I'm reading about parametric and geometric continuity using these UC Berkeley Lecture notes.
In the section "Graph of the curve", this is written:
While the conditions for parametric continuity seem stronger than geometric continuity, they are not. There are C1 curves that are not G1.
Can anyone give an example for such a curve?
Found a similar question here but the example given is of a curve which is G1 continuous but not C1 continuous.
These Cornell lectures seemingly give an example of such a curve but it is not very clear. (Page 6 of the pdf)
Consider the parameterized curve $\gamma(t)=(t^3,t^2)$. This parameterization is clearly $C^1$, because each component is (infinitely!) differentiable.
However, the graph of the curve looks like this:
As you can see, the left and right tangent vectors at the origin (where $t=0$) do not match, so the curve is not $G^1$.
This is possible because $\gamma'(0)=\left<0,0\right>$. That is, the parameterization comes gradually to a stop at $t=0$, which allows it to smoothly traverse a geometrically unsmooth curve.