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docs: the affine space chapter rewritten
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Mateusz Pusz
216
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|
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|
After Width: | Height: | Size: 25 KiB |
@ -3,13 +3,16 @@
|
||||
The affine space has two types of entities:
|
||||
|
||||
- **_Point_** - a position specified with coordinate values (e.g., location, address, etc.)
|
||||
- **_Vector_** - the difference between two points (e.g., shift, offset, displacement, duration, etc.)
|
||||
- **_Displacement vectors_** - the difference between two points (e.g., shift, offset,
|
||||
displacement, duration, etc.)
|
||||
|
||||
In the following subchapters, we will often refer to _displacement vectors_ simply as _vectors_ for
|
||||
brevity.
|
||||
|
||||
!!! 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
|
||||
The _displacement 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.md#scalars-vectors-and-tensors)
|
||||
(although, in some cases, those terms may overlap).
|
||||
|
||||
@ -18,24 +21,24 @@ The affine space has two types of entities:
|
||||
|
||||
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_
|
||||
- _Vector_ + _Point_ -> _Point_
|
||||
- _Point_ - _Vector_ -> _Point_
|
||||
- _vector_ + _vector_ -> _vector_
|
||||
- _vector_ - _vector_ -> _vector_
|
||||
- -_vector_ -> _vector_
|
||||
- _vector_ * scalar -> _vector_
|
||||
- scalar * _vector_ -> _vector_
|
||||
- _vector_ / scalar -> _vector_
|
||||
- _point_ - _point_ -> _vector_
|
||||
- _point_ + _vector_ -> _point_
|
||||
- _vector_ + _point_ -> _point_
|
||||
- _point_ - _vector_ -> _point_
|
||||
|
||||
!!! important
|
||||
|
||||
It is not possible to:
|
||||
|
||||
- add two _Points_,
|
||||
- subtract a _Point_ from a _Vector_,
|
||||
- multiply nor divide _Points_ with anything else.
|
||||
- add two _points_,
|
||||
- subtract a _point_ from a _vector_,
|
||||
- multiply nor divide _points_ with anything else.
|
||||
|
||||
|
||||
## _Points_ are more common than most of us imagine
|
||||
@ -55,7 +58,7 @@ more popular in the products we implement. They can be used to implement:
|
||||
Improving the affine space's _Points_ intuition will allow us to write better and safer software.
|
||||
|
||||
|
||||
## _Vector_ is modeled by `quantity`
|
||||
## _Displacement vector_ is modeled by `quantity`
|
||||
|
||||
Up until now, each time we used a `quantity` in our code, we were modeling some kind of a
|
||||
difference between two things:
|
||||
@ -64,15 +67,15 @@ difference between two things:
|
||||
- _duration_ between two time points,
|
||||
- the difference in _speed_ (even if relative to zero).
|
||||
|
||||
As we already know, a `quantity` type provides all operations required for a _Vector_ type in
|
||||
the affine space.
|
||||
As we already know, a `quantity` type provides all operations required for a _displacement vcector_
|
||||
abstraction in an affine space.
|
||||
|
||||
|
||||
## _Point_ is modeled by `quantity_point` and `PointOrigin`
|
||||
|
||||
In the **mp-units** library the _Point_ abstraction is modelled by:
|
||||
In the **mp-units** library, the _Point_ abstraction is modelled by:
|
||||
|
||||
- [`PointOrigin` concept](concepts.md#PointOrigin) that specifies measurement origin,
|
||||
- [`PointOrigin` concept](concepts.md#PointOrigin) that specifies measurement origin, and
|
||||
- `quantity_point` class template that specifies a _Point_ relative to a specific predefined origin.
|
||||
|
||||
|
||||
@ -90,125 +93,119 @@ 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](concepts.md#PointOriginFor)
|
||||
and specifies the origin of our measurement scale. By default, it is initialized with a quantity's
|
||||
zeroth point using the following rules:
|
||||
and specifies the origin of our measurement scale.
|
||||
|
||||
Each `quantity_point` internally stores a `quantity` object, which represents a _displacement vector_
|
||||
from the predefined origin. Thanks to this, an instantiation of a `quantity_point` can be considered
|
||||
as a model of a vector space from such an origin.
|
||||
|
||||
Forcing the user to manually predefine an origin for every domain may be cumbersome and discourage
|
||||
users from using such abstractions at all. This is why, by default, the `PO` template
|
||||
parameter is initialized with the `default_point_origin(R)` that provides the quantity points'
|
||||
scale zeroth point using the following rules:
|
||||
|
||||
- if the measurement unit of a quantity specifies its point origin in its definition
|
||||
(e.g., degree Celsius), then this point is being used,
|
||||
- otherwise, an instantiation of `zeroth_point_origin<QuantitySpec>` is being used which
|
||||
provides a zeroth point for a specific quantity type.
|
||||
provides a well-established zeroth point for a specific quantity type.
|
||||
|
||||
!!! tip
|
||||
|
||||
The `quantity_point` definition can be found in the `mp-units/quantity_point.h` header file.
|
||||
|
||||
|
||||
### Implicit point origin
|
||||
#### `zeroth_point_origin<QuantitySpec>`
|
||||
|
||||
Let's assume that Alice goes for a trip driving a car. She likes taking notes about interesting
|
||||
places that she visits on the road. For every such item, she writes down:
|
||||
`zeroth_point_origin<QuantitySpec>` is meant to be used in cases where the specific domain has
|
||||
a well-established, non-controversial zeroth point on the measurement scale. This saves the user
|
||||
from the need to write a boilerplate code that would predefine such a type for such a domain.
|
||||
|
||||
- its name,
|
||||
- a readout from the car's odometer at the location,
|
||||
- a current timestamp.
|
||||
|
||||
We can implement this in the following way:
|
||||
{style="width:80%;display: block;margin: 0 auto;"}
|
||||
|
||||
```cpp
|
||||
using std::chrono::system_clock;
|
||||
quantity_point<isq::distance[si::metre]> qp1{100 * m};
|
||||
quantity_point<isq::distance[si::metre]> qp2{120 * m};
|
||||
|
||||
struct trip_log_item {
|
||||
std::string name;
|
||||
quantity_point<isq::distance[km]> odometer;
|
||||
quantity_point<si::second> timestamp;
|
||||
};
|
||||
using trip_log = std::vector<trip_log_item>;
|
||||
assert(qp1.quantity_from_zero() == 100 * m);
|
||||
assert(qp2.quantity_from_zero() == 120 * m);
|
||||
|
||||
assert(qp2 - qp1 == 20 * m);
|
||||
assert(qp1 - qp2 == -20 * m);
|
||||
|
||||
// auto res = qp1 + qp2; // Compile-time error
|
||||
```
|
||||
|
||||
In the above code `100 * m` and `120 * m` still create two quantities that serve as _displacement
|
||||
vectors_ here. Quantity point objects can be explicitly constructed from such quantities only when
|
||||
their origin is an instantiation of the `zeroth_point_origin<QuantitySpec>`.
|
||||
|
||||
It is really important to understand that even though we can use `.quantity_from_zero()` to obtain
|
||||
the _displacement vector_ of a point from the origin, the point by itself does not represent or have
|
||||
any associated physical value. It is just a point in some space. The same point can be expressed
|
||||
with different _displacement vectors_ from different origins.
|
||||
|
||||
It is also worth mentioning that simplicity comes with a safety cost here. For some users, it
|
||||
might be surprising that the usage of `zeroth_point_origin<QuantitySpec>` makes various quantity
|
||||
point objects compatible as long as quantity types used in the origin and reference are
|
||||
compatible:
|
||||
|
||||
```cpp
|
||||
trip_log log;
|
||||
quantity_point<si::metre> qp1{isq::distance(100 * m)};
|
||||
quantity_point<si::metre> qp2{isq::height(120 * m)};
|
||||
|
||||
quantity_point timestamp_1{quantity{system_clock::now().time_since_epoch()}};
|
||||
log.emplace_back("home", quantity_point{1356 * km}, timestamp_1);
|
||||
|
||||
// some time passes
|
||||
|
||||
quantity_point timestamp_2{quantity{system_clock::now().time_since_epoch()}};
|
||||
log.emplace_back("castle", quantity_point{1401 * km}, timestamp_2);
|
||||
assert(qp2 - qp1 == 20 * m);
|
||||
assert(qp1 - qp2 == -20 * m);
|
||||
```
|
||||
|
||||
This is an excellent example of where points are helpful. There is no doubt about the correctness
|
||||
of their usage in this scenario:
|
||||
|
||||
- adding two odometer readouts or two timestamps have no physical sense, and that is why we will
|
||||
expect a compile-time error when we try to perform such operations accidentally,
|
||||
- subtracting two odometer readouts or timestamps is perfectly valid and results in a quantity
|
||||
storing the interval value between the two points.
|
||||
### Absolute _point_ origin
|
||||
|
||||
Having such a database, we can print the trip log in the following way:
|
||||
In cases where we want to implement an isolated independent space in which points are not compatible
|
||||
with other spaces, even of the same quantity type, we should manually predefine an absolute point
|
||||
origin.
|
||||
|
||||
{style="width:80%;display: block;margin: 0 auto;"}
|
||||
|
||||
```cpp
|
||||
for (const auto& item : log) {
|
||||
std::cout << "POI: " << item.name << "\n";
|
||||
std::cout << "- Distance from home: " << item.odometer - log.front().odometer << "\n";
|
||||
std::cout << "- Trip duration from start: " << (item.timestamp - log.front().timestamp).in(non_si::minute) << "\n";
|
||||
}
|
||||
inline constexpr struct origin : absolute_point_origin<origin, isq::distance> {} origin;
|
||||
|
||||
// quantity_point<si::metre, origin> qp1{100 * m}; // Compile-time error
|
||||
// quantity_point<si::metre, origin> qp2{120 * m}; // Compile-time error
|
||||
quantity_point<si::metre, origin> qp1 = origin + 100 * m;
|
||||
quantity_point<si::metre, origin> qp2 = 120 * m + origin;
|
||||
|
||||
// assert(qp1.quantity_from_zero() == 100 * m); // Compile-time error
|
||||
// assert(qp2.quantity_from_zero() == 120 * m); // Compile-time error
|
||||
assert(qp1.quantity_from(origin) == 100 * m);
|
||||
assert(qp2.quantity_from(origin) == 120 * m);
|
||||
|
||||
assert(qp1 - origin == 100 * m);
|
||||
assert(qp2 - origin == 120 * m);
|
||||
assert(origin - qp1 == -100 * m);
|
||||
assert(origin - qp2 == -120 * m);
|
||||
|
||||
assert(qp2 - qp1 == 20 * m);
|
||||
```
|
||||
|
||||
Moreover, if Alice had reset the car's trip odometer before leaving home, we could have rewritten
|
||||
one of the previous lines like that:
|
||||
This time, we can't construct a quantity point from any quantity. In order to prevent potential safety
|
||||
issues, when a custom point origin is being used, we always need to provide its object in
|
||||
an expression that results in a quantity point instantiation.
|
||||
|
||||
!!! info
|
||||
|
||||
A rationale for this longer construction syntax can be found in the
|
||||
[Why can't I create a quantity by passing a number to a constructor?](../../getting_started/faq.md#why-cant-i-create-a-quantity-by-passing-a-number-to-a-constructor)
|
||||
chapter.
|
||||
|
||||
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 a two-parameter constructor:
|
||||
|
||||
```cpp
|
||||
std::cout << "Distance from home: " << item.odometer.quantity_from_zero() << "\n";
|
||||
quantity_point qp1{100 * m, origin};
|
||||
```
|
||||
|
||||
The above always returns a quantity measured from the "ultimate" zeroth point of a scale used for
|
||||
this specific quantity type.
|
||||
|
||||
!!! tip
|
||||
|
||||
Storing _Points_ is the most efficient representation we can choose in this scenario:
|
||||
|
||||
- to store a value, we read it directly from the instrument, and no additional transformation
|
||||
is needed,
|
||||
- to print the absolute value (e.g., odometer), we have the value available right away,
|
||||
- to get any relative quantity (e.g., distance from the start, distance from the previous point,
|
||||
etc.), we have to perform a single subtraction operation.
|
||||
|
||||
If we stored _Vectors_ in our database instead, we would have to pay at runtime for additional
|
||||
operations:
|
||||
|
||||
- to store a quantity, we would have to perform the subtraction right away to get the interval
|
||||
between the current value and some reference point,
|
||||
- to print the absolute value, we would have to add the quantity to the reference point that
|
||||
we need to store somewhere in the database as well,
|
||||
- to get a relative quantity, only the currently stored one is immediate; all other values
|
||||
will require at least one quantity addition operation.
|
||||
|
||||
Now, let's assume that Bob, a friend of Alice, also keeps a log of his trips but he, of
|
||||
course, measures distances from his own home with the odometer in his car. Everything is fine as
|
||||
long as we deal with one trip at a time, but if we start to work with both at once, we may
|
||||
accidentally subtract points from different trips. The library will not prevent
|
||||
us from doing so.
|
||||
|
||||
The points from Alice's and Bob's trips should be considered separate, and to enforce it at
|
||||
compilation time, we need to introduce explicit origins.
|
||||
|
||||
|
||||
### Absolute _Point_ origin
|
||||
|
||||
The **absolute point origin** specifies the "zero" of our measurement's scale. User can
|
||||
specify such an origin by deriving from the `absolute_point_origin` class template:
|
||||
|
||||
```cpp
|
||||
enum class actor { alice, bob };
|
||||
|
||||
template<actor A>
|
||||
struct zeroth_odometer_t : absolute_point_origin<zeroth_odometer_t<A>, isq::distance> {};
|
||||
|
||||
template<actor A>
|
||||
inline constexpr zeroth_odometer_t<A> zeroth_odometer;
|
||||
```
|
||||
Again, CTAD always helps to use precisely the type we need in a current case.
|
||||
|
||||
!!! info
|
||||
|
||||
@ -225,212 +222,148 @@ inline constexpr zeroth_odometer_t<A> zeroth_odometer;
|
||||
- we can't define the above in one line of code,
|
||||
- provide the same identifier for a class and variable template.
|
||||
|
||||
Odometer is not the only one that can get an explicit point origin in our case. As timestamps are
|
||||
provided by the `std::chrono::system_clock`, their values are always relative to the epoch of this
|
||||
clock.
|
||||
Finally, please note that it is not allowed to subtract two point origins defined in terms of
|
||||
`absolute_point_origin` (e.g., `origin - origin`) as those do not contain information about the
|
||||
unit, so we cannot determine a resulting `quantity` type.
|
||||
|
||||
!!! note
|
||||
Absolute point origins are also perfect for establishing independent spaces even if the same quantity
|
||||
type and unit is being used:
|
||||
|
||||
The **mp-units** library provides means to specify
|
||||
[interoperability with other units libraries](../use_cases/interoperability_with_other_libraries.md).
|
||||
It also has built-in compatibility with `std::chrono` types, so users do not have to define
|
||||
interoperability traits or point origins for such types by themselves. Those are already
|
||||
provided in the `mp-units/systems/si/chrono.h` header file.
|
||||
|
||||
|
||||
Now, we can refactor our database to benefit from the explicit points:
|
||||
{style="width:80%;display: block;margin: 0 auto;"}
|
||||
|
||||
```cpp
|
||||
template<actor A>
|
||||
struct trip_log_item {
|
||||
std::string point_name;
|
||||
quantity_point<si::kilo<si::metre>, zeroth_odometer<A>> odometer;
|
||||
quantity_point<si::second, chrono_point_origin<system_clock>> timestamp;
|
||||
};
|
||||
inline constexpr struct origin1 : absolute_point_origin<origin1, isq::distance> {} origin1;
|
||||
inline constexpr struct origin2 : absolute_point_origin<origin2, isq::distance> {} origin2;
|
||||
|
||||
template<actor A>
|
||||
using trip_log = std::vector<trip_log_item<A>>;
|
||||
quantity_point qp1 = origin1 + 100 * m;
|
||||
quantity_point qp2 = origin2 + 120 * m;
|
||||
|
||||
assert(qp1.quantity_from(origin1) == 100 * m);
|
||||
assert(qp2.quantity_from(origin2) == 120 * m);
|
||||
|
||||
assert(qp1 - origin1 == 100 * m);
|
||||
assert(qp2 - origin2 == 120 * m);
|
||||
assert(origin1 - qp1 == -100 * m);
|
||||
assert(origin2 - qp2 == -120 * m);
|
||||
|
||||
// assert(qp2 - qp1 == 20 * m); // Compile-time error
|
||||
// assert(qp1 - origin2 == 100 * m); // Compile-time error
|
||||
// assert(qp2 - origin1 == 120 * m); // Compile-time error
|
||||
// assert(qp1.quantity_from(origin2) == 100 * m); // Compile-time error
|
||||
// assert(qp2.quantity_from(origin1) == 120 * m); // Compile-time error
|
||||
```
|
||||
|
||||
We also need to update the initialization part in our code. In the case of implicit zeroth origins,
|
||||
we could construct `quantity_point` directly from the value of a `quantity`. This is no longer
|
||||
the case.
|
||||
As a _Point_ can be represented with a _Vector_ from the origin, to improve the safety of the code
|
||||
we write, a `quantity_point` class template must be created with one of the following operations:
|
||||
|
||||
```cpp
|
||||
quantity_point qp1 = zeroth_odometer<actor::alice> + 1356 * km;
|
||||
quantity_point qp2 = 1356 * km + zeroth_odometer<actor::alice>;
|
||||
quantity_point qp3 = zeroth_odometer<actor::alice> - 1356 * km;
|
||||
```
|
||||
|
||||
Although, the `qp3` above does not have a physical sense in this specific scenario.
|
||||
|
||||
!!! note
|
||||
|
||||
[It is not allowed to subtract a _Point_ from a _Vector_](#operations-in-the-affine-space)
|
||||
thus `1356 * km - zeroth_odometer<actor::alice>` is an invalid operation.
|
||||
|
||||
!!! info
|
||||
|
||||
A rationale for this longer construction syntax can be found in the
|
||||
[Why can't I create a quantity by passing a number to a constructor?](../../getting_started/faq.md#why-cant-i-create-a-quantity-by-passing-a-number-to-a-constructor)
|
||||
chapter.
|
||||
|
||||
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 a two-parameter constructor:
|
||||
|
||||
```cpp
|
||||
quantity_point qp4{1356 * km, zeroth_odometer<actor::alice>};
|
||||
```
|
||||
|
||||
Also, as now our timestamps have a proper point origin provided in a type, we can simplify the
|
||||
previous code by directly converting `std::chrono::time_point` to our `quantity_point` type.
|
||||
|
||||
With all the above, we can refactor our initialization part to the following:
|
||||
|
||||
```cpp
|
||||
trip_log<actor::alice> alice_log;
|
||||
|
||||
alice_log.emplace_back("home", zeroth_odometer<actor::alice> + 1356 * km, system_clock::now());
|
||||
|
||||
// some time passes
|
||||
|
||||
alice_log.emplace_back("castle", zeroth_odometer<actor::alice> + 1401 * km, system_clock::now());
|
||||
```
|
||||
|
||||
|
||||
### _Point_ arithmetics
|
||||
|
||||
As another example, 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 : absolute_point_origin<home, isq::distance> {} home;
|
||||
|
||||
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 an origin or 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 a bit
|
||||
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 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
|
||||
```
|
||||
|
||||
!!! note
|
||||
|
||||
It is not allowed to subtract two point origins defined in terms of `absolute_point_origin`
|
||||
(e.g., `home - home`) as those do not contain information about the unit, so we are not able
|
||||
to determine a resulting `quantity` type.
|
||||
|
||||
|
||||
### Relative _Point_ origin
|
||||
|
||||
We often do not have only one ultimate "zero" point when we measure things.
|
||||
We often do not have only one ultimate "zero" point when we measure things. Often, we have one
|
||||
common scale, but we measure various quantities relative to different points and expect
|
||||
those points to be compatible. There are many examples here, but probably the most common are
|
||||
temperatures, timestamps, and altitudes.
|
||||
|
||||
For example, let's assume that we have the following absolute point origin:
|
||||
For such cases, relative point origins should be used:
|
||||
|
||||
{style="width:80%;display: block;margin: 0 auto;"}
|
||||
|
||||
```cpp
|
||||
constexpr struct mean_sea_level : absolute_point_origin<mean_sea_level, isq::altitude> {} mean_sea_level;
|
||||
```
|
||||
inline constexpr struct A : absolute_point_origin<A, isq::distance> {} A;
|
||||
inline constexpr struct B : relative_point_origin<A + 10 * m> {} B;
|
||||
inline constexpr struct C : relative_point_origin<B + 10 * m> {} C;
|
||||
inline constexpr struct D : relative_point_origin<A + 30 * m> {} D;
|
||||
|
||||
If we want to model a trip to Mount Everest, measuring all daily hikes from the `mean_sea_level`
|
||||
might not be efficient. We may know that we are not good climbers, so all our climbs can be
|
||||
represented with an 8-bit integer type, allowing us to save memory in our database of climbs.
|
||||
quantity_point qp1 = C + 100 * m;
|
||||
quantity_point qp2 = D + 120 * m;
|
||||
|
||||
For this purpose, we can define a `relative_point_origin` in the following way:
|
||||
assert(qp1.quantity_ref_from(qp1.point_origin) == 100 * m);
|
||||
assert(qp2.quantity_ref_from(qp2.point_origin) == 120 * m);
|
||||
|
||||
```cpp
|
||||
constexpr struct everest_base_camp : relative_point_origin<mean_sea_level + 5364 * m> {} everest_base_camp;
|
||||
```
|
||||
assert(qp1 - A == 120 * m);
|
||||
assert(qp1 - B == 110 * m);
|
||||
assert(qp1 - C == 100 * m);
|
||||
assert(qp1 - D == 90 * m);
|
||||
assert(qp1.quantity_from(A) == 120 * m);
|
||||
assert(qp1.quantity_from(B) == 110 * m);
|
||||
assert(qp1.quantity_from(C) == 100 * m);
|
||||
assert(qp1.quantity_from(D) == 90 * m);
|
||||
|
||||
The above can be used as an origin for subsequent _Points_:
|
||||
assert(qp2 - A == 150 * m);
|
||||
assert(qp2 - B == 140 * m);
|
||||
assert(qp2 - C == 130 * m);
|
||||
assert(qp2 - D == 120 * m);
|
||||
assert(qp2.quantity_from(A) == 150 * m);
|
||||
assert(qp2.quantity_from(B) == 140 * m);
|
||||
assert(qp2.quantity_from(C) == 130 * m);
|
||||
assert(qp2.quantity_from(D) == 120 * m);
|
||||
|
||||
```cpp
|
||||
constexpr quantity_point first_climb_alt = everest_base_camp + isq::altitude(std::uint8_t{42} * m);
|
||||
static_assert(first_climb_alt.quantity_from(everest_base_camp) == 42 * m);
|
||||
static_assert(first_climb_alt.quantity_from(mean_sea_level) == 5406 * m);
|
||||
static_assert(first_climb_alt.quantity_from_zero() == 5406 * m);
|
||||
assert(qp2 - qp1 == 30 * m);
|
||||
|
||||
assert(B - A == 10 * m);
|
||||
assert(C - A == 20 * m);
|
||||
assert(D - A == 30 * m);
|
||||
assert(D - C == 10 * m);
|
||||
```
|
||||
|
||||
As we can see above, the `quantity_from()` member function returns a relative distance from the
|
||||
provided point origin while the `quantity_from_zero()` returns the distance from the absolute point
|
||||
origin.
|
||||
provided point origin while the `quantity_from_zero()` always returns the distance from the
|
||||
absolute point origin.
|
||||
|
||||
Also, please note that as long as we can't subtract two absolute point origins from each other,
|
||||
it is possible to subtract relative ones or a relative and absolute one.
|
||||
|
||||
|
||||
### Converting between different representations of the same _Point_
|
||||
### 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 `quantity_point` class templates expressed in
|
||||
terms of origins relative to each other in the type system.
|
||||
As we might represent the same _point_ with _displacement vectors_ from various origins, the
|
||||
library provides facilities to convert the same _point_ to the `quantity_point` class templates
|
||||
expressed in terms of different origins.
|
||||
|
||||
For this purpose, we can use:
|
||||
{style="width:80%;display: block;margin: 0 auto;"}
|
||||
|
||||
- a converting constructor:
|
||||
For this purpose, we can use either:
|
||||
|
||||
- A converting constructor:
|
||||
|
||||
```cpp
|
||||
constexpr quantity_point<isq::altitude[m], mean_sea_level, int> qp = first_climb_alt;
|
||||
static_assert(qp.quantity_ref_from(qp.point_origin) == 5406 * m);
|
||||
quantity_point<si::metre, C> qp2C = qp2;
|
||||
assert(qp2C.quantity_ref_from(qp2C.point_origin) == 130 * m);
|
||||
```
|
||||
|
||||
- a dedicated conversion interface:
|
||||
- A dedicated conversion interface:
|
||||
|
||||
```cpp
|
||||
constexpr quantity_point qp = first_climb_alt.point_for(mean_sea_level);
|
||||
static_assert(qp.quantity_ref_from(qp.point_origin) == 5406 * m);
|
||||
quantity_point qp2B = qp2.point_for(B);
|
||||
quantity_point qp2A = qp2.point_for(A);
|
||||
|
||||
assert(qp2B.quantity_ref_from(qp2B.point_origin) == 140 * m);
|
||||
assert(qp2A.quantity_ref_from(qp2A.point_origin) == 150 * m);
|
||||
```
|
||||
|
||||
It is important to understand that the point remains the same after such a translation
|
||||
(all of them compare equal):
|
||||
|
||||
```cpp
|
||||
assert(qp2 == qp2C);
|
||||
assert(qp2 == qp2B);
|
||||
assert(qp2 == qp2A);
|
||||
```
|
||||
|
||||
!!! note
|
||||
|
||||
It is only allowed to convert between various origins defined in terms of the same
|
||||
`absolute_point_origin`. Even if it is theoretically possible to express the same _Point_ as
|
||||
a _Vector_ from another `absolute_point_origin`, the library will not allow such a conversion.
|
||||
A custom user-defined conversion function will be needed to add this functionality.
|
||||
`absolute_point_origin`. Even if it is possible to express the same _point_ as a
|
||||
_displacement vector_ 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.
|
||||
Said another way, in the library, there is no way to spell how two distinct `absolute_point_origin`
|
||||
types relate to each other.
|
||||
|
||||
|
||||
### Temperature support
|
||||
|
||||
Another important example of [relative point origins](#relative-point-origins) is the support
|
||||
of temperature quantity points. The **mp-units** library provides a few predefined point origins
|
||||
for this purpose:
|
||||
Support for temperature quantity points is probably one of the most common examples of relative
|
||||
point origins in action that we use in daily life.
|
||||
|
||||
The [@SI] definition in the library provides a few predefined point origins for this purpose:
|
||||
|
||||
```cpp
|
||||
namespace si {
|
||||
@ -459,7 +392,7 @@ The above is a great example of how point origins can be stacked on top of each
|
||||
!!! note
|
||||
|
||||
Notice that while stacking point origins, we can use not only different representation types
|
||||
but also different units for origins and a _Point_. In the above example, the relative
|
||||
but also different units for origins and a _point_. In the above example, the relative
|
||||
point origin for degree Celsius is defined in terms of `si::kelvin`, while the quantity point
|
||||
for it will use `si::degree_Celsius` as a unit.
|
||||
|
||||
@ -479,13 +412,14 @@ inline constexpr struct degree_Celsius :
|
||||
namespace usc {
|
||||
|
||||
inline constexpr struct degree_Fahrenheit :
|
||||
named_unit<{u8"°F", "`F"}, mag<ratio{5, 9}> * si::degree_Celsius, zeroth_degree_Fahrenheit> {} degree_Fahrenheit;
|
||||
named_unit<{u8"°F", "`F"}, mag<ratio{5, 9}> * si::degree_Celsius,
|
||||
zeroth_degree_Fahrenheit> {} degree_Fahrenheit;
|
||||
|
||||
}
|
||||
```
|
||||
|
||||
Now let's see how we can benefit from the above definitions. We have quite a few alternatives to
|
||||
choose from here. Depending on our needs or taste we can:
|
||||
Now, let's see how we can benefit from the above definitions. We have quite a few alternatives to
|
||||
choose from here. Depending on our needs or tastes, we can:
|
||||
|
||||
- be explicit about the unit and origin:
|
||||
|
||||
@ -511,13 +445,13 @@ choose from here. Depending on our needs or taste we can:
|
||||
quantity_point q9{20.5 * deg_C};
|
||||
```
|
||||
|
||||
*[CTAD]: Class Template Argument Deduction
|
||||
|
||||
In all of the above cases, we end up with the `quantity_point` of the same type and value.
|
||||
|
||||
To play a bit more with temperatures, we can implement a simple room AC temperature controller in
|
||||
the following way:
|
||||
|
||||
{style="width:80%;display: block;margin: 0 auto;"}
|
||||
|
||||
```cpp
|
||||
constexpr struct room_reference_temp : relative_point_origin<quantity_point{21 * deg_C}> {} room_reference_temp;
|
||||
using room_temp = quantity_point<isq::Celsius_temperature[deg_C], room_reference_temp>;
|
||||
@ -539,7 +473,7 @@ std::println("| {:<14} | {:^18} | {:^18} | {:^18} |",
|
||||
std::println("|{0:=^16}|{0:=^20}|{0:=^20}|{0:=^20}|", "");
|
||||
|
||||
auto print = [&](std::string_view label, auto v) {
|
||||
std::println("| {:<14} | {:^18} | {:^18} | {:^18{%N:.2f} %U} |", label,
|
||||
std::println("| {:<14} | {:^18} | {:^18} | {:^18:N[.2f]} |", label,
|
||||
v - room_reference_temp, (v - si::ice_point).in(deg_C), (v - si::absolute_zero).in(deg_C));
|
||||
};
|
||||
|
||||
@ -563,13 +497,20 @@ Room reference temperature: 21 °C (69.8 °F, 294.15 K)
|
||||
|
||||
### 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 prefix or postfix is
|
||||
required.
|
||||
The library does not provide a text output for quantity points. The quantity stored inside
|
||||
is just an implementation detail of this type. It is a vector from a specific origin.
|
||||
Without the knowledge of the origin, the vector by itself is useless as we can't determine
|
||||
which point it describes.
|
||||
|
||||
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
|
||||
a much better solution, but the library does not have enough information to print it that way by itself.
|
||||
In the current library design, point origin does not provide any text in its definition.
|
||||
Even if we could add such information to the point's definition, we would not
|
||||
know how to output it in the text. There may be many ways to do it. For example, should we
|
||||
prepend or append the origin part to the quantity text?
|
||||
|
||||
For example, the text output of `42 m` for a quantity point may mean many things. It may be
|
||||
an offset from the mountain top, sea level, or maybe the center of Mars.
|
||||
Printing `42 m AMSL` for altitudes above mean sea level is a much better solution, but the
|
||||
library does not have enough information to print it that way by itself.
|
||||
|
||||
|
||||
## The affine space is about type-safety
|
||||
|
||||
Reference in New Issue
Block a user