Convert Axis Angle to 4x4 homogeneous matrix?

[sarah22]

I'm stuck on how did the author managed to convert the axis angle to a 4x4 homogenous matrix.

Here's the code below. The bolded ones are the one I'm struggling at right now.

void glQuaternion::CreateMatrix(GLfloat *pMatrix)
{
// Make sure the matrix has allocated memory to store the rotation data
if(!pMatrix) return;

// First row
pMatrix[ 0] = 1.0f - 2.0f * ( m_y * m_y + m_z * m_z );
pMatrix[ 1] = 2.0f * (m_x * m_y + m_z * m_w);
pMatrix[ 2] = 2.0f * (m_x * m_z - m_y * m_w);
pMatrix[ 3] = 0.0f;

// Second row
pMatrix[ 4] = 2.0f * ( m_x * m_y - m_z * m_w );
pMatrix[ 5] = 1.0f - 2.0f * ( m_x * m_x + m_z * m_z );
pMatrix[ 6] = 2.0f * (m_z * m_y + m_x * m_w );
pMatrix[ 7] = 0.0f;

// Third row
pMatrix[ 8] = 2.0f * ( m_x * m_z + m_y * m_w );
pMatrix[ 9] = 2.0f * ( m_y * m_z - m_x * m_w );
pMatrix[10] = 1.0f - 2.0f * ( m_x * m_x + m_y * m_y );
pMatrix[11] = 0.0f;

// Fourth row
pMatrix[12] = 0;
pMatrix[13] = 0;
pMatrix[14] = 0;
pMatrix[15] = 1.0f;

// Now pMatrix[] is a 4x4 homogeneous matrix that can be applied to an OpenGL Matrix
}

Before he use the code above. He converted the axis angle to quaternion first which is the one below. I somehow managed to understand this one though. (http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/index.htm")

void glQuaternion::CreateFromAxisAngle(GLfloat x, GLfloat y, GLfloat z, GLfloat degrees)
{
// First we want to convert the degrees to radians
// since the angle is assumed to be in radians
GLfloat angle = GLfloat((degrees / 180.0f) * PI);

// Here we calculate the sin( theta / 2) once for optimization
GLfloat result = (GLfloat)sin( angle / 2.0f );

// Calcualte the w value by cos( theta / 2 )
m_w = (GLfloat)cos( angle / 2.0f );

// Calculate the x, y and z of the quaternion
m_x = GLfloat(x * result);
m_y = GLfloat(y * result);
m_z = GLfloat(z * result);
}

So my question is, how did he arrived to that formula (the bold one on the first code) ? If possible you can give me some links on how did he get that?

I'm looking all over the google but didn't get one. Maybe my tags were wrong.

Source for the code above: http://nehe.gamedev.net/data/lessons/lesson.asp?lesson=Quaternion_Camera_Class

Thank you
Sarah

 

[sarah22]

nevermind. I got it.