mp-units/docs/users_guide/framework_basics/the_affine_space.md

The Affine Space

The affine space has two types of entities:

• point - a position specified with coordinate values (i.e. location, address, etc.)
• vector - the difference between two points (i.e. shift, offset, displacement, duration, etc.)

!!! note

The _vector_ described here is specific to the affine space theory and is not the same thing
as the quantity of a vector character that we discussed in the
["Scalars, vectors, and tensors" chapter](character_of_a_quantity/#scalars-vectors-and-tensors)
(although, in some cases, those terms may overlap).

Operations in the affine space

Here are the primary operations one can do in the affine space:

• vector + vector -> vector
• vector - vector -> vector
• -vector -> vector
• vector * scalar -> vector
• scalar * vector -> vector
• vector / scalar -> vector
• point - point -> vector
• point + vector -> point
• point - vector -> point

!!! note

It is not possible to:

- add two _points_,
- subtract a _point_ from a _vector_,
- multiply nor divide _points_ with anything else.

Vector is modeled by quantity

Up until now, each time when we used a quantity in our code, we were modeling some kind of a difference between two things:

• the distance between two points
• duration between two time points
• the difference in speed (even if relative to 0)

As we already know, a quantity type provides all operations required for vector type in the affine space.

Point is modeled by quantity_point

A point is an absolute quantity with respect to an origin and is represented in the library with a quantity_point class template:

template<Reference auto R,
         PointOriginFor<get_quantity_spec(R)> auto PO = absolute_point_origin<get_quantity_spec(R)>{},
         RepresentationOf<get_quantity_spec(R).character> Rep = double>
class quantity_point;

As we can see above, the quantity_point class template exposes one additional parameter compared to quantity. The PO parameter satisfies a PointOriginFor concept and specifies the origin of our scale.

The origin

The origin specifies where the "zero" of our measurement's scale is.

Please notice that a point can be represented with a vector from the origin. This is why in the mp-units library, a quantity_point gets a quantity in its constructor. Such a quantity:

• specifies the relative distance of a specific point from the scale origin,
• is the only data member of the quantity_point class template,
• can be obtained with the relative() member function.

constexpr quantity_point<isq::altitude[m]> everest_base_camp{5364 * m};
static_assert(everest_base_camp.relative() == 5364 * m);

!!! note

As the constructor is explicit, the quantity point object can only be created from a quantity via
direct initialization. This is why the code below that uses copy initialization does not compile:

```cpp
quantity_point<isq::altitude[m]> everest_base_camp = 5364 * m;  // ERROR
```

In the mp-units library, the origin is either provided implicitly (as above) or can be predefined by the user and then provided explicitly as the quantity_point class template argument:

constexpr struct mean_sea_level : absolute_point_origin<isq::altitude> {} mean_sea_level;

constexpr quantity_point<isq::altitude[m], mean_sea_level> everest_base_camp{5364 * m};
static_assert(everest_base_camp.relative() == 5364 * m);

Stacking point origins

We often do not have one ultimate "zero" point when we measure things.

Continuing the Mount Everest trip example above, measuring all daily hikes from the mean_sea_level might not be efficient. Maybe we know that we are not good climbers, so all our climbs can be represented with an 8-bit integer type which will allow us to save memory in our database of climbs? Why not use everest_base_camp as our reference point?

It turns out that in the mp-units library, you can use a predefined at compile-time quantity_point as an origin as well:

constexpr quantity_point<isq::altitude[m], everest_base_camp, std::uint8_t> first_climb_alt{42 * m};
static_assert(first_climb_alt.relative() == 42 * m);

As we can see above, the relative() member function returns a relative distance from the current point origin. In case we would like to know the absolute altitude that we reached on this climb, we can either:

• add the two relative heights from both points

  static_assert(first_climb_alt.relative() + everest_base_camp.relative() == 5406 * m);
  

• do the same but in a slightly different way:

  static_assert(first_climb_alt.relative() + first_climb_alt.point_origin.relative() == 5406 * m);
  

• call absolute() member function

  static_assert(first_climb_alt.absolute() == 5406 * m);
  

Point arithmetics

Let's assume we are going to attend the CppCon conference that is hosted in Aurora, CO, and we want to estimate the distance we are going to travel. We have to take a taxi to a local airport, fly to DEN airport with a stopover in FRA, and in the end, get a taxi to the Gaylord Rockies Resort & Convention Center:

constexpr struct home_location : absolute_point_origin<isq::distance> {} home_location;

quantity_point<isq::distance[km], home_location> home{};
quantity_point<isq::distance[km], home_location> home_airport = home + 15 * km;
quantity_point<isq::distance[km], home_location> fra_airport = home_airport + 829 * km;
quantity_point<isq::distance[km], home_location> den_airport = fra_airport + 8115 * km;
quantity_point<isq::distance[km], home_location> cppcon_venue = den_airport + 10.1 * mi;

As we can see above, we can easily get a new point by adding a quantity to another quantity point.

If we want to find out the distance traveled between two points, we simply subtract them:

quantity<isq::distance[km]> total = cppcon_venue - home;
quantity<isq::distance[km]> flight = den_airport - home_airport;

If we would like to find out the total distance traveled by taxi as well, we have to do more calculations:

quantity<isq::distance[km]> taxi1 = home_airport - home;
quantity<isq::distance[km]> taxi2 = cppcon_venue - den_airport;
quantity<isq::distance[km]> taxi = taxi1 + taxi2;

Now it will print the results:

std::cout << "Total distance:  " << total << "\n";
std::cout << "Flight distance: " << flight << "\n";
std::cout << "Taxi distance:   " << taxi << "\n";

we will see the following output:

Total distance:  8975.25 km
Flight distance: 8944 km
Taxi distance:   31.2544 km

Temperature support

Another important example of stacking point origins is support of temperature quantity points in units different than kelvin [K].

For example, the degree Celsius scale can be implemented as follows:

constexpr auto ice_point = quantity_point<isq::thermodynamic_temperature[K]>{273.15 * K};
using Celsius_point = quantity_point<isq::Celsius_temperature[deg_C], ice_point>;

!!! note

While [stacking point origins](#stacking-point-origins) we can use not only different
representation types but also different units for an origin and a _point_.

With the above, for example, if we want to implement a room temperature controller, we can type:

constexpr Celsius_point room_reference_temperature{21 * deg_C};
using room_temperature = quantity_point<isq::Celsius_temperature[deg_C], room_reference_temperature>;

constexpr auto step_delta = isq::Celsius_temperature(0.5 * deg_C);
constexpr int number_of_steps = 6;

room_temperature room_default{};
room_temperature room_low = room_default - number_of_steps * step_delta;
room_temperature room_high = room_default + number_of_steps * step_delta;

std::cout << "Lowest temp: " << room_low.relative() << " (" << room_low - Celsius_point::zero() << ")\n";
std::cout << "Highest temp: " << room_high.relative() << " (" << room_high - Celsius_point::zero() << ")\n";

The above prints:

Lowest temp: -3 °C (18 °C)
Highest temp: 3 °C (24 °C)

The affine space is about type-safety

The following operations are not allowed in the affine space:

• add two quantity_point objects (It is physically impossible to add positions of home and Denver airports),
• subtract a quantity_point from a quantity (What would it mean to subtract DEN airport location from the distance to it?),
• multiply/divide a quantity_point with a scalar (What is the position of 2x DEN airport location?).
• multiply/divide a quantity_point with a quantity (What would multiplying the distance with the DEN airport location mean?).
• multiply/divide two quantity_point objects (What would multiplying home and DEN airport location mean?).
• mix quantity_points of different quantity kinds (It is physically impossible to subtract time from length),
• mix quantity_points of inconvertible quantities (What does it mean to subtract a distance point to DEN airport from the Mount Everest base camp altitude?),
• mix quantity_points of convertible quantities but with unrelated origins (How to subtract a point on our trip to CppCon measured relatively to our home location from a point measured relative to the center of the Solar System?).

!!! note

The usage of `quantity_point`, and affine space types in general, improves expressiveness and
type-safety of the code we write.

Handling temperature points