Origin, Vector and Length on Projective Plane or Space Do these quantities exist in a projective plane or space? How does one define them? Do they agree with the affine view?
 A: Origin is not an inherent property of the affine plane either. A linear space has a designated origin. Sure, once you pick a coordinate system you get a point you describe as $(0,0)$ in the affine plane, but that choice is arbitrary. Similarly you can pick an arbitrary point the origin of the projective plane, usually designated with $(0,0,1)$ for the standard $z=1$ embedding. But the choice is just as arbitrary.
Vector in the affine plane has two potential meanings worth distinguishing. On the one hand you can have coordinate vectors, in relation to a given basis. The projective counterparts to this would be homogeneous coordinates, which are equivalence classes of vectors with scalar multiples identified.
On the other hand a vector is a representation of a translation acting on points. While you can certainly express translations for a given embedding of the projective plane, they are not a natural concept there, but instead just one specific subset of the more natural set of projective transformations. That subset can be characterized by some properties, for example these transformations fix all points on the line at infinity, which has to be designated.
Length to my knowledge is not an inherent property of an affine plane either. You need an Euclidean plane to get an absolute metric. A natural affine concept is the ratio of lengths along a single line. This stays the same under affine transformations and changes of basis. Corresponding to this, the fundamental metric concept in projective geometry is the cross ratio, i.e. the ratio of length ratios for four given points. It is invariant under projective transformations or changes of projective basis.
One can express more Euclidean metric concepts in projective geometry by designating specific points. Specifically by designating the ideal circle points (which have coordinates $(1,\pm i,0)$ in the standard $z=1$ embedding), angles and length ratios can be measured, the latter even in different directions.
A: Origin
A projective space have no origin, and neither does an affine space. In fact, affine spaces are sometimes (aptly) described as "a vector space that has forgotten its origin": If you fix any point in an affine space, you can declare that point to be the origin, in which case you have all the structure required to make a vector space.
Vector
Projective spaces do not carry a natural vector space structure, so we should not refer to its elements as vectors. But they are closely related to vector spaces---a projective space is a projectivization $\Bbb P(\Bbb V)$ of a vector space $\Bbb V$, and we can exploit that structure to some effect.
Given a parameterized curve $[\gamma] : I \to \Bbb P(\Bbb V)$, we can consider its lifts $\gamma$, that is, the parameterized curves $\gamma : I \to \Bbb V$ such that $[\gamma] = \pi \circ \gamma$, and so the image $\gamma(t_0)$ is a vector that projects to $[\gamma](t_0)$. (Here, $\pi : \Bbb V \to \Bbb P(\Bbb V)$ is the projectivization map sending a vector $(a, b, c)$ to its span $[a, b, c]$.) Moreover, one can compute successive derivatives $\alpha'(t), \alpha''(t), \ldots$. These derivatives depend not only on the choice of parameterization (recall that the same is true for curves in Euclidean space), but also on the choice of lift.
Likewise, the elements of an affine space $\Bbb A$ are not vectors (though there is a natural "addition" action $\Bbb V \times \Bbb A \to \Bbb A$).
Length
Even affine transformations do not preserve the distance between points in Euclidean space $\Bbb E^n$, and the notion of distance certainly does not descend to a notion of distance on $\Bbb P(\Bbb E^n)$.
That said, there is an notion of arc length of a curve in projective space---that is, a measure that is preserved by all projective transformations---but its behavior is pretty unintuitive. For a nondegenerate curve $\gamma: I \to \Bbb P(V)$, one for which $\det \pmatrix{\gamma & \gamma' & \cdots & \gamma^{(n - 1)}}$ vanishes nowhere (this property is independent of the choice of lift),

*

*$(\gamma, \gamma', \ldots, \gamma^{(n - 1)})$ is a frame of $\Bbb V$ along $\gamma$, and

*there is a unique lift $\gamma$ for which $\det \pmatrix{\gamma & \gamma' & \cdots & \gamma^{(n - 1)}} = 1$.

Restricting to $n = 3$ for concreteness, some computation that I'll suppress shows that
$$\gamma''' = \kappa_0 \gamma + \kappa_1 \gamma'$$ for some functions $\kappa_0, \kappa_1$, and it turns out that we can always find a new parameterization $[\tilde \gamma]$ of $[\gamma]$ for which $\kappa_1 = 0$. This is the analogue of the fact that a curve in Euclidean space can always be reparameterized to have unit length. For any curve there are lots of parameterizations with $\kappa_1 = 0$. When we change our mind about which parameterization we want to use, $\kappa_0$ changes, so it is not invariant in the way that the curvature of a curve in Euclidean space (which does not depend on parameterization) is. However, $\kappa_0 \,dt^3$ does not depend on parameterization, so we can declare the projective arc length of a curve $[\gamma] : I \to \Bbb P(\Bbb V)$ to be $$\int_I \sqrt[3]{\kappa_0} \,dt .$$ By construction this "length" is independent of all of the choices we made, and it does not change when we apply a projective transformation.
Likewise there is notion of "affine arc length" related to projective arc length.
For more about curves in projective space (and projective arc length, and likewise with curves in affine space), see Clelland's excellent text, From Frenet to Cartan: The Method of Moving Frames; the last part of my answer is essentially a summary of some calculations in $\S$7.2 there.
