The standard construction of a projective space $P(V)$ from a vector space $V$ is to take as elements equivalence classes of $V\setminus\lbrace 0 \rbrace$ under the equivalence relation $x \sim y \iff x = \lambda y$ with $\lambda \neq 0$.
This is usually motivated at an introductory level by saying that, from a single vantage point at the origin in three-dimensional Euclidean space, a point at coordinate $\left(x_1, x_2, x_3\right)$ gets projected unto a direction, which we can assign to a point on the $2$-sphere after normalisation.
But this seemingly intuitive analogy fails at probably one of the key points of projective spaces -- namely it does not explain why antipodal points fall into the same equivalence class. Informally, it's clear why all the points 'north' from the origin are deemed equivalent, but not why they are also equivalent to all points seen when looking `south'.
My question now, is there a way to save this analogy or motivate it better to a lay audience?
To stress, I understand that the standard construction is nice from a mathematical point of view -- we can think of the projective space as the set of lines through the origin (which naturally generalises to Grassmanians), and there is of course Bezout's theorem (which leads in particular to the fact that all lines intersect in exactly one point) and a bunch of other results from AG that single out projective spaces as the best spaces to work in. However, I'm interested in creating some material for lay audiences, and I want to prevent potentially harmful lies-to-children.