state of arbitrarily high density and temperature. While the Big Bang refers to the hot, dense state in the early universe from which the expansion of the universe began, extrapolating general relativity beyond this state leads to a singularity. However, this singularity is considered a breakdown of the current theoretical models, not a physically meaningful description of the universe’s origin.

In mathematics, the Tschirnhausen cubic is a cubic plane curve defined by the polar equation$${\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right)}$$or the equivalent algebraic equation$${\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}.}$$

It is a nodal cubic, meaning that it crosses itself at one point, its node. The angle at this crossing point, inside the loop formed by the crossing, is 60°. Because the Tschirnhausen cubic has this singularity, it can be given a parametric equation, and any arc of it can be drawn as a cubic Bézier curve. It is a special case of a sinusoidal spiral, of a pursuit curve, and of a Pythagorean hodograph curve.