Torsors
Torsors are ‘relative’ values, where we can’t add them but we can find their difference. Cruicially, such ‘differences’ are not values of our torsor. For example, points in space (of any dimension) cannot be meaningfully ‘added’; but we can find their difference; and that difference is a vector (not a point).
Furthermore, we can always add such ‘differences’ to our torsor values, resulting in more torsor values. In particular, the following equation holds:
In the case of points and vectors, adding the vector to the point gives the point .
Uses
Torsors are useful for avoiding arbitrary coordinates, i.e. when there’s no natural/obvious way to define ‘zero’. We would like to build an approach to geometry which avoids the need for arbitrary choices. Projective geometry is a good starting point: we can certainly combine two points to yield the line joining them; and dually we can combine lines to yield the point where they meet (potentially ‘at infinity’). However, it’s not clear whether adding a line to a point can ‘undo’ such combinations.
We can specialise a projective space to an affine space by choosing a distinguished line (normally this is the line at infinity; but we can use any finite line as a One place we can use this is the affine plane, where there is no ‘zero’ point or line:
- The ‘difference’ between two points is a directed line segment. Note that it is not a line, since there’s no way to choose an orientation when taking a point from itself (line segments avoid this since such zero-length segments have no orientation!)
- Adding a directed line segment to a point yields another point.
We can define line segments in projective space too; but there is no way to uniquely ‘transport’ segments to coincide with arbitrary projective points. Dually, for affine lines:
- The ‘difference’ between two lines is a directed angle (where ‘angle’ refers to an intersection of lines; not any particular measure!)
- Adding a directed angle to a line yields a line. NO! It requires a point on the line, for the intersection! Otherwise we have a direction!
Again, the second property is affine, so we can Notice that these segments have no particular ‘length’ (or equivalent metric, like quadrance), since projective geometry doesn’t impose any. Likewise, these angles have no ‘arc-length’ or ‘radius’. As a consequence, we cannot ‘transport’ these objects through the space in a unique way: for example, vector addition is as easy as ‘gluing’ the start of one vector to the end of another; but this requires ‘transporting’ the vectors around (or, equivalently, redefining their origin). In projective geometry we can ‘transport’ line segments in
(Directed) line segments can interact with (directed) angles: adding an angle to a segment yields another segment: the empty angle acts as identity. I can’t think of another interaction between lines/segments and angles…
We can imagine rotating around a point, by adding an angle to a point. Adding a line segment to a point doesn’t quite represent translations, since the point is irrelevant. Scaling can use a distinguished point (as the centre), but there is no ‘unit vector’ in projective geometry, so it’s unclear how much to scale by: we need two line segments, to bring into coincidence, but doing so can also introduce a rotation.
Two angles of a distinguished point point except there’s no need for a distinguished point in that case; they
Affine
TODO