The Affine Space

The affine space has two types of entities:

• Point - a position specified with coordinate values (e.g., location, address, etc.)
• Displacement vector - 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 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 (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
• 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.

Points are more common than most of us imagine

Point abstractions should be used more often in the C++ software. They are not only about temperature or time. Points are everywhere around us and should become more popular in the products we implement. They can be used to implement:

• temperature points,
• timestamps,
• daily mass readouts from the scale,
• altitudes of mountain peaks on a map,
• current path length measured by the car's odometer,
• today's price of instruments on the market,
• and many more.

Improving the affine space's Points intuition will allow us to write better and safer software.

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:

• the distance between two points,
• 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 displacement vector abstraction in the affine space. It can be constructed with:

• the multiply syntax (works for most of the units),
• delta<Reference> construction helper (e.g., delta<isq::height[m]>(42), delta<deg_C>(3)),
• two-parameter constructor taking a number and a quantity reference/unit.

Note

The multiply syntax support is disabled for units that provide a point origin in their definition (i.e., units of temperature like K, deg_C, and deg_F).

Point is modeled by quantity_point and PointOrigin

In the mp-units library, the Point abstraction is modelled by:

• PointOrigin concept that specifies measurement origin, and
• quantity_point class template that specifies a Point relative to a specific predefined origin.

quantity_point

The quantity_point class template specifies an absolute quantity measured from a predefined origin:

template<Reference auto R,
         PointOriginFor<get_quantity_spec(R)> auto PO = default_point_origin(R),
         RepresentationOf<get_quantity_spec(R)> 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 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 origin using the following rules:

• if the measurement unit of a quantity specifies its point origin in its definition (e.g., degree Celsius), then this origin is being used,
• otherwise, an instantiation of natural_point_origin<QuantitySpec> is being used which provides a well-established natural origin for a specific quantity type.

Quantity points with default point origins may be constructed with the point construction helper or forcing an explicit conversion from the quantity:

// quantity_point qp1 = 42 * m;           // Compile-time error
// quantity_point qp2 = 42 * K;           // Compile-time error
// quantity_point qp3 = delta<deg_C>(42); // Compile-time error
quantity_point qp4(42 * m);
quantity_point qp5(42 * K);
quantity_point qp6(delta<deg_C>(42));
quantity_point qp7 = point<m>(42);
quantity_point qp8 = point<K>(42);
quantity_point qp9 = point<deg_C>(42);

Tip

The quantity_point definition can be found in the mp-units/quantity_point.h header file.

natural_point_origin<QuantitySpec>

natural_point_origin<QuantitySpec> is meant to be used in cases where the specific domain has a well-established, non-controversial, and unique natural origin on the measurement scale. This saves the user from the need to write a boilerplate code that would predefine such a type for this domain.

quantity_point<isq::distance[si::metre]> qp1(100 * m);
quantity_point<isq::distance[si::metre]> qp2 = point<m>(120);

assert(qp1.quantity_from_unit_zero() == 100 * m);
assert(qp2.quantity_from_unit_zero() == 120 * m);
assert(qp2.quantity_from(qp1) == 20 * m);
assert(qp1.quantity_from(qp2) == -20 * 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 natural_point_origin<QuantitySpec>.

It is really important to understand that even though we can use .quantity_from_unit_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 natural_point_origin<QuantitySpec> makes various quantity point objects compatible as long as quantity types used in the origin and reference are compatible:

quantity_point<si::metre> qp1{isq::distance(100 * m)};
quantity_point<si::metre> qp2 = point<isq::height[m]>(120);

assert(qp2.quantity_from(qp1) == 20 * m);
assert(qp1.quantity_from(qp2) == -20 * m);
assert(qp2 - qp1 == 20 * m);
assert(qp1 - qp2 == -20 * m);

Absolute point origin

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.

inline constexpr struct origin final : absolute_point_origin<isq::distance> {} origin;

// quantity_point<si::metre, origin> qp1{100 * m};        // Compile-time error
// quantity_point<si::metre, origin> qp2{delta<m>(120)};  // 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_unit_zero() == 100 * m);   // Compile-time error
// assert(qp2.quantity_from_unit_zero() == 120 * m);   // Compile-time error
assert(qp1.quantity_from(origin) == 100 * m);
assert(qp2.quantity_from(origin) == 120 * m);
assert(qp2.quantity_from(qp1) == 20 * m);
assert(qp1.quantity_from(qp2) == -20 * m);

assert(qp1 - origin == 100 * m);
assert(qp2 - origin == 120 * m);
assert(qp2 - qp1 == 20 * m);
assert(qp1 - qp2 == -20 * m);

assert(origin - qp1 == -100 * m);
assert(origin - qp2 == -120 * m);

// assert(origin - origin == 0 * m);   // Compile-time error

We can't construct a quantity point directly from the quantity anymore when a custom, named origin is used. To prevent potential safety and maintenance issues, we always need to explicitly provide both a compatible origin and a quantity measured from it to construct a quantity point.

Said otherwise, a quantity point defined in terms of a specific origin is the result of adding the origin and the displacement vector measured from it to the point we create.

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? chapter.

Similarly to creation of 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:

quantity_point qp1{100 * m, origin};

Again, CTAD always helps to use precisely the type we need in a current case.

Additionally, if a quantity point is defined in terms of a custom, named origin, then we can't use a quantity_from_unit_zero() member function anymore. This is to prevent surprises, as our origin may not necessarily be perceived as an absolute zero in the domain we model. Also, as we will learn soon, we can define several related origins in one space, and then it gets harder to understand which one is the zero one. This is why, to be specific and always correct about the points we use, a quantity_from(QP) member function can be used (where QP can either be an origin or another quantity point).

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.

Modeling independent spaces in one domain

Absolute point origins are also perfect for establishing independent spaces even if the same quantity type and unit is being used:

inline constexpr struct origin1 final : absolute_point_origin<isq::distance> {} origin1;
inline constexpr struct origin2 final : absolute_point_origin<isq::distance> {} origin2;

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(qp2.quantity_from(qp1) == 20 * m);       // Compile-time error
// assert(qp1.quantity_from(origin2) == 100 * m);  // Compile-time error
// assert(qp2.quantity_from(origin1) == 120 * m);  // Compile-time error

Relative Point origin

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 such cases, relative point origins should be used:

inline constexpr struct A final : absolute_point_origin<isq::distance> {} A;
inline constexpr struct B final : relative_point_origin<A + 10 * m> {} B;
inline constexpr struct C final : relative_point_origin<B + 10 * m> {} C;
inline constexpr struct D final : relative_point_origin<A + 30 * m> {} D;

quantity_point qp1 = C + 100 * m;
quantity_point qp2 = D + 120 * m;

assert(qp1.quantity_ref_from(qp1.point_origin) == 100 * m);
assert(qp2.quantity_ref_from(qp2.point_origin) == 120 * m);

assert(qp2.quantity_from(qp1) == 30 * m);
assert(qp1.quantity_from(qp2) == -30 * m);
assert(qp2 - qp1 == 30 * m);
assert(qp1 - qp2 == -30 * 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);
assert(qp1 - A == 120 * m);
assert(qp1 - B == 110 * m);
assert(qp1 - C == 100 * m);
assert(qp1 - D == 90 * 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);
assert(qp2 - A == 150 * m);
assert(qp2 - B == 140 * m);
assert(qp2 - C == 130 * m);
assert(qp2 - D == 120 * m);

assert(B - A == 10 * m);
assert(C - A == 20 * m);
assert(D - A == 30 * m);
assert(D - C == 10 * m);

assert(B - B == 0 * m);
// assert(A - A == 0 * m);  // Compile-time error

Note

Even though we can't subtract two absolute point origins from each other, it is possible to subtract relative ones or relative and absolute ones.

Converting between different representations of the same point

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 either:

• A converting constructor:

  quantity_point<si::metre, C> qp2C = qp2;
  assert(qp2C.quantity_ref_from(qp2C.point_origin) == 130 * m);
  

• A dedicated conversion interface:

  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 all such translations still describe exactly the same point (e.g., all of them compare equal):

assert(qp2 == qp2C);
assert(qp2 == qp2B);
assert(qp2 == qp2A);

Important

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 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 such a functionality.

Said another way, in the library, there is no way to spell how two distinct absolute_point_origin types relate to each other.

Range-Validated Quantity Points

Background reading

For a deeper discussion of the motivation, design trade-offs, and open questions, see the blog article Range-Validated Quantity Points.

In many domains, quantity points must stay within specific bounds. For example:

• Geographic coordinates: latitude ∈ [-90°, 90°], longitude ∈ [-180°, 180°)
• Temperature sensors: operating range [−40°C, 85°C]
• Control systems: valid input range [0, 100] units

The library provides four overflow policies that can be attached to point origins via the quantity_bounds customization point. These policies enforce bounds during construction, unit conversion, and arithmetic operations.

Production Use Case: Geographic Coordinate Systems

This feature was specifically designed to support production requirements for geographic/navigation systems, where different angle types require different wrapping behaviors:

• Latitude (symmetric): reflects at ±90° boundaries (can't go past poles)
• Longitude (mirrored): wraps circularly at -180° to +180°
• Elevation (symmetric): reflects at ±90° like latitude
• Geometric Azimuth (mirrored): 0° = East, counter-clockwise, wraps [-180°, 180°)
• Bearing (mirrored): 0° = North, clockwise, wraps [-180°, 180°)
  • Conversion: bearing = 90° - geometric_azimuth (involves negation)
  • Cannot use relative_point_origin (requires separate absolute_point_origin)
• Heading (mirrored): 0° = North, counter-clockwise, wraps [-180°, 180°)
  • Conversion: heading = geometric_azimuth - 90° (simple offset)
  • Uses relative_point_origin<east + delta<geometric_azimuth[degree]>(-90.)>

These requirements came from companies working with coordinate reference systems (CRS), navigation, and geodesy. See the complete discussion in GitHub #782.

Key Insight: relative_point_origin can only model offset transformations (addition/subtraction). Transformations involving negation or sign flips (like bearing ↔ azimuth) require separate absolute_point_origin types with explicit conversion functions. This ensures type safety while allowing the framework to handle appropriate coordinate transformations automatically.

Type Safety with is_kind: All geographic quantity specs use is_kind to prevent accidental mixing of different angle types (e.g., latitude + longitude, bearing + heading). This requires explicit conversion when using trigonometric functions:

// Convert to plain angular_measure for trig functions
const quantity<angular_measure> lat_angle = isq::angular_measure(lat.quantity_from_unit_zero());
const quantity result = sin(lat_angle);  // Works with angular_measure

The current framework handles all wrapping schemes and coordinate transformations expressible as offsets. Advanced geodetic calculations (ellipsoid parameters, distance formulas for WGS84, etc.) require additional domain-specific extensions beyond the core library. For a complete working example, see geographic.h and the UAV: Multiple Altitude Reference Systems example, which demonstrates multiple point origins, bounds checking policies, overflow handling, and coordinate reference conversions.

Available Overflow Policies

Policy	Behavior	Error Handling	Use Case
check_in_range	Reports violation via constraint_violation_handler or MP_UNITS_EXPECTS	Depends on rep type	Bounds checking with customizable error behavior
clamp_to_range	Clamps to nearest boundary: clamp(value, min, max)	Silent correction	Saturating arithmetic, sensor limits, UI controls
wrap_to_range	Wraps circularly to [min, max): modulo arithmetic	Value transformation	Periodic quantities (angles, time-of-day, longitude)
reflect_in_range	Reflects at boundaries (like a bouncing ball)	Value transformation	Geographic latitude, physical boundaries
check_non_negative	Reports violation if value < 0	Depends on rep type	Inherently non-negative quantities (length, mass)
clamp_non_negative	Clamps negative values to zero	Silent correction	FP rounding noise in non-negative domains

Important

The four bounded-range policies (check_in_range, clamp_to_range, wrap_to_range, reflect_in_range) expose public min and max data members, enabling std::numeric_limits specializations to reflect the constrained bounds. check_non_negative and clamp_non_negative are halfline policies with no upper bound and therefore do not expose max. See the std::numeric_limits note below for details.

Error behavior of check_in_range

check_in_range delegates error reporting to the representation type:

• If the rep has a constraint_violation_handler specialization (e.g. constrained<double, throw_policy>), the handler's on_violation() is called, providing guaranteed enforcement regardless of build mode.
• Otherwise, falls back to MP_UNITS_EXPECTS, which may be disabled in release builds.

See Tutorial: Custom Contract Handlers for a step-by-step guide to implementing custom error policies, or How to: Ensure Ultimate Safety for a complete example of combining check_in_range with constrained rep types.

Example Usage

// Define quantity specifications and origins
inline constexpr struct geo_latitude final : quantity_spec<isq::angular_measure> {} geo_latitude;
inline constexpr struct geo_longitude final : quantity_spec<isq::angular_measure> {} geo_longitude;

inline constexpr struct equator final : absolute_point_origin<geo_latitude> {} equator;
inline constexpr struct prime_meridian final : absolute_point_origin<geo_longitude> {} prime_meridian;

// Attach bounds to origins
template<>
inline constexpr auto quantity_bounds<equator> = reflect_in_range{-90 * si::degree, 90 * si::degree};

template<>
inline constexpr auto quantity_bounds<prime_meridian> = wrap_to_range{-180 * si::degree, 180 * si::degree};

// Define bounded quantity point types
template<typename T = double> using latitude = quantity_point<si::degree, equator, T>;
template<typename T = double> using longitude = quantity_point<si::degree, prime_meridian, T>;

void example()
{
  latitude lat = equator + 45.0 * deg;           // Valid: within [-90, 90]
  lat = equator + 95.0 * deg;                    // Reflects to 85° (reflect_in_range mirrors at boundary)

  longitude lon = prime_meridian + 270.0 * deg;  // Wraps to -90° (wrap_to_range treats range as circular)
  lon += 200.0 * deg;                            // Result wraps: -90 + 200 = 110°
}

How It Works?

Bounds are enforced at these points:

1. Construction from a quantity: quantity_point{value * unit, origin}
2. Unit conversion: qp.in(other_unit), qp.force_in(other_unit)
3. Arithmetic operations: operator+=, operator-=, operator++, operator--
4. Origin conversion: qp.point_for(new_origin)

After each mutation the library transparently applies the policy specified in quantity_bounds<origin>.

std::numeric_limits integration

When a quantity_point has a quantity_bounds policy attached to its origin, the std::numeric_limits specialization automatically reflects the constrained bounds:

template<>
inline constexpr auto quantity_bounds<prime_meridian> = wrap_to_range{-180 * deg, 180 * deg};

using longitude = quantity_point<geo_longitude[deg], prime_meridian>;

quantity lon_min = std::numeric_limits<longitude>::min();  // prime_meridian - 180°
quantity lon_max = std::numeric_limits<longitude>::max();  // prime_meridian + 180°

This works because the policy struct exposes public min and max data members, which quantity_point::min() and max() access to provide meaningful bounds. Without a quantity_bounds policy, std::numeric_limits delegates to the representation type's limits.

Custom Policies (One-Sided Bounds)

The library ships two built-in one-sided halfline policies for [0, +∞) domains: check_non_negative (reports a violation when the value is negative) and clamp_non_negative (silently clamps negative values to zero). These are automatically applied to any quantity_point whose origin is a natural_point_origin of a non-negative quantity spec (such as isq::length, isq::mass, or isq::duration).

For other one-sided constraints, you can create a fully custom policy:

template<Quantity Q>
struct clamp_bottom {
  Q min;  // Public member enables std::numeric_limits support

  template<Quantity V>
  constexpr V operator()(V v) const { return v < min ? min : v; }
};

// Usage: hydraulic system with minimum operating pressure
inline constexpr struct atmospheric_pressure final : absolute_point_origin<isq::pressure> {} atmospheric_pressure;
template<>
inline constexpr auto quantity_bounds<atmospheric_pressure> = clamp_bottom{1000 * si::kilo<si::pascal>};

using hydraulic_pressure = quantity_point<si::kilo<si::pascal>, atmospheric_pressure>;
hydraulic_pressure p1 = atmospheric_pressure + 2000 * kPa;  // OK: 2000 kPa
hydraulic_pressure p2 = atmospheric_pressure + 500 * kPa;   // Clamped to 1000 kPa (min operating pressure)

Hierarchical bounds validation

When a relative_point_origin defines bounds and its parent origin also has bounds, the library enforces at compile time that the child's bounds fit within the parent's bounds (after translating to the parent's reference frame).

This hierarchical validation ensures semantic correctness: if you model origin B as relative to origin A, and A has operational constraints, then B inheriting those constraints is physically and logically appropriate (e.g., a room's temperature range shouldn't exceed the building HVAC system's capability, a drone's altitude above launch point shouldn't exceed airspace authorization limits).

Absolute origin bounds represent physical constraints (e.g., temperature can't go below absolute zero) and are always enforced—this is non-negotiable.

Relative origin bounds hierarchical enforcement represents our current design decision based on typical use cases. If you encounter scenarios where nested relative origins need independent bounds (not constrained by their parent), please share your feedback so we can evaluate adding opt-in flexibility.

Implementation reference

See geographic.h and overflow_policies.h for complete examples.

Temperature support

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:

namespace si {

inline constexpr struct absolute_zero final : absolute_point_origin<isq::thermodynamic_temperature> {} absolute_zero;

inline constexpr struct ice_point final : relative_point_origin<point<milli<kelvin>>(273'150)}> {} ice_point;

}

namespace usc {

inline constexpr struct fahrenheit_zero final :
  relative_point_origin<point<mag_ratio<5, 9> * si::degree_Celsius>(-32)> {} fahrenheit_zero;

}

The above is a great example of how point origins can be stacked on top of each other:

• usc::fahrenheit_zero is defined relative to si::ice_point
• si::ice_point is defined relative to si::absolute_zero.

Note

Notice that while stacking point origins, we can use different representation types and 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.

The temperature point origins defined above are provided explicitly in the respective units' definitions:

namespace si {

inline constexpr struct kelvin final :
    named_unit<"K", kind_of<isq::thermodynamic_temperature>, absolute_zero> {} kelvin;
inline constexpr struct degree_Celsius final :
    named_unit<{u8"℃", "`C"}, kelvin, ice_point> {} degree_Celsius;

}

namespace usc {

inline constexpr struct degree_Fahrenheit final :
    named_unit<{u8"℉", "`F"}, mag_ratio<5, 9> * si::degree_Celsius,
               fahrenheit_zero> {} degree_Fahrenheit;

}

As it was described above, default_point_origin(R) returns a natural_point_origin<QuantitySpec> when a unit does not provide any origin in its definition. As of today, the units of temperature are the only ones in the entire mp-units library that provide such origins.

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:

  quantity_point<si::degree_Celsius, si::ice_point> q1 = si::ice_point + delta<deg_C>(20.5);
  quantity_point<si::degree_Celsius, si::ice_point> q2{delta<deg_C>(20.5), si::ice_point};
  quantity_point<si::degree_Celsius, si::ice_point> q3{delta<deg_C>(20.5)};
  quantity_point<si::degree_Celsius, si::ice_point> q4 = point<deg_C>(20.5);
  

• specify a unit and use its point origin implicitly:

  quantity_point<si::degree_Celsius> q5 = si::ice_point + delta<deg_C>(20.5);
  quantity_point<si::degree_Celsius> q6{delta<deg_C>(20.5), si::ice_point};
  quantity_point<si::degree_Celsius> q7{delta<deg_C>(20.5)};
  quantity_point<si::degree_Celsius> q8 = point<deg_C>(20.5);
  

• benefit from CTAD:

  quantity_point q9 = si::ice_point + delta<deg_C>(20.5);
  quantity_point q10{delta<deg_C>(20.5), si::ice_point};
  quantity_point q11{delta<deg_C>(20.5)};
  quantity_point q12 = point<deg_C>(20.5);
  

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:

Now that the Range-Validated Quantity Points infrastructure is available, we can attach a clamp_to_range policy so the controller silently enforces the comfort band:

constexpr struct room_reference_temp final : relative_point_origin<point<deg_C>(21)> {} room_reference_temp;
template<>
inline constexpr auto mp_units::quantity_bounds<room_reference_temp> = clamp_to_range{delta<deg_C>(-3), delta<deg_C>(3)};
using room_temp = quantity_point<deg_C, room_reference_temp>;

constexpr auto step_delta = delta<deg_C>(0.5);
constexpr int number_of_steps = 6;

room_temp room_ref{};
room_temp room_low  = room_ref - number_of_steps * step_delta;  // −3 °C: exactly at lower bound
room_temp room_high = room_ref + number_of_steps * step_delta;  // +3 °C: exactly at upper bound

// Any attempt to exceed the ±3 °C comfort range is silently clamped
room_temp clamped = room_ref + delta<deg_C>(10.0);              // → auto-clamped to +3 °C from reference

std::println("Room reference temperature: {} ({}, {::N[.2f]})\n",
             room_ref.quantity_from_unit_zero(),
             room_ref.in(usc::degree_Fahrenheit).quantity_from_unit_zero(),
             room_ref.in(si::kelvin).quantity_from_unit_zero());

std::println("| {:<18} | {:^18} | {:^18} | {:^18} |",
             "Temperature delta", "Room reference", "Ice point", "Absolute zero");
std::println("|{0:=^20}|{0:=^20}|{0:=^20}|{0:=^20}|", "");

auto print_temp = [&](std::string_view label, auto v) {
  std::println("| {:<18} | {:^18} | {:^18} | {:^18:N[.2f]} |", label,
               v - room_reference_temp, (v - si::ice_point).in(deg_C), (v - si::absolute_zero).in(deg_C));
};

print_temp("Lowest", room_low);
print_temp("Default", room_ref);
print_temp("Highest", room_high);
print_temp("Clamped to max", clamped);

The above prints:

Room reference temperature: 21 ℃ (69.8 ℉, 294.15 K)

| Temperature delta  |   Room reference   |     Ice point      |   Absolute zero    |
|====================|====================|====================|====================|
| Lowest             |       -3 ℃        |       18 ℃        |     291.15 ℃      |
| Default            |        0 ℃        |       21 ℃        |     294.15 ℃      |
| Highest            |        3 ℃        |       24 ℃        |     297.15 ℃      |
| Clamped to max     |        3 ℃        |       24 ℃        |     297.15 ℃      |

No text output for Points

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.

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 \(42\ \mathrm{m}\) 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 \(42\ \mathrm{m}\ \mathrm{AMSL}\) 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

The following operations are not allowed in the affine space:

• 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 the 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 subtracting a distance point to DEN airport from the Mount Everest base camp altitude mean?
• mixing quantity_points of convertible quantities but with unrelated origins
  • How do we 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?

Important: The affine space improves safety

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