as - almost something

as (short for almost-something) is an attempt at creating an easy to use header-only math library for prototype and hobby projects.

Disclaimer

This library is really just a dumb experimental thing meant for my own personal amusement. If you're curious please do have a poke around but I would strongly advise not using it for anything serious.

Deficiencies include but are not limited to:

• The API is not stable (I'm still iterating on it)
• The test coverage is currently woefully inadequate (I'm working on improving this)
  • I now technically have 100% test coverage! 🥳 I'm sure not all corner cases are covered but given reasonable inputs things seem to be behaving as expected.
• I have used this in a few projects now
  • Vulkan-Fruit (Twitter link) - Experiment to learn more about Vulkan
  • ToyMeshPathTracer - Fork of Aras Pranckevičius' (aras_p's) interview task/assignment.
  • as-camera - Simple camera representation
  • as-camera-input - Interesting camera behaviours, built on-top of as-camera.
  • sdl-bgfx-imgui-marching-cubes - Marching Cubes implementation
  • sdl-bgfx-imgui-1d-nonlinear-transforms - Random experiments with interpolation and noise
• There's bound to be bugs!
• The performance is likely not very good either (I'm working on improving this, for example creating template specializations for the common dimensions).
• No SIMD (yet? we'll see... 🤔)
• I've probably made some horrible mistake somewhere which I'll be terribly embarrassed about once brought to my attention.

Using and/or installing the library

as is pretty simple to drop into your project if you want to give it a try. I'll describe several options if you're using CMake, and offer some advice if you're sans-CMake.

CMake

FetchContent (preferred)

The easiest way to use the library is with CMake's FetchContent module.

Add these commands to the top of your CMakeLists.txt file:

include(FetchContent)
FetchContent_Declare(
    as
    GIT_REPOSITORY https://github.com/pr0g/as.git
    GIT_TAG        v2.0.2) # most recent stable release
FetchContent_MakeAvailable(as)

Then just link against as in the target_link_libraries command:

target_link_libraries(
    ${PROJECT_NAME} PUBLIC as)

You can then simply include as files like so:

#include "as/as-math-ops.hpp"
// etc...

Full Include

It is also possible to drop the repo into your project as-is and use add_subdirectory

|-- root
    |-- as #added (<root>)
        |-- CMakeLists.txt
        |-- ...
    |-- main.cpp
    |-- CMakeLists.txt

Full example:

cmake_minimum_required(VERSION 3.15)
project(app VERSION 0.0.1 LANGUAGES CXX)

# required for as
add_subdirectory(as) # note: folder name is 'as'

add_executable(${PROJECT_NAME} main.cpp)

# required for as
target_compile_definitions(
    ${PROJECT_NAME} PRIVATE
    AS_PRECISION_FLOAT AS_COL_MAJOR)

# required for as (as is not imported so we're not using as::as)
target_link_libraries(${PROJECT_NAME} PUBLIC as)

Partial Include

If this all looks like a lot of hassle another option is just to copy-paste the as folder (the one under the include/ directory in the repo) into your project directory (maybe pop it in an external or 3rdParty folder or some such?)

|-- root
    |-- as #added (<root>/include/as)
    |-- main.cpp
    |-- CMakeLists.txt

or

|-- root
    |-- external
        |-- as #added (<root>/include/as)
        |-- another-useful-lib
    |-- main.cpp
    |-- CMakeLists.txt

Your CMakeLists.txt file can then look like this:

cmake_minimum_required(VERSION 3.15)
project(app VERSION 0.0.1 LANGUAGES CXX)

add_executable(${PROJECT_NAME} main.cpp)

# required for as
target_compile_definitions(
    ${PROJECT_NAME} PRIVATE
    AS_PRECISION_FLOAT AS_COL_MAJOR)

# required for as (we need to set this ourselves now)
target_compile_features(
    ${PROJECT_NAME} PRIVATE cxx_std_11)

# needed if adding as to a nested folder called 'external'
target_include_directories(
    ${PROJECT_NAME} PRIVATE external) # optional

Non-CMake

Anything non-CMake is more complicated to cover as this technically entails any/every potential build system under the sun 🤔

My advice would be to use something similar to the Partial Include section above but just set the include directories in Visual Studio or XCode or whatever other build system you might be using (Bazel, waf, ninja, make, Premake etc...)

Origins

The inspiration for this project came from reading an excellent blog post by Nathan Reed (@Reedbeta) about writing a vector math library. You can find the post titled 'On Vector Math Libraries' here (it's well worth a read!).

The post has some interesting advice on how one might write a math library and goes into some detail about the pros and cons of various approaches.

I decided to follow the advice outlined (template heavy, parameterized on dimensions and type). This gives a high degree of flexibility at a cost of some performance and compile time.

I've tried something a bit different with the interface (which is a little unusual) but I've grown quite fond of it over time. I'll try and persuade you the approach I've taken is a good idea not completely stupid 🙃.

Update: It turns out the namespace approach was a terrible idea and wasn't fun or easy to use at all. There's now just a totally flat namespace (as) and everything is either a free function with a type name prefix or a static function on a type (which may be removed in future).

Design

A lot of the design is informed by the article I mention in the Origins section (maybe read that first?) but there are some things I'd like to try and explain.

Names and Namespaces

An aim I had early on for the API was to attempt to prioritize ease of use and to make functions easy to discover.

As all the types I'm creating are pretty simple I decided not to hide any internal state, so all members are public. This was a design choice to make writing expressions such as v.x += value; easier than v.set_x(v.get_x() + value). This may well be a decision I regret and I know there are downsides to this (harder to know when a value changes, you lose a bit of abstraction) but for my use case this is a trade off I'm willing to make (I've slowly learnt everything in programming is a tradeoff and nothing is perfect 😖).

All Most functions are implemented as free functions. I wanted to avoid using static functions because you get into problems with circular dependencies (for example when static functions on one type need to include a type in another file and you'd like the API to be symmetrical).

Update: Currently there's limited use of static functions to reduce boilerplate when writing template functions, they might wind up being removed to further simplify the API. The do not rely on any other types.

For example if we want to be able to create a quaternion from a 3 by 3 matrix and vice versa, the api might look like this:

mat3 m = // ... build some rotation
quat q = m.to_quaternion(); // member
// or
quat q = mat3::to_quaternion(m); // static

...

quat q = // ... build some rotation
mat3 m = q.to_mat3(); // member
// or
mat3 m = quat::to_mat3(q); // static

The big problem with this is we get into a situation where mat3.h can't include quat.h and quat.h can't include mat3.h at the same time (we can get around this if we stick the implementation in a .cpp file and forward declare the type, but then we lose the ability to inline (forgetting LTO for the moment) and in doing so the ability to have a header-only library 😫).

So we're back to free functions. The lovely property about this is we can keep the <type>.h files quite light (we pretty much just have the special members and overloaded operators there) and all other functions can go into a separate file, which includes whatever types it likes. So we now have:

mat3 m1 = // ... build some rotation
quat q1 = quat_from_mat3(m1); // free
quat q2 = // ... build some rotation
mat3 m2 = mat3_from_quat(q2); // free

When I first tried this, a big problem I ran into was how to logically group functions. One advantage to having static functions on a type is it's easy in most modern IDEs to type the name, hit the '.' operator, and see a list of available methods member/static functions. This advantage is really important and I wanted to try and find a way to do something similar.

I realized I could take advantage of C++ namespaces to group functions by type. Irritatingly using the name vec3 or mat4 for the namespace means I lose the ability to use them as type names 😖 however a workaround I decided to borrow from C and C++ was to add a _t postfix to indicate the type itself and use the unadorned name as the namespace (I know any name ending with _t is technically reserved, in C at least, but as I'm keeping everything inside my own namespace I think I should be safe - failing that I could use _s instead for struct but I'm sticking with _t for now).

Update: The approach above actually wound up making the library cumbersome and difficult to use (basically it was hard to remember what namespace something actually lived in) so I've now opted to just use a single flat namespace and use type prefixes on function names.

With this approach I can now do this:

mat3_t m = // ... build some rotation
quat_t q = quat::from_mat3(m); // free (update: this was rubbish)

quat q = quat_from_mat3(m); // free (update: now looks like this)

When you type mat3:: in an IDE you'll get the list of all operations supported for mat3_t. Another idea (which is perhaps more contentious) is to put all common/generic matrix operations in the mat namespace, and specific operations (like rotation_xyz) in the mat3 namespace. This might actually make things slightly harder to find in certain cases but I like the preciseness of the grouping so again this will remain until I've had a chance to start hating it 😉.

Update: I started hating it.

Specializations

The core idea of the library is to create types that are (pretty) generic and write all functions in terms of those generic versions. This is so in theory you could have a vec<int, 10> and all most of the operations you get with a friendly vec3 all still work.

The reality is 99.9% of the time people just want to use vec3 and mat3/mat4. To make these common cases more ergonomic, I provide explicit (full) template specializations for the class templates vec2/3/4 and mat3/4. The reason for doing this is purely for ease of use. I provide accessors for fields such as x, y, z and useful constructor overloads (I'll get on to function template specializations later which are done for performance reasons).

This is most commonly done by using a union in C/C++.

template<typename T> struct vec<T, 3>
{
    union
    {
        T data[3];
        struct { T x; T y; T z; };
        vec<T, 2> xy;
    };

    T& operator[](int i) { return data[i]; }
}

There are a few problems with this approach unfortunately. First, C++ does not support anonymous structs. There's a pretty good chance your compiler will support this but you might get a warning or it might not compile. The next problem with this approach is if you write to the field data[1] and then later read from y, you've technically invoked undefined behaviour in C++ (type punning like this is legal in C but not strictly in C++). I was totally unaware of this until relatively recently and I have Sy Brand (aka @TartanLlama) to thank for pointing this out to me. To play devil's advocate chances are everything will work fine but there's no guarantee - you might be able to avoid it by enabling -fno-strict-aliasing but perhaps you're in an environment where you (or your user) can't.

For the record the following approach can also technically lead to undefined behaviour:

template<typename T>struct vec<T, 3>
{
    T x; T y; T z;
    T& operator[](int i) { return ((&x)[i]); }
};

I spent a while digging into possible alternatives and eventually came across a post on gamedev.net about this very topic. In it there's what I believe to be quite an elegant solution to the problem which is standard compliant! 😄

template<typename T> struct vec<T, 3>
{
    T x, y, z;
    T& operator[](size_t i) { return this->*elem[i]; }
    const T& operator[](size_t i) const { return this->*elem[i]; }

private:
    static T vec::*elem[3];
};

using vec3 = vec<float, 3>;

template<typename T>
T vec3<T>::*vec3<T>::elem[3] = { &vec3<T>::x, &vec3<T>::y, &vec3<T>::z }; // .cpp file

This definitely looks more complicated than the previous snippets but it isn't as bad as it looks at first glance. Essentially what's happening is we introduce 3 new member variables (x, y, z) for this specialization which is where the actual data is stored per instance (no surprises there).

The first unusual bit of code to examine is the declaration of the static member float vec::*elem[3];. The syntax may be unfamiliar but this is declaring an array of member pointers (not just regular pointers) indicated by the vec:: prefix before *elem[3]. Because it's static we need to initialize it in the .cpp file. Here were setup each member pointer to point to the address of the member variables of the class (so elem[0] corresponds to the address/offset of x in vec3).

The last line of interest is return this->*elem[i]; used in the operator[] overloads. Remember that in a normal member function there's always an implicit this pointer:

struct vec
{
    float x, y, z;
    float get_x()
    {
        return x; // actually return this->x;
    }
};

So our expression return this->*elem[i]; is really syntactic sugar for return this->*this->elem[i]; In this case the first this is the instance/object we're calling the [] operator on, and then we dereference the pointer to the data member for that instance. It's easier to think about if we think how we'd do this outside the class.

struct Person
{
    int age_;
};

void Example()
{
    int Person::*age_ptr = &Person::age_; // declare a member pointer to age
    Person person;
    person.*age_ptr = 30; // dereference the member pointer on an instance of Person
    printf("%d\n", person.*age_ptr);
}

We're doing the exact same thing but just using this as the object instead of a named instance like in the Person example.

One drawback to this approach is if you provide a full template specialization you have to put the static definitions in the .cpp file, which sort of throws a spanner in the works with creating a header-only library 😫. The good news is if you only partially specialize the class template you can put the definition of the static variables in the header and the linker won't complain 🙂 This is because the definition of the static data member is itself a template 🥳.

The last thing to mention is the function template specializations. These were made specifically for performance reasons. It's unfortunate but the reality is the hand crafted implementations of the various functions for vec3 are considerably faster than their generic counterparts. This came to my attention when using the as math library for the ToyMeshPathTracer. I dropped in the as library in place of the existing simple float3 type and was disappointed to see my version taking a lot longer to render the scene 😔. I spent a while trying to figure out why things were so much slower (thank you Nathan Reed for your ideas and suggestions) and in the end it all seemed to come down to the code the compiler would generate for the generic versions vs the handcrafted ones. I suppose this finding does slightly negate the point of writing a generic math library in the first place but I still think having a standard interface for all types is useful (you don't fall into the trap of having a slightly lacking Vector2 type which hasn't received as much love) and you can get up and running with the generic versions and write custom specializations later only if you need to for performance reasons.

Row/Column Major

• Storage: Row major
  • Basis axes data is contiguous
  • All hardcoded matrices (e.g. projection matrices) are in row major format

// e.g.

// expanded
[x0, y0, z0, w0, x1, y1, z1, w1, x2, y2, z2, w2, x3, y3, z3, w3] 

// wrapped
[x0, y0, z0, w0, // x axis
 x1, y1, z1, w1, // y axis
 x2, y2, z2, w2, // z axis
 x3, y3, z3, w3] // translation

• Convention: Row or column
  • Library uses traversal order of rows/columns to switch

I wanted it to be possible to use the library with either row or column vectors and not get them accidentally mixed up. Different graphics APIs use different conventions (OpenGL uses column-major and DirectX uses row-major). Now the layout in memory (as far as C++ is concerned) is always row-major (I recommend reading this article by Sean Middleditch (@stmiddleditch) for a great explanation of this subject).

With row-major, when a vector is transformed by a matrix, the notation is to place the vector to the left of the matrix (so you multiply reading left to right) but with column-major the notation is to place the vector being transformed to the right of the matrix (in this case you multiply reading right to left). Now under the hood the multiplication is actually exactly the same, but I make sure to only enable the right overload for operator * when multiplying a vector and a matrix to make sure you can't get them the wrong way round (I do this by forcing you to pick the convention you want to use before you start using the library with a #define - you must define either AS_COL_MAJOR or AS_ROW_MAJOR).

In the case of multiplying two matrices the multiplication order is different to account for the left-to-right or right-to-left convention when combining matrices.

UPDATE: I realized attempting to support the row/column toggle with non-square matrices was actually impossible. I could either have square matrices with the row/column switch (TLDR: I picked this), non-square matrices with row-major layout only, or have non-square row-major matrices and square only column-major (I didn't really like this mismatch).

The reason I wasn't able to support non-square column major matrices boils down to how you multiply matrices. Technically matrix multiplication is multiplying each row of the left matrix by each column of the right matrix. The resulting matrix will have the same number of rows as the left matrix and same number of columns as the right matrix - e.g. a 2x4 * 4x2 = 2x2 matrix and a 4x2 * 2x4 = 4x4 matrix). For matrix multiplication to be well defined (possible) the number of columns in the left matrix must equal the number of rows in the right matrix (see earlier examples). When trying to do things column-major, certain matrix sizes don't really work, you have to always start with the row in the left matrix. When AS_COL_MAJOR is defined I pull a bit of a trick under the hood where the data layout of the matrix doesn't change but the order in which elements are visited is (we multiply columns in the left matrix by rows in the right matrix) but this trick only really works when the matrices are square. It's the same effect as if the matrices were transposed and you did the normal row * col multiply. Remember to get the same answer as when in row-major you also have to swap the order of the matrices as A * B != B * A for matrices.

As square matrices are the most common and I wanted to keep the switch I've removed the ability to select row/column sizes in the matrix types and instead you just set a dimension (d).

Also as a random aside this article by Sebastian Sylvan (@ssylvan) is pure genius and well worth a read to help simplify the naming of your transforms 😎

real

I decided to also provide an abstraction over float and double for the common types (vec3, mat4 etc.) so you can easily switch between precisions by either defining AS_PRECISION_FLOAT or AS_PRECISION_DOUBLE. I've wrapped all the base math functions with r versions (e.g. fabsr replaces either fabsf if floating point precision is used or fabs if double precision is used).

Update: It turned out this was totally unnecessary as if you you std::abs etc... (in the std:: namespace) they are overloaded for float and double so the correct version will be selected at compile time.

Miscellaneous

Physical design

Even though this is a header-only library I still like the separation of interface and implementation when looking at the code. To achieve this all implementation code is in .inl files that are included at the bottom of all the .h files. You should never need to include these explicitly.

Other Approaches

There are a million other math libraries out there that are more complete, more reliable, more efficient and likely more interesting. If this has in any way piqued your interest I suggest checking out these libraries and articles.

Libraries

• OpenGL Mathematics (glm)
• DirectXMath
• Eigen
• Tuesday
• HandmadeMath
• RealTimeMath
• as-c-math

Articles

• How To Write A Maths Library In 2016
• On Vector Math Libraries

Related Links

• GameDev.net - A slick trick in c