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+# 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{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:
+
+```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{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:
+
+```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.relative() == 5406 * m);
+    ```
+
+- do the same but in a slightly different way:
+
+    ```cpp
+    static_assert(first_climb_alt.relative() + first_climb_alt.point_origin.relative() == 5406 * m);
+    ```
+
+- call `absolute()` member function
+
+    ```cpp
+    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:
+
+```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 it 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 [stacking point origins](#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:
+
+```cpp
+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:
+
+```cpp
+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:
+
+```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.
+
+
+## Handling temperature points
+