# 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:

```cpp
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](../basic_concepts/#pointoriginfor)
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.

```cpp
constexpr quantity_point<isq::altitude[m]> everest_base_camp_alt{5364 * m};
static_assert(everest_base_camp_alt.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_alt = 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:

```cpp
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_alt{5364 * m};
static_assert(everest_base_camp_alt.relative() == 5364 * m);
```

!!! note

    The `mean_sea_level` and the default `absolute_point_origin<isq::altitude>` origins are distinct from
    each other, which means that _points_ defined with them are not compatible (can't be subtracted or
    compared).


### Class Template Argument Deduction (CTAD)

Typing the entire `quantity_point` type may sometimes be quite verbose. Also, please note that we
"accidentally" used `double` as a representation type in the above examples, even though we operated
only on integral values. This was done for the convenience of saving typing.

To improve the developer's experience, the `quantity_point` class template comes with the user-defined
class template argument deduction guides. Thanks to them, the above definitions can be rewritten as
follows:

- implicit default origin

    ```cpp
    constexpr quantity_point everest_base_camp_alt{isq::altitude(5364 * m)};
    ```

- explicit origin

    ```cpp
    constexpr quantity_point everest_base_camp_alt{isq::altitude(5364 * m), mean_sea_level};
    ```


### Relative _point_ origins

We often do not have only 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_alt` as our reference point?

For this purpose, we can define a `relative_point_origin` in the following way:

```cpp
constexpr struct everest_base_camp : relative_point_origin<everest_base_camp_alt> {} everest_base_camp;
```

The above can be used as an origin for subsequent _points_:

```cpp
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_

    ```cpp
    static_assert(first_climb_alt.relative() + everest_base_camp_alt.relative() == 5406 * m);
    ```

- subtract the "zero altitude" _point_ from the current _point_

    ```cpp
    static_assert(first_climb_alt - quantity_point{0 * m, mean_sea_level} == 5406 * m);
    ```

- call `absolute()` member function on the current _point_

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


### Converting between different representations of the same _point_

As we might represent the same _point_ with _vectors_ from various origins, the **mp-units** library
provides facilities to convert the _point_ to the `quantity_point` class templates expressed in terms
of different origins.

For this purpose, we can either use:

- a converting constructor:

    ```cpp
    static_assert(quantity_point<isq::altitude[m], mean_sea_level>{first_climb_alt}.relative() == 5406 * m);
    ```

- a dedicated conversion interface:

    ```cpp
    constexpr QuantityPoint auto qp = first_climb_alt.point_from(mean_sea_level);
    static_assert(qp.relative() == 5406 * m);
    ```

!!! note

    It is allowed to only covert between various origins defined in terms of the same
    `absolute_point_origin`. Even if it is possible to express the same _point_ as a _vector_
    from another `absolute_point_origin`, the **mp-units** library will not allow it, and
    a custom user-defined conversion function will be needed to provide such a functionality.

    Said otherwise, in the **mp-units** library, there is no way to spell how two distinct
    `absolute_point_origin` types relate to each other.


### _Point_ arithmetics

Let's assume we will attend the CppCon conference hosted in Aurora, CO, and we want to estimate
the distance we will 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 cab to the Gaylord Rockies Resort & Convention Center:

```cpp
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:

```cpp
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:

```cpp
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 if we will print the results:

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

we will see the following output:

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


### Temperature support

Another important example of [relative point origins](#relative-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:

```cpp
constexpr struct ice_point : relative_point_origin<quantity_point<isq::thermodynamic_temperature[K]>{273.15 * K}> {} ice_point;
using Celsius_point = quantity_point<isq::thermodynamic_temperature[deg_C], ice_point>;
```

!!! note

    Notice that while 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:

```cpp
constexpr struct room_reference_temperature : relative_point_origin<Celsius_point{21 * deg_C}> {} room_reference_temperature;
using room_temperature = quantity_point<isq::thermodynamic_temperature[deg_C], room_reference_temperature>;

constexpr auto step_delta = isq::thermodynamic_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:

```text
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.