Character of a Quantity¶
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Scalars, vectors, and tensors¶
ISO 800002
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 crossproduct 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 mpunits do not support the operations mentioned above. They expose only multiplication and division operators, which do not work for linear algebrabased 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 compiletime 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 800001:2009
In deriving the dimension of a quantity, no account is taken of its scalar, vector, or tensor character.
Also, it explicitly states that:
ISO 800002
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:
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):
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).character> 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.
To enable the usage of a userdefined type as a representation type for vector or tensor quantities,
we need to provide a partial specialization of is_vector
or is_tensor
customization points.
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>;
template<>
constexpr bool mp_units::is_vector<la_vector> = true;
With the above, we can use la_vector
as a representation type for our quantity:
In case there is an ambiguity of operator*
between mpunits and a linear algebra library, we can
either:

use twoparameter constructor

provide a dedicated overload of
operator*
that will resolve the ambiguity and wrap the above
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.
Hacking the character¶
Sometimes we want to use a vector quantity, but we don't care about its direction. For example, the standard gravity acceleration constant always points down, so we might not care about this in a particular scenario. In such a case, we may want to "hack" the library to allow scalar types to be used as a representation type for scalar quantities.
For example, we can do the following:
which says that every type that can be used as a scalar representation is also allowed for vector quantities.
Doing the above is actually not such a big "hack" as the ISO 80000 explicitly allows it:
ISO 800002
A vector is a tensor of the first order and a scalar is a tensor of order zero.
Despite it being allowed by ISO 80000, for typesafety reasons, we do not allow such a behavior by default, and a user has to opt into such scenarios explicitly.