Points are not deltas

The unit checker from the last episode is happy to add two temperature readings: 21 °C + 17 °C, same unit and same quantity kind, gives 38 °C. Nothing in the world is 38 °C because of it. The sum of this morning’s temperature and this afternoon’s is not a physical quantity at all, and no amount of dimensional analysis will flag it.

Mathematics has had a name for this distinction for a century: the affine space. A point is a position against an origin, such as a temperature reading, a timestamp, or an altitude. A delta is the difference between two points: a temperature change, a duration, a climb. Deltas form a vector space, so you can add them freely. Points are not deltas, and the operations reflect that. Subtracting two points gives a delta, and adding a delta to a point gives a new point; those are the only combinations the algebra permits.

std::chrono users already know this split as time_point vs duration. mp-units generalizes it to every quantity.

quantity_point vs quantity

The API names map straight onto the physics: point<deg_C>(17.0) is a reading, delta<deg_C>(4.5) is a change. Adding the change to the reading yields a new reading, subtracting two readings yields a change, and afternoon.quantity_from(si::ice_point) answers the only question an absolute reading can honestly answer: how far from a stated origin? The two commented lines, adding two readings and assigning a point to a delta, do not compile.

Temperatures make the sharpest example because their everyday unit hides an origin: 0 °C is a point, the ice point. The same modeling covers timestamps against durations, positions against displacements, and altitudes against climbs. Anywhere the code passes “a metre” around, ask whether it means a metre from where or a metre of difference. mp-units lets the signature say which, and the compiler holds the line.

This is the third post in the run of well-typed numbers that are still wrong, after wrong units and wrong kinds.

Sources: mp-units documentation: the affine space.