# L-System coordinate conversion (as opposed to drawing): Extending the Hilbert Space Filling Curve

I have been reading for some time about L-Systems, and specifically the Hilbert Space filling curve. I am interested in writing a function to convert upper-triangular matrix coordinates into an 1-dimensional space filling curve coordinates.

The usual Hilbert curve is derived with a first-order curve:

Then a series of production rules are iteratively applied to obtain a 1-dimensional coordinate system that is invertible.

I have successfully implemented the mapping algorithm listed on Wikipedia,

//rotate/flip a quadrant appropriately
void rot(int n, int *x, int *y, int rx, int ry) {
if (ry == 0) {
if (rx == 1) {
*x = n-1 - *x;
*y = n-1 - *y;
}

//Swap x and y
int t  = *x;
*x = *y;
*y = t;
}
}

//convert (x,y) to d
int xy2d (int n, int x, int y) {
int rx, ry, s, d=0;
for (s=n/2; s>0; s/=2) {
rx = (x & s) > 0;
ry = (y & s) > 0;
d += s * s * ((3 * rx) ^ ry);
rot(n, &x, &y, rx, ry);
}
return d;
}

//convert d to (x,y)
void d2xy(int n, int d, int *x, int *y) {
int rx, ry, s, t=d;
*x = *y = 0;
for (s=1; s<n; s*=2) {
rx = 1 & (t/2);
ry = 1 & (t ^ rx);
rot(s, x, y, rx, ry);
*x += s * rx;
*y += s * ry;
t /= 4;
}
}


However, I need to add two more first0order curves to this process: one that is a right angle and the other which is a wedge. These two new curves do not need rotations or reflection, and in combination with the usual Hilbert 1st-order curve (with rotation and reflection), one can fill an upper triangular matrix (see my sketch below):

That is, the upper left of the red right-angle again becomes a right angle, the upper right becomes a Hilbert curve, and the lower right part of the right-angle becomes a wedge, and so on.

I would like to implement the above proposed L-System conversion, but every single tutorial on L-Systems is about STARTING AT ZERO and then DRAWING A LINE that fills the space without crossing itself (i.e. writing a string of steps to trace the whole space, instead of converting a subset of coordinates in the space to there 1-D counterpart).

Can anyone offer any intuition on how I might incorporate these new fundamental curves into the existing code? Are there any resources that explain L-System conversions (as opposed to drawing systems), and how to construct them in code??