# Is the gradient of the self-intersections of a curve zero?

Suppose a curve with self-intersections can be described by $$\phi(x,y)=0$$. Suppose the intersections are $$T_i$$, $$i=1,2,...$$ and the gradient $$\nabla \phi$$ at those intersections are well defined. Then is it true that $$\nabla\phi(T_i)=0$$ for all $$i$$? In other words, are the gradients at those intersections all zero?

If we agree that $$\phi$$ is continuously differentiable (so $$\nabla \phi(x,y)$$ is a continuous function of $$x$$ and $$y$$), then yes, this must be true.

The reason is that, if $$\nabla \phi(x_0, y_0) \neq 0$$ for some $$(x_0, y_0)$$, then the implicit function theorem guarantees that (locally) we can write $$y$$ as a function of $$x$$ or $$x$$ as a function of $$y$$. However, at a self-intersection $$T_i$$, our curve fails the horizontal and vertical line tests, so we cannot express $$x$$ as a function of $$y$$ or $$y$$ as a function of $$x$$.

Assuming $$\phi(x,y)$$ is continuously differentiable in a neighbourhood of $$T_i$$, yes, because otherwise you could use the Implicit Function Theorem to get a unique curve in a neighourhood of $$T_i$$ satisfying $$\phi(x,y) = 0$$.

## A different argument

(For the case where two branches of the curve have distinct tangents at the intersection point)

Since $$\phi$$'s values do not change along the curve, we know that if $$\phi$$ is continuously differentiable then any of its directional derivatives along the curve is $$0$$.

At an intersection point, we would have a null directional derivative along two linearly independent directions. Having projection $$0$$ along two linearly independent vectors, we know the gradient is $$0$$.

• This assumes the two branches of the curve have distinct tangents at the intersection point. But what if they have the same tangent there? – Robert Israel Apr 21 at 18:16
• @RobertIsrael Thanks for the observation! I hadn't thought about that, but it seems the argument does not apply in that case. I'll try to think if it can be expanded to cover it. – dafinguzman Apr 22 at 20:03