I'm learning projective geometry and need help with the following exercise :

Let $p_1, p_2$ be two points of the projective plane $\mathbb{RP}^2$ with homogeneous coordinates $[1 : 1 : 1]$ and $[-2 : 1 : 3]$ respectively. $(1)$ Find the points of the plane $\mathbb{A}^2 = \{z = 1\}$ which belongs to the equivalence classes of $p_1$ and $p_2$ respectively. $(2)$ Are they unique?

Here are my thoughts :

$(1)$ The equivalence classe of the point $p_1$ are all the points of $\mathbb{R}^3 \setminus \{0\}$ which satisfies the equivalence relation $(x, y, z) \sim (1, 1, 1) \iff \exists k \neq 0 \in \mathbb{R} : (x, y, z) = k(1, 1, 1)$, i.e. the equivalence classe of $p_1$ is the line passing through the origin and the point $(1, 1, 1)$. This line intersects the plane $A^2 = \{z=1\}$ at the point $(1, 1, 1)$.

Similarly, the equivalence classe of the point $p_2$ is the line passing through the origin and the point $(-2, 1, 3)$. This line intersects the plane $A^2 = \{z=1\}$ at the point $(-\frac{2}{3}, \frac{1}{3}, 1)$.

$(2)$ Concerning uniqueness I would say that since a line in $\mathbb{R^3}$ passing through the origin intersects the plane $\mathbb{A}^2 = \{z = 1\}$ at a unique point then the points we found in $(1)$ are unique.

I still don't know how to correctly write an answer to this exercise. I feel my answer/explanation is "too messy" and doesn't really answer the questions being asked. I would like some help to improve it and correct my errors if I have made any.

  • 2
    $\begingroup$ I don't see any issue. Your answer looks good for me. $\endgroup$ – Huy Dang Jan 2 '18 at 21:51

I'd be happy with your answer.

For a slightly different angle, you could take a more algorithmic approach: describe what you did to come up with $(-\frac23,\frac13,1)$, namely divide by the third coordinate. This makes uniqueness more obvious: there is a single third coordinate, and a single result for a division of two numbers.

But while this highlights the “I know what to do” aspect of learning, it misses on the “I understand why I'm doing it like this”, so I'd say your original answer to (1) in particular demonstrates understanding and you should have confidence in it.

  • $\begingroup$ Thank you for taking the time to verify my work and for your positive feedback. It was exactly the boost I needed. $\endgroup$ – user347616 Jan 2 '18 at 23:34

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