Can one construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$ so that

everyorthogonal projection onto a two dimensional plane is a unit disc?

It is easy to construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$ so that one orthogonal projection is a unit disc: take a Peano-type curve $\gamma=(\gamma_1,\gamma_2):[0,1]\to\mathbb{R}^2$ that fills the unit disc and define $\Gamma(t)=(\gamma_1(t),\gamma_2(t),t):[0,1]\to\mathbb{R}^3$. Clearly $\Gamma$ is one-to-one so it is an embedding and its projection onto the first two coordinates fills the disc. It remains now to "glue" the ends to make it an embedding of $\mathbb{S}^1$.