Quantity Specification Conversions

Every time a quantity is copied into a differently typed one, passed through the call operator of a quantity specification, or put through quantity_cast, the library has to answer one question: how are these two quantity specifications related? The answer decides whether the code compiles as written, needs an explicit conversion, needs a cast, or is rejected altogether.

Systems of Quantities shows the four outcomes on the length hierarchy, which is all that everyday code needs. This chapter gives the complete rules, including what happens when quantity equations and quantity kinds are involved.

Vocabulary

A quantity specification comes in one of three shapes:

• A named quantity is defined with quantity_spec. It has a dimension, an optional parent, an optional equation, and may be marked with is_kind.
• A derived quantity is a product of powers of named quantities, such as isq::mass * pow<2>(isq::length) / pow<2>(isq::time). The library keeps it normalized, so equal names are merged, zero exponents disappear, and dimensionless is the identity (isq::length / isq::length is dimensionless, while isq::width / isq::length stays a derived quantity). A factor may itself be a kind, spelled kind_of_<K>, when a kind and a typed quantity met in one equation (see Kinds).
• A kind, spelled kind_of<K>, where K is a named quantity that is its own kind root, or a product of such quantities, stands for any quantity of the hierarchy below K. This is what a bare unit carries: 42 * m has the quantity specification kind_of<isq::length>.

The rules refer to a few facts about a named quantity:

• Its tree is the chain of parents. The root is the top of that chain. Two named quantities are in the same tree when they share a root.
• Its kind root is the nearest quantity on the way up, itself included, that was declared with is_kind. If there is none, the kind root is the root. A quantity declared with is_kind is a declared kind. Angular measure and storage capacity are the usual examples, see Nested quantity kinds.
• Its equation is the one given in its definition or, if none was given, its parent's, recursively. A base quantity has none. A quantity defines its equation when it has one that it didn't take over from its parent: it is a root, its parent has no equation, or its equation differs from its parent's. A child that restates its parent's equation doesn't define it. The defining ancestor of a quantity is the nearest quantity on the way up, itself included, that defines the equation the quantity has. dimensionless is a root whose equation is the empty product, so rotation, a child of dimensionless without an equation of its own, has dimensionless as its defining ancestor.
• Its depth says how many levels of equations lie below it. A quantity with no equation at all, such as a base quantity, has depth 0. A quantity with an equation has depth one more than the deepest factor of that equation, so speed (length / duration) has depth 1 and acceleration (velocity / duration) has depth 2. The empty product is an equation too, one without factors, so dimensionless and rotation, which takes it over, have depth 1. A derived quantity has the depth of its deepest factor.
• Its carried kinds are the declared kinds it carries, each with a count that may be negative or fractional. The count comes from two places. First, the quantity itself and every quantity on its way up count once each if they are declared kinds, so phase angle carries phase angle once and angular measure once. Second, every factor of its equation adds its own carried kinds multiplied by the factor's exponent, so a factor in a denominator subtracts. Angular velocity, defined over an angular displacement, carries angular measure once this way. Counts of one kind add up and a count of zero is dropped, so speed carries nothing. A derived quantity carries the sum over its factors: isq::angular_velocity * isq::radius / isq::angular_measure / isq::speed carries nothing: its numerator angle and its denominator angle cancel.

For the definitions below, where dimensionless is the library's own:

inline constexpr struct length : quantity_spec<dim_length> {} length;
inline constexpr struct width : quantity_spec<length> {} width;
inline constexpr struct radius : quantity_spec<width> {} radius;
inline constexpr struct path_length : quantity_spec<length> {} path_length;
inline constexpr auto arc_length = path_length;
inline constexpr struct duration : quantity_spec<dim_time> {} duration;
inline constexpr struct mass : quantity_spec<dim_mass> {} mass;
inline constexpr struct speed : quantity_spec<length / duration> {} speed;
inline constexpr struct special_speed : quantity_spec<speed> {} special_speed;
inline constexpr struct displacement : quantity_spec<length, quantity_tensor_order::vector> {} displacement;
inline constexpr struct velocity : quantity_spec<speed, displacement / duration> {} velocity;
inline constexpr struct acceleration : quantity_spec<velocity / duration> {} acceleration;
inline constexpr struct force : quantity_spec<mass * acceleration> {} force;
inline constexpr struct angular_measure : quantity_spec<dimensionless, arc_length / radius, is_kind> {} angular_measure;
inline constexpr struct phase_angle : quantity_spec<angular_measure, is_kind> {} phase_angle;
inline constexpr struct rotational_displacement : quantity_spec<angular_measure, path_length / radius> {} rotational_displacement;
inline constexpr struct rotation : quantity_spec<dimensionless> {} rotation;

the facts are:

Quantity	Root	Kind root	Equation	Defining ancestor	Depth	Carried kinds
width	length	length	none	none	0	none
speed	speed	speed	length / duration	speed	1	none
special_speed	speed	speed	length / duration	speed	1	none
velocity	speed	speed	displacement / duration	velocity	1	none
acceleration	acceleration	acceleration	velocity / duration	acceleration	2	none
force	force	force	mass * acceleration	force	3	none
angular_measure	dimensionless	angular_measure	arc_length / radius	angular_measure	1	angular_measure
phase_angle	dimensionless	phase_angle	arc_length / radius	angular_measure	1	phase_angle, angular_measure
rotational_displacement	dimensionless	angular_measure	path_length / radius	angular_measure	1	angular_measure
rotation	dimensionless	dimensionless	(empty)	dimensionless	1	none

Special speed and phase angle take their equation over from the parent, and rotational displacement restates its parent's (path_length is arc_length), so none of the three defines one. Force has depth 3 because its equation reaches acceleration, whose equation reaches velocity.

The character of a quantity plays no part in these rules.

Outcomes

A conversion gets one of four ranks, ordered from worst to best:

Rank	Meaning
none	the quantities have nothing in common
cast	same kind, different branches (quantity_cast is required)
explicit	the target is more specific than the source, or a declared kind is lost or gained
implicit	every source quantity is a target quantity, or the source is an uncommitted kind that the target specializes

utility::implicitly_convertible() is true for implicit, utility::explicitly_convertible() for implicit and explicit, and utility::castable() for all but none.

A conversion is explicit for one of two reasons: the target is more specific than the source (a downcast), or the conversion drops or gains a declared kind. The two reasons lead to the same rank with one exception, a source that is a kind, such as the kind_of<isq::length> that 42 * m carries (see Kinds). A kind hasn't committed to any quantity of its tree yet, so a downcast from it takes nothing away and is implicit: kind_of<inverse(isq::time)> converts to isq::frequency implicitly. Dropping a declared kind loses the kind whatever the source is, so kind_of<isq::angular_measure / isq::time> to dimensionless / isq::time stays explicit.

A conversion is decided by a sequence of operations, described below. Each operation has a rank, and the conversion gets the lowest of them: inverse(isq::time) to isq::frequency expands frequency with implicit and pairs time with period_duration, which ranks explicit, so the conversion is explicit. When the operations can be applied in more than one way, the conversion gets the best rank over all of them.

Overview of the process

The process runs in up to three steps. Step 1 picks the rule that applies. Between two single named quantities, or a named quantity and a kind, the tree rule or the kind rules decide the rank on their own. When a derived quantity is involved, the factors common to both sides cancel first. If a single named quantity is left on each side, the same rules decide. Otherwise the equation rule continues with steps 2 and 3, with K in place of a kind kind_of<K>:

flowchart TD
    start([step 1: source → target]) --> same{same type?}
    same -- yes --> r_yes([implicit])
    same -- no --> dim{same dimension?}
    dim -- no --> r_no([none])
    dim -- yes --> named{both sides a single named<br>quantity, or a kind of one?}
    named -- yes --> kind{a kind on<br>either side?}
    kind -- yes --> kinds([kind rules])
    kind -- no --> tree([tree rule])
    named -- no --> common[cancel the factors<br>common to both sides]
    common --> left{a single named quantity<br>left on both sides?}
    left -- yes --> kind
    left -- no --> eq([equation rule:<br>go to step 2])

Step 2 makes one operation at a time. After each operation it starts again from the top, until both sides are the same: both empty, or the same product right after an exact equation:

flowchart TD
    left{step 2:<br>both sides the same?} -- yes --> done([go to step 3])
    left -- no --> single{exact equation, or<br>a product of its own tree?}
    single -- no --> pair[pair the factors of every<br>tree found on both sides,<br>except siblings still waiting]
    single -- yes --> sub[expand it]
    pair --> apart{a pair ranking none, or<br>different roots of one<br>dimension left unpaired,<br>neither equation mentioning<br>the other's tree?}
    apart -- yes --> r_no([none])
    apart -- no --> paired{anything paired?}
    paired -- no --> cancel{the other side empty,<br>and self-cancel possible?}
    cancel -- yes --> cancelled[self-cancel]
    cancel -- no --> deep{a factor with<br>an equation?}
    deep -- no --> r_no
    deep -- yes --> reduce[expand the deepest,<br>or the shallowest target<br>factor while siblings wait]
    sub --> left
    paired -- yes --> left
    cancelled --> left
    reduce --> left

Step 3 takes the lowest rank of the operations made in step 2, and the kind boundary may lower it:

flowchart TD
    rank([step 3:<br>lowest operation rank]) --> both{a kind lost<br>and one gained?}
    both -- yes --> r_no([none])
    both -- no --> change{a kind lost, or one gained<br>by a source that is not a kind?}
    change -- yes --> expl([at most<br>explicit])
    change -- no --> keep([unchanged])

Checks that come first

Identical quantity specifications convert with implicit. Quantity specifications of different dimensions never convert, and quantity_cast won't force them either:

static_assert(!utility::castable(isq::time, isq::length));

Everything below assumes equal dimensions.

Kinds

A kind kind_of<K> names a quantity somewhere in the tree below the kind root K without saying where. A conversion between a kind of a named quantity and a named quantity never ranks cast.

A kind as the source

When the source is a kind, the target is implicit if it lies in the tree below K, sub-kinds included, unless its equation lacks a declared kind that K carries, which makes it explicit. A child of angular velocity defined as inverse(duration) is reached only explicitly. If the target is above K, the kind has to leave its declared kind and the rank is explicit. Anything else is none:

static_assert(utility::implicitly_convertible(kind_of<isq::length>, isq::width));
static_assert(utility::implicitly_convertible(kind_of<dimensionless>, isq::angular_measure));
static_assert(!utility::implicitly_convertible(kind_of<isq::angular_measure>, dimensionless));
static_assert(utility::explicitly_convertible(kind_of<isq::angular_measure>, dimensionless));
static_assert(!utility::castable(kind_of<isq::frequency>, isq::activity));

This is why quantity<isq::height[m]> tower_height = 42 * m; compiles: the right-hand side is a quantity of kind length and height is a length.

A kind as the target

When the target is a kind, asking for "any quantity of kind K" is the same as asking for K itself. A source below K is implicit, or explicit if one of them carries a declared kind the other doesn't, lying on the way up, the source included and K excluded, or carried by an equation. A source above K is explicit, like any other downcast. Any other source, from another branch of the same tree or from a different tree, is none:

static_assert(utility::implicitly_convertible(isq::width, kind_of<isq::length>));
static_assert(utility::explicitly_convertible(isq::angular_measure, kind_of<dimensionless>));
static_assert(!utility::implicitly_convertible(isq::angular_measure, kind_of<dimensionless>));
static_assert(utility::explicitly_convertible(dimensionless, kind_of<isq::angular_measure>));
static_assert(!utility::implicitly_convertible(dimensionless, kind_of<isq::angular_measure>));

Kinds in equations

When a derived quantity is involved, the equation rules below run with K in place of the kind. Every downcast step from a source kind ranks implicit, while dropping a declared kind stays explicit:

static_assert(utility::implicitly_convertible(kind_of<inverse(isq::time)>, isq::frequency));
static_assert(utility::explicitly_convertible(inverse(isq::time), kind_of<isq::frequency>));
static_assert(!utility::implicitly_convertible(inverse(isq::time), kind_of<isq::frequency>));

The first line against the last two shows the asymmetry. 1 / (2 * s) is a kind and becomes a frequency implicitly. 1 / isq::time(2 * s) is a typed inverse(time), and frequency is defined over period duration, so this one needs isq::frequency(...).

A kind that meets a typed quantity in one equation stays a kind factor of the result, spelled kind_of_<K> inside the derived quantity, and the typed factor stays typed. The rules above then apply to each factor on its own: a kind factor on the source side converts as a kind does, a kind factor on the target side pairs as K does, and a kind factor and a typed factor of one tree are different factors, like isq::width and isq::length:

inline constexpr struct rate_of_climb : quantity_spec<isq::speed, isq::height / isq::duration> {} rate_of_climb;

static_assert(utility::implicitly_convertible(kind_of<isq::length> / isq::duration, rate_of_climb));
static_assert(!utility::implicitly_convertible(isq::length / isq::duration, rate_of_climb));
static_assert(utility::implicitly_convertible(isq::height / kind_of<isq::duration>, isq::height / isq::period_duration));
static_assert(!utility::implicitly_convertible(isq::height / isq::duration, isq::height / isq::period_duration));
static_assert(utility::implicitly_convertible(kind_of<isq::length> / isq::length, dimensionless));
static_assert(utility::implicitly_convertible(kind_of<isq::length> * isq::length, isq::height * isq::length));
static_assert(!utility::implicitly_convertible(pow<2>(isq::length), isq::height * isq::length));

The first two lines are 42 * m / isq::duration(2 * s) against isq::length(42 * m) / isq::duration(2 * s): only the first may be a rate of climb because only the first has a kind for its length. The fifth line is the ratio of two lengths, and the last two keep the product apart against the merged pow<2>(isq::length). Only a result whose factors are all kinds is a kind itself, kind_of<isq::length / isq::duration> for 42 * m / (2 * s).

The unit of a product

The unit of q1 * q2 is U1 * U2, and the product converts to it as to every unit whose kind converts implicitly to its specification. A unit's kind is the product of its factors' kinds, and only some named units carry a kind of their own, mainly the base units, the angle units, and the ones the SI brochure keeps for one quantity, Hz, Bq, Gy, Sv and kat. The FAQ has the full list and the reasons. Every other named unit has the kind of its definition: N is kg * m / s², J is N * m and W is J / s, so the kind of N * m * W is a product of base kinds, and it converts to torque * power the way kg * m² / s² converts to an energy. That the joule is not written for a moment of force is a convention ISO 80000-1:2009 states and the library leaves to the author. What the brochure keeps for one quantity stays so alone and inside a product:

static_assert(UnitOf<decltype(J), isq::torque>);
static_assert(!UnitOf<decltype(Hz), isq::activity>);
static_assert(!UnitOf<decltype(Bq / m), isq::frequency / isq::length>);

Two named quantities

Between two named quantities their positions in the hierarchy decide. Equations are consulted only for the declared kinds they carry. The kind boundary described in Derived quantities lowers the rank of every rule below where a declared kind is lost or gained.

1. Different trees: none. Frequency and activity both have the dimension of inverse time and have nothing in common.
2. The source is a descendant of the target: implicit, unless one carries a declared kind the other doesn't, which makes it explicit. A kind the source carries may lie on the way up, the source included and the target excluded, or come from the source's equation: phase coefficient is defined as phase_angle / path_length, so it carries a phase angle that its parent propagation coefficient doesn't. The reverse happens too. A child defined as inverse(duration) under angular velocity lacks the angle its parent's equation carries, so its upcast is explicit as well.
3. The target is a descendant of the source: explicit.
4. Neither: cast if both have the same kind root, otherwise none. Two such quantities that each carry a declared kind the other lacks are none as well.

static_assert(!utility::castable(isq::frequency, isq::activity));                     // 1
static_assert(utility::implicitly_convertible(isq::width, isq::length));              // 2
static_assert(utility::explicitly_convertible(isq::angular_measure, dimensionless));  // 2, kind loss
static_assert(!utility::implicitly_convertible(isq::angular_measure, dimensionless));
static_assert(utility::explicitly_convertible(isq::phase_coefficient, isq::propagation_coefficient));  // 2
static_assert(!utility::implicitly_convertible(isq::phase_coefficient, isq::propagation_coefficient));
static_assert(utility::explicitly_convertible(isq::length, isq::width));              // 3
static_assert(!utility::implicitly_convertible(isq::length, isq::width));
static_assert(utility::castable(isq::height, isq::width));                            // 4
static_assert(!utility::explicitly_convertible(isq::height, isq::width));
static_assert(!utility::castable(isq::angular_measure, isq::solid_angular_measure));  // 4, different kinds

Derived quantities

As soon as one side is a derived quantity, the question becomes whether the source can be rewritten into the target. Both sides are treated as lists of factors with exponents, and the library remembers which side each factor came from. A named quantity is a one-factor list, dimensionless is the empty list.

Three operations are available. Each one removes or rewrites factors, and each has a rank.

Pair a source factor with a target factor from the same tree, numerator with numerator or denominator with denominator. The common exponent is consumed and the pair ranks like the two named quantities would. An upcast ranks implicit, or explicit when a declared kind is left on the way. A downcast ranks explicit, or implicit when the source is a kind, and siblings are cast. The direction is always source to target, so 1 / isq::width to 1 / isq::length is implicit. A pair ranking none ends the conversion with none.

Self-cancel a numerator and a denominator factor of the same dimension on the same side. A source-side self-cancel ranks implicit, because a ratio of two quantities of one dimension is a number: height / width or distance / wavelength. A target-side self-cancel ranks explicit, or implicit when the source is a kind, because it asserts that a number is one particular ratio, which is a downcast. A declared kind lost on the way is handled by the kind boundary below.

Expand a factor into the equation of its defining ancestor. Expanding a source factor ranks implicit, because every quantity satisfies its own equation, inherited or not. Expanding a target factor ranks as the downcast from that ancestor to the factor, which is implicit when they are the same quantity or the source is a kind, and explicit otherwise.

The kind boundary. The carried kinds of the source and the target are compared by parts, between the two sides once their common factors have canceled and independently of the operations. For each declared kind, the numerator excess and the denominator excess are compared separately, an absent kind counting as zero. If either part is larger on the source side, the conversion ranks at best explicit, because the kind would be lost: 1 / isq::angular_measure to dimensionless loses a denominator angle. If either part is larger on the target side, it ranks at best explicit, because the kind would be gained. If a kind is lost and one gained, the conversion is none, as between two unrelated kinds. This includes the same kind lost in one part and gained in the other. A kind as the source (kind_of<K>, which has no position below K yet) is exempt from the second cap. The comparison lowers the final rank, and the result doesn't depend on which operations are taken:

static_assert(!utility::implicitly_convertible(isq::arc_length / isq::radius, isq::angular_measure));
static_assert(utility::explicitly_convertible(isq::arc_length / isq::radius, isq::angular_measure));
static_assert(!utility::implicitly_convertible(isq::angular_measure * isq::radius, isq::arc_length));
static_assert(utility::explicitly_convertible(isq::angular_measure * isq::radius, isq::arc_length));
static_assert(!utility::implicitly_convertible(isq::angular_velocity / isq::frequency, dimensionless));
static_assert(utility::implicitly_convertible(isq::angular_velocity / isq::frequency, isq::angular_measure));
static_assert(utility::implicitly_convertible(kind_of<dimensionless>, isq::angular_measure));

The last two lines: ω / f is 2π rad, an angle, and kind_of<dimensionless>, the quantity specification of a bare 3 * one, may become any quantity of its tree, an angle included.

The rules say which operation applies when more than one could:

• Common factors cancel first. A factor present on both sides, in the same part, is canceled before any operation. The rest is then decided as if it were converted on its own: isq::phase_coefficient * isq::mass to isq::propagation_coefficient * isq::mass is explicit, like the bare pair.
• Exact equation first. If one side is a single named quantity, possibly raised to a power, and the other side is exactly the equation of that quantity's defining ancestor raised to the same power, that factor is expanded before any other operation. isq::instantaneous_power * isq::duration / isq::period to isq::active_power is such a case: both are children of power, and without this rule they would pair as siblings.
• A product of its own tree. A single named quantity whose equation is non-empty and consists only of quantities from its own tree is expanded before pairing. In the ISQ this happens in the dimensionless tree, where every pure number is a sibling of every other, and for isq::intrinsic_carrier_density, defined as sqrt(electron_density * hole_density) within the number density tree. With pixel_x and pixel_y children of pixel_count, and pixel_area a dimensionless quantity defined as pixel_x * pixel_y but not a kind, pixel_area to pixel_count * pixel_y is implicit and the reverse is explicit, instead of both being cast between siblings.
• Pairing. Otherwise, whenever a source factor and a target factor share a tree, they are paired. Two factors of one tree relate by their positions in the hierarchy, not by their equations. A pair of siblings waits while some other factor still has an equation that mentions their tree, so that siblings are paired only once every factor of their tree is present. When a tree offers several ways to pair, the rank of the pairing is the best over all pairings of that tree's factors.
• Different roots stay apart. Two named quantities of the same dimension from different trees, left over after pairing, end the conversion with none, unless the equation of either mentions the other's tree, in which case the equations decide. This keeps roots of one dimension apart inside products: isq::frequency * isq::width to isq::activity * isq::length is none, because neither equation mentions the other's tree. An equation that does mention the other tree still resolves: isq::radiant_intensity * isq::solid_angular_measure to isq::radiant_flux is implicit.
• Self-cancel only against an empty side. A side may self-cancel only once the other side is empty. Otherwise isq::arc_length / isq::radius to isq::angular_measure would cancel the source into a number before the equation is used.

• Reduce the deepest factor. When nothing else applies, the factor with the greatest depth is expanded. Ties go, in this order, to the source numerator, the source denominator, the target numerator, the target denominator. Within that list every factor of that depth is tried and the best rank is kept, so the names of the quantities never decide the result. The factor with the greatest depth is reduced first, so both sides meet at the most specific common equation.

  While siblings wait, the factor of the target with the least depth that has an equation is expanded instead, numerator first. We want the angle of kg * m² * rad / s to reach the angular velocity in angular_velocity * moment_of_inertia * rotation. The problem is that the deeper moment of inertia, defined over angular velocity, would expand first and cancel it, so the angle would pair with the rotation and the conversion would end in none. Expanding the shallowest factor first lets the factor that brings the tree in meet the waiting siblings before anything cancels it.

The process stops when both lists are the same: both empty, or the same product right after an exact equation. Its rank is the lowest rank of the operations taken, or implicit if no operation was needed, lowered further by the kind boundary where it applies. It stops with none when a pair ranks none, or when factors remain and no operation applies.

Worked examples

Each diagram below follows one conversion through the procedure. A box shows what is left to convert, an arrow is one operation with its rank, and the conversion gets the lowest rank of its arrows. When the kind boundary applies, it is the last arrow, and it can only lower the result.

isq::width * isq::height → isq::area: implicit

flowchart TD
    A["width · height → area"]
    B["width · height → length²"]
    C["height → length"]
    D["<b>implicit</b>"]
    A -->|"expand area (target, defines its equation): <b>implicit</b>"| B
    B -->|"pair width with length (upcast): <b>implicit</b>"| C
    C -->|"pair height with length (upcast): <b>implicit</b>"| D

static_assert(utility::implicitly_convertible(isq::width * isq::height, isq::area));

isq::area → isq::width * isq::height: explicit

flowchart TD
    A["area → width · height"]
    B["length² → width · height"]
    C["length → height"]
    D["<b>explicit</b>"]
    A -->|"expand area (source): <b>implicit</b>"| B
    B -->|"pair length with width (downcast): <b>explicit</b>"| C
    C -->|"pair length with height (downcast): <b>explicit</b>"| D

static_assert(!utility::implicitly_convertible(isq::area, isq::width * isq::height));
static_assert(utility::explicitly_convertible(isq::area, isq::width * isq::height));

inverse(isq::time) → isq::frequency: explicit

isq::time is an alias of isq::duration.

flowchart TD
    A["1 / time → frequency"]
    B["1 / time → 1 / period_duration"]
    C["<b>explicit</b>"]
    A -->|"expand frequency (target, defines its equation): <b>implicit</b>"| B
    B -->|"pair time with period_duration (downcast): <b>explicit</b>"| C

static_assert(!utility::implicitly_convertible(inverse(isq::time), isq::frequency));
static_assert(utility::explicitly_convertible(inverse(isq::time), isq::frequency));

isq::frequency → inverse(isq::time): implicit

flowchart TD
    A["frequency → 1 / time"]
    B["1 / period_duration → 1 / time"]
    C["<b>implicit</b>"]
    A -->|"expand frequency (source): <b>implicit</b>"| B
    B -->|"pair period_duration with time (upcast): <b>implicit</b>"| C

static_assert(utility::implicitly_convertible(isq::frequency, inverse(isq::time)));

width / duration → special_speed: explicit

An expansion on the target side isn't always implicit. With the names from the vocabulary section, speed defines length / duration, so every length per duration is a speed and converts to it implicitly. special_speed only inherits that equation, and not every speed is a special speed. The equation alone can't make a special speed implicitly, so expanding special_speed on the target side ranks explicit, like the downcast from speed.

flowchart TD
    A["width / duration → special_speed"]
    B["width / duration → length / duration"]
    C["width → length"]
    D["<b>explicit</b>"]
    A -->|"expand special_speed (target, inherits the equation of speed): <b>explicit</b>"| B
    B -->|"pair duration with duration (same quantity): <b>implicit</b>"| C
    C -->|"pair width with length (upcast): <b>implicit</b>"| D

The same conversion to speed is implicit, because speed defines its equation:

static_assert(!utility::implicitly_convertible(width / duration, special_speed));
static_assert(utility::explicitly_convertible(width / duration, special_speed));
static_assert(utility::implicitly_convertible(width / duration, speed));

inverse(isq::repetency) → isq::height: cast

Repetency is defined as inverse(wavelength), and wavelength and height are different branches of the tree of length:

flowchart TD
    A["1 / repetency → height"]
    B["wavelength → height"]
    C["<b>cast</b>"]
    A -->|"expand repetency (source): <b>implicit</b>"| B
    B -->|"pair wavelength with height (different branches): <b>cast</b>"| C

static_assert(utility::castable(inverse(isq::repetency), isq::height));
static_assert(!utility::explicitly_convertible(inverse(isq::repetency), isq::height));

Flattening both sides to length would have ranked this explicit and treated a wavelength as a height.

isq::angular_velocity * isq::radius / isq::speed → dimensionless: explicit

In ISQ angular_velocity is angular_displacement / duration, and angular_displacement is an alias of rotational_displacement, which restates the equation path_length / radius of angular measure. Every operation is implicit, but the source carries angular measure and the target carries nothing, so the kind boundary makes the conversion explicit:

flowchart TD
    A["angular_velocity · radius / speed → dimensionless"]
    B["rotational_displacement · radius / (duration · speed) → dimensionless"]
    C["path_length / (duration · speed) → dimensionless"]
    D["path_length / length → dimensionless"]
    E["nothing left"]
    F["<b>explicit</b>"]
    A -->|"expand angular_velocity (source): <b>implicit</b>"| B
    B -->|"expand rotational_displacement (source): <b>implicit</b>, the two radius factors cancel, the source leaves the kind angular_measure, noted for the kind boundary"| C
    C -->|"expand speed (source): <b>implicit</b>, the two duration factors cancel"| D
    D -->|"self-cancel path_length against length (source): <b>implicit</b>"| E
    E -->|"kind boundary: the source left the kind angular_measure: <b>explicit</b>"| F

static_assert(!utility::implicitly_convertible(isq::angular_velocity * isq::radius / isq::speed, dimensionless));
static_assert(utility::explicitly_convertible(isq::angular_velocity * isq::radius / isq::speed, dimensionless));

The same divided by isq::angular_measure: implicit

The source carries the angle once in the numerator and once in the denominator, so the carried kinds balance:

flowchart TD
    A["angular_velocity · radius / (angular_measure · speed) → dimensionless"]
    B["rotational_displacement · radius / (duration · angular_measure · speed) → dimensionless"]
    C["radius / (duration · speed) → dimensionless"]
    D["radius / length → dimensionless"]
    E["nothing left"]
    F["<b>implicit</b>"]
    A -->|"expand angular_velocity (source): <b>implicit</b>"| B
    B -->|"self-cancel rotational_displacement against angular_measure (source): <b>implicit</b>, the numerator and the denominator both leave the kind angular_measure, both noted for the kind boundary"| C
    C -->|"expand speed (source): <b>implicit</b>, the two duration factors cancel"| D
    D -->|"self-cancel radius against length (source): <b>implicit</b>"| E
    E -->|"kind boundary: the source carries angular_measure in its numerator and its denominator, which cancel, so no kind is lost: <b>implicit</b>"| F

static_assert(utility::implicitly_convertible(isq::angular_velocity * isq::radius / isq::angular_measure / isq::speed,
                                     dimensionless));

Consequences

• A factor common to both sides doesn't change how they relate: A * X to B * X is decided exactly as A to B, as long as no factor of X cancels against A or B (isq::height / isq::width * isq::width is just isq::height). Pairing comes before the equations, so if isq::energy converts to isq::kinetic_energy only explicitly, so does isq::energy * isq::time / isq::period_duration to isq::kinetic_energy.
• Pairing first makes reciprocal ratios of plain quantities castable: isq::height / isq::width to isq::width / isq::height is cast. A declared kind is different: it behaves much like a dimension, and like a dimension it doesn't convert to its inverse, so isq::angular_measure to inverse(isq::angular_measure) is none.
• A declared kind is a boundary in both directions. isq::angular_measure to isq::arc_length / isq::radius is explicit, and so is the reverse, however the equation is rearranged: quantity<isq::arc_length[m]>(angle * r), isq::angular_measure(s / r). The same holds for declared kinds with a dimension, such as a user's fluid head under height: convert it with isq::height(h) before using it in a formula.

• Under plain isq, ISQ defines torque and angular momentum without an angle but angular velocity with one, so isq::torque * isq::angular_velocity to isq::power and isq::moment_of_inertia * pow<2>(isq::angular_velocity) to isq::kinetic_energy are explicit. The explicit conversion also has to name a unit without the angle, because the call operator and quantity_cast accept only a unit of the target's kind, and rad / s carries the angle:

  quantity power = quantity<isq::power[W]>(torque * omega);
  quantity energy = quantity<isq::kinetic_energy[J]>(inertia * pow<2>(omega));
  

  The FAQ shows the same for angular momentum.

  Tip

  The isq_angle system defines torque with the angle, so there the angle cancels and both conversions stay implicit, provided the quantities use the angular units. See Strong Angular System.

• A kind converts implicitly into sub-kinds of its tree. quantity<isq::angular_measure[rad]> angle = 3 * m / (2 * m); compiles, since 3 * m / (2 * m) is a kind of dimensionless and angular measure sits in that tree.

• A named quantity is implicitly convertible from its own equation, but not from a more generic one. isq::length / isq::time converts implicitly to speed, and inverse(isq::period_duration) to frequency, while inverse(isq::time) converts to frequency only explicitly. Two exceptions: a declared kind is gained implicitly only from a source kind, and a source kind kind_of<K> converts implicitly to anything below K that keeps the kinds K carries, so 1 / (2 * s), a kind of inverse(isq::time), converts to frequency implicitly.