Consider two paths (mappings) $$\vec{d}_1(k)=(\sin{k},\cos{k}),\\\vec{d}_2(k)=(\sin{2k},t\cos{2k}),$$ and $k$ is from $-\pi$ to $\pi$ and $t>0$.
Such closed paths are homeomorphic to $S^1$. And the winding number of $\vec{d}_1$ is 1 (when $k$ runs from $-\pi$ to $\pi$, $\vec{d}_1$ winds around the origin once) and the winding number of $\vec{d}_2$ is 2. Then we may say $d_1$ and $d_2$ belong to different homotopy class in the fundamental group $\pi_1(S^1)=Z$.
Question: What is the homotopy class for the mapping $\vec{d}(k)=\vec{d}_1(k)+\vec{d}_2(k)$?
Of course, we can work out the winding number and will find when $t>1$, the winding number of $\vec{d}$ is 2, while the winding number is 1 when $1>t>0$. Further, we can also find when $t>1$, $|\vec{d}_2(k)|\geq|\vec{d}_1{k}|$, and when $0<t<1$, $|\vec{d}_2(k)|\leq|\vec{d}_1{k}|$. From this observation, it seems that the winding number (homotopy class) of $\vec{d}$ is somehow determined by the larger component between $\vec{d}_1$ and $\vec{d}_2$.
Is this always true for any such paths summation? Say $\vec{d}(k)=\vec{d}_1(k)+\vec{d}_2(k)+\cdots$, if there is a largest component $|\vec{d}_i(k)|\geq$ others, then the homotopy class of $\vec{d}$ is the same as $\vec{d}_i(k)$.