## Saturday, 21 February 2009

### Quaternions

Quaternions, discovered in 1843 by Hamilton, are the extension of complex numbers to four dimensions. As it turns out, they are very useful in computer graphics and can represent any rotation in 3D space in a compact and computationally cheap form. Another advantage of quaternions is that they avoid gimbal lock as they can describe a rotation in one operation as opposed to Euler angles that combine yaw, pitch and roll in separate operations.

A quaternion can be written as \hat{\mathbf{q}}=iq_x+jq_y+kq_z+q_w=\mathbf{q}_v+q_w where i,j,k are all different square roots of -1 such that, \begin{align*}ij & = k, & \qquad ji & = -k, \\jk & = i, & kj & = -i, \\ki & = j, & ik & = -j.\end{align*} The vector \hat{\mathbf{q}}_v is closely related to the axis of rotation and the angle of rotation affects all parts of the quaternion as we shall see below.

A unit quaternion, \hat{\mathbf{q}}, can represent a rotation of 2\phi radians around an axis, \mathbf{u}, by \hat{\mathbf{q}}=(\sin \phi \mathbf{u}, \cos \phi). To rotate a point or vector, \mathbf{p}, by this quaternion is, \hat{\mathbf{q}}\mathbf{p}\hat{\mathbf{q}}^{-1}. It can be shown that the inverse, \hat{\mathbf{q}}^{-1}, for a unit quaternion, \hat{\mathbf{q}}, is actually the conjugate where this is defined as, \hat{\mathbf{q}}^{*} = (-\mathbf{q}_v, q_w).

The product of two quaternions is determined by the product of the basis elements and the distributive law, \begin{align*}\hat{\mathbf{q}}\hat{\mathbf{r}} & = (iq_x+jq_y+kq_z+q_w)(ir_x+jr_y+kr_z+r_w)\\& = i(q_y r_z-q_z r_y+r_w q_x+q_w r_x)\\& \quad + j(q_z r_x-q_x r_z+r_w q_y+q_w r_y)\\& \quad + k(q_x r_y-q_y r_x+r_w q_z+q_w r_z)\\& \quad + q_w r_w-q_x r_x-q_y r_y-q_z r_z\\& = (\mathbf{q}_v \times \mathbf{r}_v + r_w \mathbf{q}_v + q_w \mathbf{r}_v,q_w r_w - \mathbf{q}_v \cdot \mathbf{r}_v).\end{align*} The transformation expressed by a unit quaternion can also, and more usefully in computer graphics, be represented by the matrix, M =\begin{pmatrix} 1-2(q_y^2+q_z^2) & 2(q_xq_y-q_wq_z) & 2(q_xq_z+q_wq_y) & 0 \\ 2(q_xq_y+q_wq_z) & 1-2(q_x^2+q_z^2) & 2(q_yq_z-q_wq_x) & 0 \\ 2(q_xq_z-q_wq_y) & 2(q_yq_z+q_wq_x) & 1-2(q_x^2+q_y^2) & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}

## Monday, 9 February 2009

### Text rendering

I was looking for a good library for text rendering. Something that would support truetype faces, antialiasing and kerning and be relatively easy to use. I noticed freetype in the credits to Braid (which I really need to play when the PC version appears - the demo is awesome). So after a bit of research, I found it had everything I was looking for and decided to implement it in my renderer.

Most of the time I spent in implementing this was actually writing the parts of the renderer I needed. I used the approach of using freetype to render a string to a texture and then rendering a quad with this texture. However, I feel uneasy about this as it makes the assumption that the strings I'm rendering don't change much (how fast is freetype at rendering?) and that it will probably take up more texture memory than storing individual glyphs as textures. This is something I'd like to investigate more but at the moment don't have the time. Anyway here's something to look at:

## Friday, 6 February 2009

### Logging

It's often useful to use logging messages or error messages in case something abnormal happens. Or just for fun. A useful trick is to send them straight to the visual studio output window. Using a small amount of code we can even override std::cout and std::cerr which is pretty cool. I'm just going to dive straight into some code:

class logbuf : public std::streambuf
{
public:
virtual std::streamsize xsputn( const char *s, std::streamsize n )
{
// Output to visual studio output window
OutputDebugStringW(widen(s).c_str());
return n;
}
virtual int overflow(int c = EOF)
{
if(c != EOF) OutputDebugString(widen(std::string(1, c)).c_str());
return 1;
}
private:
int indent_;
char last_;
};

int main(int argc, const char* argv[])
{
// Create the new stream to redirect cout and cerr output to vs.
logbuf merr;
std::ostream out(&merr);

// Smart class that will swap streambufs and replace them
// when object goes out of scope.
class StreamBuf_Swapper
{
public:
StreamBuf_Swapper( std::ostream& orig, std::ostream& replacement )
: buf_(orig.rdbuf()), str_(orig)
{ orig.rdbuf( replacement.rdbuf() ); }
~StreamBuf_Swapper() { str_.rdbuf( buf_ ); }
private:
std::streambuf* buf_;
std::ostream& str_;
} cerrswap(std::cerr, out), coutswap(std::cout, out);

try
{
// ... Lots of code
}
catch( const std::exception& e )
{
std::cerr << e.what();
}

return 0;
}


This code takes the streambuf associated with cout and cerr and simply swaps it with a streambuf derived class (vsstream) that prints anything to the visual studio output window. Simple and useful. One other thing to note is that if the output window contains a line formatted:

<filename>(<line number>) : <message>

then it will automatically display that file and line in the editor when double clicked. A few more things you could do with it would be to log errors to a file as well or maybe have something differentiate between normal log messages and error messages (in fact, I'll probably have to do this at some point).