Matrices/Matrix (Maths for Game Developers)

Edit on Github | Updated: 17th October 2020

Introduction to matrices

A Matrix is just a table of numbers, thats all it is, really! By puttting numbers in a table like this we can do some very cool stuff which really shines when doing 3D game programming or even AI.

Things you can do or represent with Matrices:

  • Add/subtract/multiply 2 together
  • Transpose/Scale/Rotate Matrices
  • LookAt Matrix
  • Perspective Matrix
  • Identity Matrix
  • Frustum

A Vector is basically a Matrix with just 1 row, you can find out more about Vectors in our post on them.

Perspective Matrix


Nintendo DS

The Nintendo DS Operating System has a basic Matrix library defined by the header file IrisMTX.h. This file was leaked as part of the September 2020 “Platinum leak” as it is part of the Nintendo DS Boot ROM.

Here are the types it provides to the developer:

typedef s32 MtxRow3_t[3];
typedef struct {
    MtxRow3_t r;
} MtxRow3;

typedef MtxRow3   MtxRow;
typedef MtxRow3   Mtx33Row;

typedef vl MtxRow    vMtxRow;
typedef vl MtxRow3   vMtxRow3;
typedef vl Mtx33Row  vMtx33Row;

typedef s32 MtxRow4_t[4];
typedef struct {
    MtxRow4_t r;
} MtxRow4;

typedef MtxRow4   Mtx44Row;

typedef vl MtxRow4   vMtxRow4;
typedef vl Mtx44Row  vMtx44Row;

typedef s32 Mtx_t[4][3];
typedef union {
    Mtx_t m;
} Mtx;
typedef  vl Mtx   vMtx;

typedef s32 Mtx33_t[3][3];
typedef union {
    Mtx33_t m;
} Mtx33;
typedef  vl Mtx33   vMtx33;

// used for projection matrix
typedef s32 Mtx44_t[4][4];
typedef union {
    Mtx44_t m;
} Mtx44;
typedef  vl Mtx44   vMtx44;

Here are a all of the functions it provides:


void MTX_Identity(Mtx *dstp);
void MTX33_Identity(Mtx33 *dstp);
void MTX44_Identity(Mtx44 *dstp);

void MTX_Copy(      const Mtx *srcp, Mtx   *dstp);
void MTX_Copy2Mtx33(const Mtx *srcp, Mtx33 *dstp);
void MTX_Copy2Mtx44(const Mtx *srcp, Mtx44 *dstp);

void MTX33_Copy(      const Mtx33 *srcp, Mtx33 *dstp);
void MTX33_Copy2Mtx(  const Mtx33 *srcp, Mtx   *dstp);
void MTX33_Copy2Mtx44(const Mtx33 *srcp, Mtx44 *dstp);

void MTX44_Copy(      const Mtx44 *srcp, Mtx44 *dstp);
void MTX44_Copy2Mtx(  const Mtx44 *srcp, Mtx   *dstp);
void MTX44_Copy2Mtx33(const Mtx44 *srcp, Mtx33 *dstp);

void MTX_Concat(Mtx *src0p, Mtx *src1p, Mtx *dstp);
void MTX33_Concat(Mtx33 *src0p, Mtx33 *src1p, Mtx33 *dstp);
void MTX44_Concat(Mtx44 *src0p, Mtx44 *src1p, Mtx44 *dstp);

void MTX_Transpose( Mtx *srcp,  Mtx *dstp);
void MTX33_Transpose( Mtx33 *srcp,  Mtx33 *dstp);
void MTX44_Transpose( Mtx44 *srcp,  Mtx44 *dstp);

void MTX_Translate(Mtx *dstp, s32 x, s32 y, s32 z);
void MTX44_Translate(Mtx44 *dstp, s32 x, s32 y, s32 z);

void MTX_TranslateBefore(Mtx *srcp, Mtx *dstp, s32 x, s32 y, s32 z);
void MTX44_TranslateBefore(Mtx44 *srcp, Mtx44 *dstp, s32 x, s32 y, s32 z);

void MTX_TranslateAfter(Mtx *srcp, Mtx *dstp, s32 x, s32 y, s32 z);
void MTX44_TranslateAfter(Mtx44 *srcp, Mtx44 *dstp, s32 x, s32 y, s32 z);

void MTX_Scale(Mtx *dstp, s32 xS, s32 yS, s32 zS);
void MTX33_Scale(Mtx33 *dstp, s32 xS, s32 yS, s32 zS);
void MTX44_Scale(Mtx44 *dstp, s32 xS, s32 yS, s32 zS);

void MTX_ScaleBefore(Mtx *srcp, Mtx *dstp, s32 xS, s32 yS, s32 zS);
void MTX33_ScaleBefore(Mtx33 *srcp, Mtx33 *dstp, s32 xS, s32 yS, s32 zS);
void MTX44_ScaleBefore(Mtx44 *srcp, Mtx44 *dstp, s32 xS, s32 yS, s32 zS);

void MTX_ScaleAfter(Mtx *srcp, Mtx *dstp, s32 xS, s32 yS, s32 zS);
void MTX33_ScaleAfter(Mtx33 *srcp, Mtx33 *dstp, s32 xS, s32 yS, s32 zS);
void MTX44_ScaleAfter(Mtx44 *srcp, Mtx44 *dstp, s32 xS, s32 yS, s32 zS);

#define MTX_RotateX(dstp, theta)        MTXxx_RotatePriv(_,   X, SIN_NDIV_DEFAULT, dstp, theta)
#define MTX_RotateY(dstp, theta)        MTXxx_RotatePriv(_,   Y, SIN_NDIV_DEFAULT, dstp, theta)
#define MTX_RotateZ(dstp, theta)        MTXxx_RotatePriv(_,   Z, SIN_NDIV_DEFAULT, dstp, theta)

#define MTX33_RotateX(dstp, theta)        MTXxx_RotatePriv(33_, X, SIN_NDIV_DEFAULT, dstp, theta)
#define MTX33_RotateY(dstp, theta)        MTXxx_RotatePriv(33_, Y, SIN_NDIV_DEFAULT, dstp, theta)
#define MTX33_RotateZ(dstp, theta)        MTXxx_RotatePriv(33_, Z, SIN_NDIV_DEFAULT, dstp, theta)

#define MTX44_RotateX(dstp, theta)        MTXxx_RotatePriv(44_, X, SIN_NDIV_DEFAULT, dstp, theta)
#define MTX44_RotateY(dstp, theta)        MTXxx_RotatePriv(44_, Y, SIN_NDIV_DEFAULT, dstp, theta)
#define MTX44_RotateZ(dstp, theta)        MTXxx_RotatePriv(44_, Z, SIN_NDIV_DEFAULT, dstp, theta)

void MTX_RotateX256(Mtx *dstp, u32 theta);
void MTX_RotateY256(Mtx *dstp, u32 theta);
void MTX_RotateZ256(Mtx *dstp, u32 theta);
void MTX_RotateX1024(Mtx *dstp, u32 theta);
void MTX_RotateY1024(Mtx *dstp, u32 theta);
void MTX_RotateZ1024(Mtx *dstp, u32 theta);
void MTX_RotateX4096(Mtx *dstp, u32 theta);
void MTX_RotateY4096(Mtx *dstp, u32 theta);
void MTX_RotateZ4096(Mtx *dstp, u32 theta);

void MTX33_RotateX256(Mtx33 *dstp, u32 theta);
void MTX33_RotateY256(Mtx33 *dstp, u32 theta);
void MTX33_RotateZ256(Mtx33 *dstp, u32 theta);
void MTX33_RotateX1024(Mtx33 *dstp, u32 theta);
void MTX33_RotateY1024(Mtx33 *dstp, u32 theta);
void MTX33_RotateZ1024(Mtx33 *dstp, u32 theta);
void MTX33_RotateX4096(Mtx33 *dstp, u32 theta);
void MTX33_RotateY4096(Mtx33 *dstp, u32 theta);
void MTX33_RotateZ4096(Mtx33 *dstp, u32 theta);

void MTX44_RotateX256(Mtx44 *dstp, u32 theta);
void MTX44_RotateY256(Mtx44 *dstp, u32 theta);
void MTX44_RotateZ256(Mtx44 *dstp, u32 theta);
void MTX44_RotateX1024(Mtx44 *dstp, u32 theta);
void MTX44_RotateY1024(Mtx44 *dstp, u32 theta);
void MTX44_RotateZ1024(Mtx44 *dstp, u32 theta);
void MTX44_RotateX4096(Mtx44 *dstp, u32 theta);
void MTX44_RotateY4096(Mtx44 *dstp, u32 theta);
void MTX44_RotateZ4096(Mtx44 *dstp, u32 theta);

#define MTXxx_RotatePriv(xx_, axis, ndiv, dstp, theta)  MTXxx_RotateNDiv(xx_, axis, ndiv, dstp, theta)
#define MTXxx_RotateNDiv(xx_, axis, ndiv, dstp, theta)  MTX##xx_##Rotate##axis##ndiv(     dstp, theta)

void MTX_RotateXSinCos(Mtx *dstp, s32 sinA, s32 cosA);
void MTX_RotateYSinCos(Mtx *dstp, s32 sinA, s32 cosA);
void MTX_RotateZSinCos(Mtx *dstp, s32 sinA, s32 cosA);

void MTX33_RotateXSinCos(Mtx33 *dstp, s32 sinA, s32 cosA);
void MTX33_RotateYSinCos(Mtx33 *dstp, s32 sinA, s32 cosA);
void MTX33_RotateZSinCos(Mtx33 *dstp, s32 sinA, s32 cosA);

void MTX44_RotateXSinCos(Mtx44 *dstp, s32 sinA, s32 cosA);
void MTX44_RotateYSinCos(Mtx44 *dstp, s32 sinA, s32 cosA);
void MTX44_RotateZSinCos(Mtx44 *dstp, s32 sinA, s32 cosA);

#define MTX_RotateAxis(    dstp, axisp, heta)   MTXxx_RotateAxisPriv(    _,   SIN_NDIV_DEFAULT, dstp, axisp, theta)
#define MTX_RotateAxisFast(dstp, axisp, theta)    MTXxx_RotateAxisPrivFast(_,   SIN_NDIV_DEFAULT, dstp, axisp, theta)

#define MTX33_RotateAxis(    dstp, axisp, heta)   MTXxx_RotateAxisPriv(    33_, SIN_NDIV_DEFAULT, dstp, axisp, theta)
#define MTX33_RotateAxisFast(dstp, axisp, theta)  MTXxx_RotateAxisPrivFast(33_, SIN_NDIV_DEFAULT, dstp, axisp, theta)

#define MTX44_RotateAxis(    dstp, axisp, heta)   MTXxx_RotateAxisPriv(    44_, SIN_NDIV_DEFAULT, dstp, axisp, theta)
#define MTX44_RotateAxisFast(dstp, axisp, theta)  MTXxx_RotateAxisPrivFast(44_, SIN_NDIV_DEFAULT, dstp, axisp, theta)

void MTX_RotateAxis256(Mtx *dstp, const Vec *axisp, u32 theta);
void MTX_RotateAxis256Fast(Mtx *dstp, const Vec *axisp, u32 theta);
void MTX_RotateAxis1024(Mtx *dstp, const Vec *axisp, u32 theta);
void MTX_RotateAxis1024Fast(Mtx *dstp, const Vec *axisp, u32 theta);
void MTX_RotateAxis4096(Mtx *dstp, const Vec *axisp, u32 theta);
void MTX_RotateAxis4096Fast(Mtx *dstp, const Vec *axisp, u32 theta);

void MTX33_RotateAxis256(Mtx33 *dstp, const Vec *axisp, u32 theta);
void MTX33_RotateAxis256Fast(Mtx33 *dstp, const Vec *axisp, u32 theta);
void MTX33_RotateAxis1024(Mtx33 *dstp, const Vec *axisp, u32 theta);
void MTX33_RotateAxis1024Fast(Mtx33 *dstp, const Vec *axisp, u32 theta);
void MTX33_RotateAxis4096(Mtx33 *dstp, const Vec *axisp, u32 theta);
void MTX33_RotateAxis4096Fast(Mtx33 *dstp, const Vec *axisp, u32 theta);

void MTX44_RotateAxis256(Mtx44 *dstp, const Vec *axisp, u32 theta);
void MTX44_RotateAxis256Fast(Mtx44 *dstp, const Vec *axisp, u32 theta);
void MTX44_RotateAxis1024(Mtx44 *dstp, const Vec *axisp, u32 theta);
void MTX44_RotateAxis1024Fast(Mtx44 *dstp, const Vec *axisp, u32 theta);
void MTX44_RotateAxis4096(Mtx44 *dstp, const Vec *axisp, u32 theta);
void MTX44_RotateAxis4096Fast(Mtx44 *dstp, const Vec *axisp, u32 theta);

#define MTXxx_RotateAxisPriv(    xx_, ndiv, dstp, axisp, theta) MTXxx_RotateAxisNDiv(    xx_, ndiv, dstp, axisp, theta)
#define MTXxx_RotateAxisPrivFast(xx_, ndiv, dstp, axisp, theta) MTXxx_RotateAxisNDivFast(xx_, ndiv, dstp, axisp, theta)
#define MTXxx_RotateAxisNDiv(    xx_, ndiv, dstp, axisp, theta) MTX##xx_##RotateAxis##ndiv(         dstp, axisp, theta)
#define MTXxx_RotateAxisNDivFast(xx_, ndiv, dstp, axisp, theta) MTX##xx_##RotateAxis##ndiv##Fast(   dstp, axisp, theta)


void MTX_RotateAxisSinCos(Mtx *dstp, const Vec *axisp, s32 sinA, s32 cosA);
void MTX_RotateAxisSinCosFast(Mtx *dstp, const Vec *axisp, s32 sinA, s32 cosA);

void MTX33_RotateAxisSinCos(Mtx33 *dstp, const Vec *axisp, s32 sinA, s32 cosA);
void MTX33_RotateAxisSinCosFast(Mtx33 *dstp, const Vec *axisp, s32 sinA, s32 cosA);

void MTX44_RotateAxisSinCos(Mtx44 *dstp, const Vec *axisp, s32 sinA, s32 cosA);
void MTX44_RotateAxisSinCosFast(Mtx44 *dstp, const Vec *axisp, s32 sinA, s32 cosA);

void MTX_AlignZ(Mtx *dstp, const Vec *vLookInvp);
void MTX33_AlignZ(Mtx33 *dstp, const Vec *vLookInvp);
void MTX44_AlignZ(Mtx44 *dstp, const Vec *vLookInvp);

void MTX_QuatMtx(Mtx *dstp, const Quat *quatp);
void MTX_QuatMtxFast(Mtx *dstp, const Quat *quatp);

void MTX33_QuatMtx(Mtx33 *dstp, const Quat *quatp);
void MTX33_QuatMtxFast(Mtx33 *dstp, const Quat *quatp);

void MTX44_QuatMtx(Mtx44 *dstp, const Quat *quatp);
void MTX44_QuatMtxFast(Mtx44 *dstp, const Quat *quatp);

void MTX_LookAt(    Mtx *dstp, const Pos *eye, const Pos *at, const Vec *vUp, Vec *vDst);
void MTX_LookAtFast(Mtx *dstp, const Pos *eye, const Pos *at, const Vec *vUp, Vec *vDst);

#define MTX44_Perspective(dstp, fovy, aspect, near, far, scaleW)    MTX44_PerspectivePriv(SIN_NDIV_DEFAULT, dstp, fovy, aspect, near, far, scaleW)

void MTX44_Perspective256(Mtx44 *dstp, u32 fovy, s32 aspect, s32 near, s32 far, s32 scaleW);
void MTX44_Perspective1024(Mtx44 *dstp, u32 fovy, s32 aspect, s32 near, s32 far, s32 scaleW);
void MTX44_Perspective4096(Mtx44 *dstp, u32 fovy, s32 aspect, s32 near, s32 far, s32 scaleW);

#define MTX44_PerspectivePriv(ndiv, dstp, fovy, aspect, near, far, scaleW)  MTX44_PerspectiveNDiv(ndiv, dstp, fovy, aspect, near, far, scaleW)
#define MTX44_PerspectiveNDiv(ndiv, dstp, fovy, aspect, near, far, scaleW)  MTX44_Perspective##ndiv(    dstp, fovy, aspect, near, far, scaleW)

void MTX44_PerspectiveSinCos(Mtx44 *dstp, s32 sinA, s32 aspect, s32 near, s32 far, s32 scaleW, s32 cosA);

void MTX44_Frustum(Mtx44 *dstp, s32 t, s32 b, s32 l, s32 r, s32 n, s32 f, s32 scaleW);

void MTX44_Ortho(  Mtx44 *dstp, s32 t, s32 b, s32 l, s32 r, s32 n, s32 f, s32 scaleW);

void MTX_MultVec(  const Mtx   *mult, Vec *srcp, Vec *dstp);
void MTX33_MultVec(const Mtx33 *mult, Vec *srcp, Vec *dstp);
void MTX44_MultVec(const Mtx44 *mult, Vec *srcp, Vec *dstp);

void MTX_MultVecArray(  const Mtx   *mult, Vec *srcBasep, Vec *dstBasep, u32 count);
void MTX33_MultVecArray(const Mtx33 *mult, Vec *srcBasep, Vec *dstBasep, u32 count);
void MTX44_MultVecArray(const Mtx44 *mult, Vec *srcBasep, Vec *dstBasep, u32 count);

void MTX_MultVecSR(  const Mtx   *mult, Vec *srcp, Vec *dstp);
void MTX44_MultVecSR(const Mtx44 *mult, Vec *srcp, Vec *dstp);

void MTX_MultVecArraySR(  const Mtx   *mult, Vec *srcBasep, Vec *dstBasep, u32 count);
void MTX44_MultVecArraySR(const Mtx44 *mult, Vec *srcBasep, Vec *dstBasep, u32 count);