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Solution of your original question: A smooth curve cannot intersect infinitely many times lines in all directions.

Let $\gamma: T\to \Gamma$ be the parametrization of your curve, where $T$ is the unit circle, $\gamma\in C^\infty$ and $\gamma'(t)\neq 0$ for all $t$. Then $$f(t)=\frac{\gamma'(t)}{|\gamma'(t)|}$$ is a $C^\infty$ map $T\to T$. Applying Sard's Lemma, we obtain that almost every value of $f$ is regular, therefore, the $f$-preimage of almost every value is finite.

On the other hand, if the curve intersects some lines in direction $\theta\in T$ infinitely many times, then it must have infinitely many points where the direction of the tangent line is $\theta$.

Indeed, suppose that $t_k$ are the points where $\gamma(t_k)$ belong to some line $\ell$ with direction $\theta$, wlog $\theta$ is the horizontal direction and $\ell$ is the $x$-axis. Let $t^*$ be an accumulation point of $t_k$. Then in a neighborhood of $t_k$, $\Gamma$ is a graph of a smooth function, $\phi(x)$ which intersect the horizontal axis infinitely many times in any neighborhood of $\gamma(t^*)$,
so by Rolle's theorem, $\phi'(t)=0$ infinitely many times in a neighborhood of $t^*$.

The best reference for Sard's Lemma is J. Milnor, Topology from the differentiab;e viewpoint, UP of Virginia, Charlottesville, 1965,

Solution of your original question: A smooth curve cannot intersect infinitely many times lines in all directions.

Let $\gamma: T\to \Gamma$ be the parametrization of your curve, where $T$ is the unit circle, $\gamma\in C^\infty$ and $\gamma'(t)\neq 0$ for all $t$. Then $$f(t)=\frac{\gamma'(t)}{|\gamma'(t)|}$$ is a $C^\infty$ map $T\to T$. Applying Sard's Lemma, we obtain that almost every value of $f$ is regular, therefore, the $f$-preimage of almost every value is finite.

On the other hand, if the curve intersects a line in direction $\theta\in T$ infinitely many times, then it must have infinitely many points where the direction of the tangent line to $\Gamma$ is $\theta$.

Indeed, suppose that $t_k$ are the points where $\gamma(t_k)$ intersects some line $\ell$ with direction $\theta$, wlog $\theta$ is the horizontal direction and $\ell$ is the $x$-axis. Let $t^*$ be an accumulation point of $t_k$. Then in a neighborhood of $t_k$, $\Gamma$ is a graph of a smooth function, $\phi(x)$ which intersect the horizontal axis infinitely many times in any neighborhood of $\gamma(t^*)$,
so by Rolle's theorem, $\phi'(t)=0$ infinitely many times in a neighborhood of $t^*$.

The best reference for Sard's Lemma is J. Milnor, Topology from the differentiab;e viewpoint, UP of Virginia, Charlottesville, 1965,

Solution of your original question: A smooth curve cannot intersect infinitely many times lines in all directions.

Let $\gamma: T\to \Gamma$ be the parametrization of your curve, where $T$ is the unit circle, $\gamma\in C^\infty$ and $\gamma'(t)\neq 0$ for all $t$. Then $$f(t)=\frac{\gamma'(t)}{|\gamma'(t)|}$$ is a $C^\infty$ map $T\to T$. Applying Sard's Lemma, we obtain that almost every value of $f$ is regular, therefore, the $f$-preimage of almost every value is finite.

On the other hand, if the curve intersects some lines in direction $\theta\in T$ infinitely many times, then it must have infinitely many points where the direction of the tangent line is $\theta$.

Indeed, suppose that $t_k$ are the points where $\gamma(t_k)$ belong to some lines with direction $\theta$, wlog $\theta$ is the horizontal direction. Let $t^*$ be an accumulation point of $t_k$. Then in a neighborhood of $t_k$, $\Gamma$ is a graph of function, $\phi(x)$ which intersect the horizontal axis infinitely many times,
so by Rolle's theorem, $\phi'(t)=0$ infinitely many times in a neighborhood of $t^*$.

The best reference for Sard's Lemma is J. Milnor, Topology from the differentiab;e viewpoint, UP of Virginia, Charlottesville, 1965,

Solution of your original question: A smooth curve cannot intersect infinitely many times lines in all directions.

Let $\gamma: T\to \Gamma$ be the parametrization of your curve, where $T$ is the unit circle, $\gamma\in C^\infty$ and $\gamma'(t)\neq 0$ for all $t$. Then $$f(t)=\frac{\gamma'(t)}{|\gamma'(t)|}$$ is a $C^\infty$ map $T\to T$. Applying Sard's Lemma, we obtain that almost every value of $f$ is regular, therefore, the $f$-preimage of almost every value is finite.

On the other hand, if the curve intersects some lines in direction $\theta\in T$ infinitely many times, then it must have infinitely many points where the direction of the tangent line is $\theta$.

Indeed, suppose that $t_k$ are the points where $\gamma(t_k)$ belong to some line $\ell$ with direction $\theta$, wlog $\theta$ is the horizontal direction and $\ell$ is the $x$-axis. Let $t^*$ be an accumulation point of $t_k$. Then in a neighborhood of $t_k$, $\Gamma$ is a graph of a smooth function, $\phi(x)$ which intersect the horizontal axis infinitely many times in any neighborhood of $\gamma(t^*)$,
so by Rolle's theorem, $\phi'(t)=0$ infinitely many times in a neighborhood of $t^*$.

The best reference for Sard's Lemma is J. Milnor, Topology from the differentiab;e viewpoint, UP of Virginia, Charlottesville, 1965,

Solution of your original question. Let $\gamma: T\to \Gamma$ be the parametrization of your curve, where $T$ is the unit circle, $\gamma\in C^\infty$ and $\gamma'(t)\neq 0$ for all $t$. Then $$f(t)=\frac{\gamma'(t)}{|\gamma'(t)|}$$ is a $C^\infty$ map $T\to T$. Applying Sard's Lemma, we obtain that almost every value of $f$ is regular, therefore, the $f$-preimage of almost every value is finite.

On the other hand, if the curve intersects some lines in direction $\theta\in T$ infinitely many times, then it must have infinitely many points where the direction of the tangent line is $\theta$.

Indeed, suppose that $t_k$ are the points where $\gamma(t_k)$ belong to some lines with direction $\theta$, wlog $\theta$ is the horizontal direction. Let $t^*$ be an accumulation point of $t_k$. Then in a neighborhood of $t_k$, $\Gamma$ is a graph of function, $\phi(x)$ which intersect the horizontal axis infinitely many times,
so by Rolle's theorem, $\phi'(t)=0$ infinitely many times in a neighborhood of $t^*$.

The best reference for Sard's Lemma is J. Milnor, Topology from the differentiab;e viewpoint, UP of Virginia, Charlottesville, 1965,

Solution of your original question: A smooth curve cannot intersect infinitely many times lines in all directions.

Let $\gamma: T\to \Gamma$ be the parametrization of your curve, where $T$ is the unit circle, $\gamma\in C^\infty$ and $\gamma'(t)\neq 0$ for all $t$. Then $$f(t)=\frac{\gamma'(t)}{|\gamma'(t)|}$$ is a $C^\infty$ map $T\to T$. Applying Sard's Lemma, we obtain that almost every value of $f$ is regular, therefore, the $f$-preimage of almost every value is finite.

On the other hand, if the curve intersects some lines in direction $\theta\in T$ infinitely many times, then it must have infinitely many points where the direction of the tangent line is $\theta$.

Indeed, suppose that $t_k$ are the points where $\gamma(t_k)$ belong to some lines with direction $\theta$, wlog $\theta$ is the horizontal direction. Let $t^*$ be an accumulation point of $t_k$. Then in a neighborhood of $t_k$, $\Gamma$ is a graph of function, $\phi(x)$ which intersect the horizontal axis infinitely many times,
so by Rolle's theorem, $\phi'(t)=0$ infinitely many times in a neighborhood of $t^*$.

The best reference for Sard's Lemma is J. Milnor, Topology from the differentiab;e viewpoint, UP of Virginia, Charlottesville, 1965,