Let $\gamma(t)$ be a $C^r$ smooth curve in the plane. Suppose $r\ge 2$, so that one can define the curvature $\kappa(t)$ at $\gamma(t)$. For example, $\kappa(t)=0$ means that the curve is kind of flat at that point.
What does it look like if $\kappa(t)=\infty$? Does it always correspond a corner?
Thanks!
Not a corner, in fact recall at a point, $$\tau = 1/ r $$ where r is the radius of the circle of best fit. So I don't really think you could have infinite curvature, rather you would approach infinite curvature as the radius approached zero. Think of a single point, or the singularity before the big bang.
If you are looking towards the problem in a different way just noting that the curvature is a property of the second derivative.
Think about $f(t)=|t|^{3/2}$; it is $C^1$, yes, but also $f″(0)=+\infty.$
To turn this into a surface with infinite curvature, note that $X(u,v)=(u,|u|^{3/2},v)$ so you can see just extending this curve in the z direction, we are forming a sheet. You will find the normal curvature at the origin in the direction normal to the xy-plane is in fact infinite.