Let $X=(x_1,x_2,x_3)$ and $Y=(y_1,y_2,y_3)$ be two points on the unit sphere $S^2=\{(x_1,x_2,x_3)\,|\,x_1^2+x_2^2+x_3^2=1\}$. Is there a "nice" necessary and sufficient condition on the coordinates $x_i$ and $y_i$ for $X$ and $Y$ to be on the same longitude? (I am hoping to avoid the spherical coordinate system, if possible.) For latitude $x_3=y_3$ will do; but how about the longitude case?
How about generation to the $(n-1)$-dimensional sphere (unit sphere of $\mathbb{R}^n$)?
Same longitude means the same projection to the equator, or the same homogeneous first 2 coordinates, i.e., $x_1:x_2=y_1:y_2$, i.e., either $x_2=y_2=0$ or neither is 0 and $\frac{x_1}{x_2}=\frac{y_1}{y_2}$.
Higher dimensions can be handled similarly by using equatorial projection. Note that the spherical coordinates on $S^{n-1}$ are defined recursively: $\theta_{n-1}$ is defined as the complement of the angle between $x$ and the $n$th axis (i.e., $\sin^{-1}x_n$), and the other coordinates are those of $$\frac{(x_1...x_{n-1})}{\sqrt{|x|^2-|x_n|^2}}$$ in $S^{n-2}$. Therefore two points $(x_1...x_n)$ and $(y_1...y_n)$ have the same $(\theta_1...\theta_{k-1})$ iff $$\frac{(x_1,...,x_k)}{\sqrt{\sum_{i=1}^kx_i^2}} =\frac{(y_1,...,y_k)}{\sqrt{\sum_{i=1}^ky_i^2}}$$