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docs: "The Affine Space" chapter updated with the latest changes

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Mateusz Pusz
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docs/users_guide/framework_basics/the_affine_space.md
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@@ -26,6 +26,7 @@ Here are the primary operations one can do in the affine space:
- _vector_ / scalar -> _vector_
- _point_ - _point_ -> _vector_
- _point_ + _vector_ -> _point_
- _vector_ + _point_ -> _point_
- _point_ - _vector_ -> _point_
!!! note
@@ -50,95 +51,76 @@ As we already know, a `quantity` type provides all operations required for _vect
the affine space.
## _Point_ is modeled by `quantity_point`
## _Point_ is modeled by `PointOrigin` and `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:
In the **mp-units** library the _point_ abstraction is modelled by:
- [`PointOrigin` concept](basic_concepts.md#PointOrigin) that specifies measurement origin,
- `quantity_point` class template that specifies a _point_ relative to a specific predefined origin.
### Absolute _point_ origin
The **absolute point origin** specifies where the "zero" of our measurement's scale is. User can
specify such an origin by deriving from the `absolute_point_origin` class template:
```cpp
constexpr struct mean_sea_level : absolute_point_origin<isq::altitude> {} mean_sea_level;
```
### `quantity_point`
The `quantity_point` class template specifies an absolute quantity with respect to an origin:
```cpp
template<Reference auto R,
PointOriginFor<get_quantity_spec(R)> auto PO = absolute_point_origin<get_quantity_spec(R)>{},
PointOriginFor<get_quantity_spec(R)> auto PO,
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.md#pointoriginfor)
and specifies the origin of our scale.
to `quantity`. The `PO` parameter satisfies a [`PointOriginFor` concept](basic_concepts.md#PointOriginFor)
and specifies the origin of our measurement 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.
As a _point_ can be represented with a _vector_ from the origin, a `quantity_point` class
template can be created with the following operations:
```cpp
constexpr quantity_point<isq::altitude[m]> everest_base_camp_alt{5364 * m};
static_assert(everest_base_camp_alt.relative() == 5364 * m);
quantity_point qp1 = mean_sea_level + 42 * m;
quantity_point qp2 = 42 * m + mean_sea_level;
quantity_point qp3 = mean_sea_level - 42 * 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:
[It is not allowed to subtract a _point_ from a _vector_](#operations-in-the-affine-space)
thus `42 * m - mean_sea_level` is an invalid operation.
```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:
Similarly to [creation of a quantity](../../getting_started/quick_start.md#creating-a-quantity),
if someone does not like the operator-based syntax to create a `quantity_point`, the same results
can be achieved with `make_quantity_point` factory function:
```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);
quantity_point qp4 = make_quantity_point<mean_sea_level>(42 * m);
quantity_point qp5 = make_quantity_point<mean_sea_level>(-42 * m);
```
!!! note
The provided `quantity` representing an offset from the origin is stored inside the `quantity_point`
class template and can be obtained with a `quantity_from_origin()` member function:
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).
```cpp
constexpr quantity_point everest_base_camp_alt = mean_sea_level + isq::altitude(5364 * m);
static_assert(everest_base_camp_alt.quantity_from_origin() == 5364 * m);
```
### 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
### Relative _point_ origin
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?
represented with an 8-bit integer type allowing 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:
@@ -150,31 +132,18 @@ constexpr struct everest_base_camp : relative_point_origin<everest_base_camp_alt
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);
constexpr quantity_point first_climb_alt = everest_base_camp + isq::altitude(std::uint8_t{42} * m);
static_assert(first_climb_alt.quantity_from_origin() == 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:
As we can see above, the `quantity_from_origin()` 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 subtract the absolute point origin from the current _point_:
- 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);
```
```cpp
static_assert(first_climb_alt - mean_sea_level == 5406 * m);
static_assert(first_climb_alt - first_climb_alt.absolute_point_origin == 5406 * m);
```
### Converting between different representations of the same _point_
@@ -188,22 +157,23 @@ 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);
constexpr quantity_point<isq::altitude[m], mean_sea_level, int> qp = first_climb_alt;
static_assert(qp.quantity_from_origin() == 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);
constexpr quantity_point qp = first_climb_alt.point_for(mean_sea_level);
static_assert(qp.quantity_from_origin() == 5406 * m);
```
!!! note
It is allowed to only covert between various origins defined in terms of the same
It is only allowed to convert 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.
from another `absolute_point_origin`, the library will not provide such a conversion.
A custom user-defined conversion function will be needed to add this 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.
@@ -213,19 +183,19 @@ For this purpose, we can either use:
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:
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;
constexpr struct home : absolute_point_origin<isq::distance> {} home;
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;
quantity_point<isq::distance[km], home> home_airport = home + 15 * km;
quantity_point<isq::distance[km], home> fra_airport = home_airport + 829 * km;
quantity_point<isq::distance[km], home> den_airport = fra_airport + 8115 * km;
quantity_point<isq::distance[km], home> 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.
As we can see above, we can easily get a new point by adding a quantity to an origin or another
quantity point.
If we want to find out the distance traveled between two points, we simply subtract them:
@@ -234,8 +204,8 @@ 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:
If we would like to find out the total distance traveled by taxi as well, we have to do a bit
more calculations:
```cpp
quantity<isq::distance[km]> taxi1 = home_airport - home;
@@ -243,7 +213,7 @@ quantity<isq::distance[km]> taxi2 = cppcon_venue - den_airport;
quantity<isq::distance[km]> taxi = taxi1 + taxi2;
```
Now if we will print the results:
Now, if we print the results:
```cpp
std::cout << "Total distance: " << total << "\n";
@@ -259,53 +229,85 @@ Flight distance: 8944 km
Taxi distance: 31.2544 km
```
!!! note
It is not allowed to subtract two point origins defined in terms of `absolute_point_origin`
(i.e. `mean_sea_level - mean_sea_level`) as those do not contain information about the unit
so we are not able to determine a resulting `quantity` type.
### 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:
The [SI](../../appendix/glossary.md#si) system definition in the **mp-units** library provides
two predefined point origins:
```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>;
namespace mp_units::si {
inline constexpr struct absolute_zero : absolute_point_origin<isq::thermodynamic_temperature> {} absolute_zero;
inline constexpr struct ice_point : relative_point_origin<absolute_zero + 273.15 * kelvin> {} ice_point;
}
```
With the above, we can be explicit what is the origin of our temperature point. For example, if
we want to implement the degree Celsius scale we can do it as follows:
```cpp
using Celsius_point = quantity_point<isq::Celsius_temperature[deg_C], si::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_.
but also different units for an origin and a _point_. In the above example, the relative
point origin is defined in terms of `si::kelvin`, while the quantity point uses
`si::degree_Celsius`.
With the above, for example, if we want to implement a room temperature controller, we can type:
To play a bit w temperatures we can implement a simple room's AC temperature controller in
the following way:
```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 struct room_reference_temp : relative_point_origin<si::ice_point + 21 * deg_C> {} room_reference_temp;
using room_temp = quantity_point<isq::Celsius_temperature[deg_C], room_reference_temp>;
constexpr auto step_delta = isq::thermodynamic_temperature(0.5 * deg_C);
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;
room_temp room_low = room_reference_temp - number_of_steps * step_delta;
room_temp room_high = room_reference_temp + 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";
std::println("| {:<14} | {:^18} | {:^18} | {:^18} |", "Temperature", "Room reference", "Ice point", "Absolute zero");
std::println("|{0:=^16}|{0:=^20}|{0:=^20}|{0:=^20}|", "");
auto print = [&](std::string_view label, auto v){
std::println("| {:<14} | {:^18} | {:^18} | {:^18} |",
label, v - room_reference_temp, v - si::ice_point, v - si::absolute_zero);
};
print("Lowest", room_low);
print("Default", room_reference_temp);
print("Highest", room_high);
```
The above prints:
```text
Lowest temp: -3 °C (18 °C)
Highest temp: 3 °C (24 °C)
| Temperature | Room reference | Ice point | Absolute zero |
|================|====================|====================|====================|
| Lowest | -3 °C | 18 °C | 291.15 °C |
| Default | 0 °C | 21 °C | 294.15 °C |
| Highest | 3 °C | 24 °C | 297.15 °C |
```
### No text output for _points_
The library does not provide a text output for quantity points, as printing just a number and a unit
is not enough to adequately describe a quantity point. Often an additional postfix is required.
is not enough to adequately describe a quantity point. Often, an additional postfix is required.
For example, the text output of `42 m` may mean many things and can also be confused with an output
of a regular quantity. On the other hand, printing `42 m AMSL` for altitudes above mean sea level is
@@ -316,23 +318,26 @@ a much better solution, but the library does not have enough information to prin
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?).
- **adding** two `quantity_point` objects
- It is physically impossible to add positions of home and Denver airports.
- **subtracting** a `quantity_point` from a `quantity`
- What would it mean to subtract DEN airport location from the distance to it?
- **multiplying/dividing** a `quantity_point` with a scalar
- What is the position of `2 *` DEN airport location?
- **multiplying/dividing** a `quantity_point` with a quantity
- What would multiplying the distance with the DEN airport location mean?
- **multiplying/dividing** two `quantity_point` objects
- What would multiplying home and DEN airport location mean?
- **mixing** `quantity_points` of different quantity kinds
- It is physically impossible to subtract time from length.
- **mixing** `quantity_points` of inconvertible quantities
- What does it mean to subtract a distance point to DEN airport from the Mount Everest base camp
altitude?
- **mixing** `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
The usage of `quantity_point` and affine space types in general, improves expressiveness and
type-safety of the code we write.