For every integer$k\geq 2$and every$A\subseteq \mathbb{N}$, we define the$k$-directions setsof$A$as$D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$and$D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$, where$\Vert \cdot \Vert$is the Euclidean norm and$A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$. Via an appropriate homeomorphism,$D^{k}(A)$is a generalisation of theratio set$R(A):=\{a/b:a,b\in A\}$. We study$D^{k}(A)$and$D^{\text{}\underline{k}}(A)$as subspaces of$S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets$X\subseteq S^{k-1}$such that there exists$A\subseteq \mathbb{N}$satisfying$D^{\text{}\underline{k}}(A)^{\prime }=X$, where$Y^{\prime }$denotes the set of accumulation points of$Y$. Moreover, we provide a simple sufficient condition for$D^{k}(A)$to be dense in$S^{k-1}$. We conclude with questions for further research.