# 3D Translation using only distance and angles

So I'm starting to foray into 3D game programming, but I've hit something of a snag.

I have an object I want to move around. It should always move either forward or backward relative to its local Z axis, and otherwise change directions by changing its local X and Y rotations.

Now doing this in 2D is no problem. When I only consider the X-Z plane, the following code works fine:

m_Position.x -= (float)sin(m_Rotation.y * Pi / 180) * Speed;
m_Position.z -= (float)cos(m_Rotation.y * Pi / 180) * Speed;


And when only considering the Y-Z plane, the following works fine:

m_Position.y += (float)sin(m_Rotation.x * Pi / 180) * Speed;
m_Position.z -= (float)cos(m_Rotation.x * Pi / 180) * Speed;


But when combining the two, they don't work. Now I didn't expect them to work just straight up stuck together like that, since the Z value is getting conflicting signals. But I'm not sure where to go from here.

I've read a bit about matrix math, and that seems the way to go, but the tutorials I have found are either difficult to follow, or they focus on rotating a point around the origin (or translating if you already have a translation vector, which is the thing I'm looking for). The clearest one I've found is this, but I'm not sure how to adapt the rotate-around-origin matrix math to the move-in-direction-of-angle matrix math.

My extremely rudimentary understanding of matrix math would suggest that translations Ta and Tb along angles A and B, if A is a rotation around the Y axis and B around the X axis, would be represented as:

Xa = x*sinA + y*0 + z*0 + w*0
Ya = x*0 + y*1 + z*0 + w*0
Za = x*0 + y*0 + z*cosA + w*0
Wa = x*0 + y*0 + z*0 + w*1

Xb = x*1 + y*0 + z*0 + w*0
Yb = x*0 + y*sinB + z*0 + w*0
Zb = x*0 + y*0 + z*cosB + w*0
Wb = x*0 + y*0 + z*0 + w*1


But I don't know whether that's right, and what I have to do with them. Multiply them together? Add the resulting vectors? Am I even close?

Can someone point me in the direction of good resources for learning this sort of thing?

EDIT: Just to be clear, I have the object's current (X,Y,Z) and its (Roll, Yaw, Pitch). I want it to move in from the former in the direction defined by the latter. I know I can use OpenGL to do this transformation on a purely visual level by just rotating first and then translating, but I need to have access to the new (X,Y,Z) for things like collision detection, indexing, etc.

EDIT 2: I've created the following code after finding this, but I'm not 100% sure if it's correct, since there are also issues with my visual rotations at the moment. It seems to work, at least superficially. Thoughts?

float X = (float)sin(m_Rotation.y * Pi / 180) * (float)cos(m_Rotation.x * Pi / 180);
float Y = (float)sin(-m_Rotation.x * Pi / 180);
float Z = (float)cos(m_Rotation.y * Pi / 180) * (float)cos(m_Rotation.x * Pi / 180);


Basically you have to:

1. Create a rotation matrix from Yaw/Pitch and Roll
2. Multiply your local direction vector (z-axis) by that matrix and multiply the resulting vector by speed, that is the translation

I added a code example. There are 2 convenience classes that do most of the ugly work: vector and matrix. Scroll directly down to main, to see the implementation of 1) 2) and 3)

#include <cmath>
#include <cassert>
#include <cstdio>

#define NEARLY_EQUAL_EPS_F( a, b, eps )     (fabsf( (a)-(b) ) < (eps))
#define ASSERT_VECTOR( v )                  (assert( NEARLY_EQUAL_EPS_F(0.0f, (v)[3], 1.e-7)))
#define FLOAT_FORMATOR                      "%10.6f"

namespace demo
{
class vector
{
public:
vector() {}
explicit vector( const float _v[4] )                    { v[0]=_v[0]; v[1]=_v[1]; v[2]=_v[2]; v[3]=_v[3]; }
vector( const vector &_v )                              { v[0]=_v[0]; v[1]=_v[1]; v[2]=_v[2]; v[3]=_v[3]; }
vector( float _x, float _y, float _z, float _w )        { v[0]=_x; v[1]=_y; v[2]=_z; v[3]=_w; }

float           &operator[]( int i )                    { return v[i]; }
const float     &operator[]( int i ) const              { return v[i]; }

// useful for glVertex3fv, glNormal3fv
const float*    data_ptr() const                        { return v; }
vector          operator+(const vector &b ) const       { ASSERT_VECTOR( b ); return vector( v[0]+b[0], v[1]+b[1], v[2]+b[2], v[3]+b[3] ); }
// scale vector
vector          operator*( float scale ) const          { return vector( v[0]*scale, v[1]*scale, v[2]*scale, v[3]*scale ); }

private:
float v[4];
};

class matrix
{
public:
matrix() {}

vector&                 operator[]( int i )             { return cols[i]; }
const vector&           operator[]( int i ) const       { return cols[i]; }

const float*            data_ptr() const                { return cols[0].data_ptr(); }
matrix&                 rotateY( float a );
matrix&                 rotateX( float a );
matrix&                 rotateZ( float a );

// return an identity matrix
static matrix           identity();
// create a rotation matrix around y-axis
static matrix           rotate_y( float a )             { return matrix::identity().rotateY( a ); }
// create a rotation matrix around y-axis
static matrix           rotate_x( float a )             { return matrix::identity().rotateX( a ); }
// create a rotation matrix around y-axis
static matrix           rotate_z( float a )             { return matrix::identity().rotateZ( a ); }

private:
vector cols[4];
};

inline
matrix
matrix::identity()
{
matrix m;
m[0] = vector( 1.0f, 0.0f, 0.0f, 0.0f );
m[1] = vector( 0.0f, 1.0f, 0.0f, 0.0f );
m[2] = vector( 0.0f, 0.0f, 1.0f, 0.0f );
m[3] = vector( 0.0f, 0.0f, 0.0f, 1.0f );
return m;
}

inline
matrix&
matrix::rotateY( float a )
{
cols[0][0] = cos(a);    cols[0][1] = 0.0f;  cols[0][2] = -sin(a);   cols[0][3] = 0.0f;
cols[1][0] = 0.0f;      cols[1][1] = 1.0f;  cols[1][2] = 0.0f;      cols[1][3] = 0.0f;
cols[2][0] = sin(a);    cols[2][1] = 0.0f;  cols[2][2] = cos(a);    cols[2][3] = 0.0f;
cols[3][0] =            cols[3][1] =        cols[3][2] = 0.0f;      cols[3][3] = 1.0f;
return *this;
}

inline
matrix&
matrix::rotateX( float a )
{
cols[0][0] = 1.0f;      cols[0][1] = 0.0f;      cols[0][2] = 0.0f;     cols[0][3] = 0.0f;
cols[1][0] = 0.0f;      cols[1][1] = cos(a);    cols[1][2] = sin(a);   cols[1][3] = 0.0f;
cols[2][0] = 0.0f;      cols[2][1] = -sin(a);   cols[2][2] = cos(a);   cols[2][3] = 0.0f;
cols[3][0] =            cols[3][1] =            cols[3][2] = 0.0f;     cols[3][3] = 1.0f;
return *this;
}

inline
matrix&
matrix::rotateZ( float a )
{
cols[0][0] = cos(a);    cols[0][1] = sin(a);    cols[0][2] = 0.0f;      cols[0][3] = 0.0f;
cols[1][0] = -sin(a);   cols[1][1] = cos(a);    cols[1][2] = 0.0f;      cols[1][3] = 0.0f;
cols[2][0] = 0.0f;      cols[2][1] = 0.0f;      cols[2][2] = 1.0f;      cols[2][3] = 0.0f;
cols[3][0] =            cols[3][1] =            cols[3][2] = 0.0f;      cols[3][3] = 1.0f;
return *this;
}
// matrix-matrix multiplication
inline
matrix operator*( const matrix &a, const matrix &b )
{
matrix result;

register unsigned int r;
register unsigned int c;

for( r = 0; r < 4; ++r )
{
for( c = 0; c < 4; ++c )
{
result[c][r] = a[0][r] * b[c][0] +
a[1][r] * b[c][1] +
a[2][r] * b[c][2] +
a[3][r] * b[c][3];
}
}
return result;
}

// matrix-vector multiplication
inline
vector operator*( const matrix &m, const vector &v )
{
vector ret;
ret[0] = v[0] * m[0][0] + v[1] * m[1][0] + v[2] * m[2][0] + v[3] * m[3][0];
ret[1] = v[0] * m[0][1] + v[1] * m[1][1] + v[2] * m[2][1] + v[3] * m[3][1];
ret[2] = v[0] * m[0][2] + v[1] * m[1][2] + v[2] * m[2][2] + v[3] * m[3][2];
ret[3] = v[0] * m[0][3] + v[1] * m[1][3] + v[2] * m[2][3] + v[3] * m[3][3];
return ret;
}

inline void DumpVectorf( FILE *file, const char *msg, const vector &p )  { fprintf( file, "%-15s: "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR"\n", msg, p[0], p[1], p[2], p[3] ); fflush( file ); }

inline void DumpMatrixf( FILE *file, const char *msg, const matrix &m )
{
fprintf( file, "%-15s:\n", msg );
fprintf( file, "  row0: "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR"\n", m[0][0], m[1][0], m[2][0], m[3][0] );
fprintf( file, "  row1: "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR"\n", m[0][1], m[1][1], m[2][1], m[3][1] );
fprintf( file, "  row2: "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR"\n", m[0][2], m[1][2], m[2][2], m[3][2] );
fprintf( file, "  row3: "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR" "FLOAT_FORMATOR"\n", m[0][3], m[1][3], m[2][3], m[3][3] );
fflush( file );
}
}

using namespace demo;

int main()
{
vector m_Rotation(1.0f, 0.5f, 0.0f, 0.0f);
vector m_Position(2.0f, 3.0f, 0.0f, 1.0f);
float speed = 3.0f;

// 1) first calculate the rotation matrix from m_Rotation
// !Mind the rotation order, here we use X, Y, Z, if you use a different one, reshuffle this
// We create 3 matrices, each one for a rotation around 1 of the axes,
// then we multiply them to find the final rotation matrix
matrix rot = matrix::rotate_x(m_Rotation[0]) * matrix::rotate_y(m_Rotation[1]) * matrix::rotate_z(m_Rotation[2]);
DumpMatrixf(stdout, "matrix rot", rot);

DumpVectorf(stdout, "position start", m_Position);

// 2) now calculate the translation vector by multiplying the local-space walking direction with that matrix to find the world-space walking direction
// I assume here your nomal walking direction in object space is {0, 0, -1} change it if you use a different one
vector translation = rot * vector(0.0f, 0.0f, -1.0f, 0.0f);
// multiply it with the "Speed"
translation = translation * speed;
DumpVectorf(stdout, "translation", translation);

// 3) add the final translation to the position
m_Position = m_Position + translation;
DumpVectorf(stdout, "position final", m_Position);

return 0;
}


The output of the example:

matrix rot     :
row0:   0.877583   0.000000   0.479426   0.000000
row1:   0.403423   0.540302  -0.738460   0.000000
row2:  -0.259035   0.841471   0.474160   0.000000
row3:   0.000000   0.000000   0.000000   1.000000
position start :   2.000000   3.000000   0.000000   1.000000
translation    :  -1.438277   2.215381  -1.422480   0.000000
position final :   0.561723   5.215381  -1.422480   1.000000


Basically, it should move in whatever direction the camera is looking.

Assuming you have a view matrix representing the inverted position/orientation of the camera, You already have this direction without needing to calculate it.

Matrix myNewMatrix = view.Invert();

float3 directionToMovePlayer = new float3(-myNewMatrix.m31, -myNewMatrix.m32, -myNewMatrix.m33);

• 31, 32, 33 (3rd row) are for DirectX matrices, OpenGL would use 13, 23, 33 (3rd column) – Steve H Mar 14 '12 at 12:52
• Sorry, I didn't think that through - while the current code does have the object moving wherever the camera is looking, I intend to add other objects that move around independently of the camera according to the same system. I also need to know their XYZ coordinates, in order to calculate how far they are from other objects, collisions, etc. – GarrickW Mar 14 '12 at 13:06
• @Steve H for OpenGL it is the other way around, the translation is m[3][0], m[3][1] and m[3][2], the left number is the column, the right number is the row, that is why it is called column-major form. Look at my answer here, there is also code that shows how to easily verify it. – Maik Semder Mar 14 '12 at 17:26
• Thanks, thats good to know. I don't do alot of OpenGL. I wasn't conveying the translation vector to the OP, I was conveying the Z Basis Vector which is the negate of the camera's viewing direction. Which is the 3rd column. If I follow your lead there it is m[2][0], m[2][1], m[2][2]. My point was that he doesn't have to do what you describe in the 1st & 2nd step on your answer because that resulting vector is already inherent in the camera's view matrix. Since the camera is oriented in the direction he wants to go, that direction vector is one of the view matrix's basis vectors, but inverted. – Steve H Mar 14 '12 at 21:39
• But since the OP seems to be also interested in a more generic tool to move objects around that may not be tied to the camera's direction, your whole answer is certainly best for that. – Steve H Mar 14 '12 at 21:47