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Character of a Quantity

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Scalars, vectors, and tensors

ISO 80000-2

Scalars, vectors and tensors are mathematical objects that can be used to denote certain physical quantities and their values. They are as such independent of the particular choice of a coordinate system, whereas each scalar component of a vector or a tensor and each component vector and component tensor depend on that choice.

Such distinction is important because each quantity character represents different properties and allows different operations to be done on its quantities.

For example, imagine a physical units library that allows the creation of a \(speed\) quantity from both \(length / time\) and \(length * time\). It wouldn't be too safe to use such a product, right?

Now we have to realize that both of the above operations (multiplication and division) are not even mathematically defined for linear algebra types such as vectors or tensors. On the other hand, two vectors can be passed as arguments to dot and cross-product operations. The result of the first one is a scalar. The second one results in a vector that is perpendicular to both vectors passed as arguments. Again, it wouldn't be safe to allow replacing those two operations with each other or expect the same results from both cases. This simply can't work.

ISQ defines quantities of all characters

While defining quantities ISO 80000 explicitly mentions when a specific quantity has a vector or tensor character. Here are some examples:

Quantity Character Quantity Equation
\(duration\) scalar {base quantity}
\(mass\) scalar {base quantity}
\(length\) scalar {base quantity}
\(path\; length\) scalar {base quantity}
\(radius\) scalar {base quantity}
\(position\; vector\) vector {base quantity}
\(velocity\) vector \(position\; vector / duration\)
\(acceleration\) vector \(velocity / duration\)
\(force\) vector \(mass * acceleration\)
\(power\) scalar \(force \cdot velocity\)
\(moment\; of\; force\) vector \(position\; vector \times force\)
\(torque\) scalar \(moment\; of\; force \cdot \{unit\; vector\}\)
\(surface\; tension\) scalar \(\lvert force \rvert / length\)
\(angular\; displacement\) scalar \(path\; length / radius\)
\(angular\; velocity\) vector \(angular\; displacement / duration * \{unit\; vector\}\)
\(momentum\) vector \(mass * velocity\)
\(angular\; momentum\) vector \(position\; vector \times momentum\)
\(moment\; of\; inertia\) tensor \(angular\; momentum \otimes angular\; velocity\)

In the above equations:

  • \(a * b\) - regular multiplication where one of the arguments has to be scalar
  • \(a / b\) - regular division where the divisor has to be scalar
  • \(a \cdot b\) - dot product of two vectors
  • \(a \times b\) - cross product of two vectors
  • \(\lvert a \rvert\) - magnitude of a vector
  • \(\{unit\; vector\}\) - a special vector with the magnitude of \(1\)
  • \(a \otimes b\) - tensor product of two vectors or tensors

Note

As of now, all of the C++ physical units libraries on the market besides mp-units do not support the operations mentioned above. They expose only multiplication and division operators, which do not work for linear algebra-based representation types. If a user of those libraries would like to create the quantities provided in the above table properly, this would result in a compile-time error stating that multiplication and division of two linear algebra vectors is impossible.

Characters don't apply to dimensions and units

ISO 80000 explicitly states that dimensions are orthogonal to quantity characters:

ISO 80000-1:2009

In deriving the dimension of a quantity, no account is taken of its scalar, vector, or tensor character.

Also, it explicitly states that:

ISO 80000-2

All units are scalars.

Defining vector and tensor quantities

To specify that a specific quantity has a vector or tensor character a value of quantity_character enumeration can be appended to the quantity_spec describing such a quantity type:

inline constexpr struct displacement final : quantity_spec<length, quantity_character::vector> {} displacement;
inline constexpr struct position_vector final : quantity_spec<displacement> {} position_vector;

inline constexpr struct displacement final : quantity_spec<displacement, length, quantity_character::vector> {} displacement;
inline constexpr struct position_vector final : quantity_spec<position_vector, displacement> {} position_vector;

QUANTITY_SPEC(displacement, length, quantity_character::vector);
QUANTITY_SPEC(position_vector, displacement);

With the above, all the quantities derived from position_vector or displacement will have a correct character determined according to the kind of operations included in the quantity equation defining a derived quantity.

For example, velocity in the below definition will be defined as a vector quantity (no explicit character override is needed):

inline constexpr struct velocity final : quantity_spec<speed, displacement / duration> {} velocity;

inline constexpr struct velocity final : quantity_spec<velocity, speed, displacement / duration> {} velocity;

QUANTITY_SPEC(velocity, speed, displacement / duration);

Representation types for vector and tensor quantities

As we remember, the quantity class template is defined as follows:

template<Reference auto R,
         RepresentationOf<get_quantity_spec(R)> Rep = double>
class quantity;

The second template parameter is constrained with a RepresentationOf concept that checks if the provided representation type satisfies the requirements for the character associated with this quantity type.

Note

The current version of the C++ Standard Library does not provide any types that could be used as a representation type for vector and tensor quantities. This is why users are on their own here 😟.

To provide examples and implement unit tests, our library uses the types proposed in the P1385 and available as a Conan package in the Conan Center. However, thanks to the provided customization points, any linear algebra library types can be used as a vector or tensor quantity representation type.

For example, here is how it can be done for the P1385 types:

#include <matrix>

using la_vector = STD_LA::fixed_size_column_vector<double, 3>;

Quantity auto q = la_vector{1, 2, 3} * isq::velocity[m / s];

In case there is an ambiguity of operator* between mp-units and a linear algebra library, we can either:

  • use two-parameter constructor

    Quantity auto q = quantity{la_vector{1, 2, 3}, isq::velocity[m / s]};

  • provide a dedicated overload of operator* that will resolve the ambiguity and wrap the above

    template<Reference R>
Quantity auto operator*(la_vector rep, R)
{
  return quantity{rep, R{}};
}

Note

The following does not work:

Quantity auto q1 = la_vector{1, 2, 3} * m / s;
Quantity auto q2 = isq::velocity(la_vector{1, 2, 3} * m / s);
quantity<isq::velocity[m/s]> q3{la_vector{1, 2, 3} * m / s};

In all the cases above, the SI unit m / s has an associated scalar quantity of isq::length / isq::time. la_vector is not a correct representation type for a scalar quantity so the construction fails.