Choosing a sphere with finitely many intersections with a curve

Question

Let $\Phi:U\mapsto V$ be a $C^1$-diffeomorphism between two open subsets of $\mathbb{R}^n$. for two points $x,y \in U$ let $$\gamma(t) = tx + (1-t)y$$ be a curve in U. Can i always choose a $r\in\mathbb{R}$, such that $$\partial B_r(\Phi(y))\cap im(\Phi\circ\gamma)$$ is finite?