How do I initialize the t-variables in “A Fast Voxel Traversal Algorithm for Ray Tracing”?

Question

I am trying to implement the algorithm explained on this paper, used to traverse grid cells in order following a straight line, which is useful for ray tracing:

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Answer 1

Initialization for X-coordinate variables (the same for Y)

  DX = X2 - X1
  tDeltaX = GridCellWidth / DX
  tMaxX = tDeltaX * (1.0 - Frac(X1 / GridCellWidth)) 
  //Frac if fractional part of float, for example, Frac(1.3) = 0.3, Frac(-1.7)=0.3

Example:

  GridCellWidth, Height = 20
  X1 = 5, X2 = 105 
  Y1 = 5, Y2 = 55 
  DX = 100, DY  = 50
  tDeltaX = 0.2, tDeltaY = 0.4 
  tMaxX = 0.2 * (1.0 - 0.25) = 0.15  //ray will meet first vertical line at this param
  tMaxY = 0.4 * (1.0 - 0.25) = 0.3   //ray will meet first horizontal line at this param

We can see that first cell border will be met at parameter t = 0.15

Answer 2

The one that worked for me in 3D for both positive and negative directions (in CUDA C):

#define SIGN(x) (x > 0 ? 1 : (x < 0 ? -1 : 0))
#define FRAC0(x) (x - floorf(x))
#define FRAC1(x) (1 - x + floorf(x))

float tMaxX, tMaxY, tMaxZ, tDeltaX, tDeltaY, tDeltaZ;
int3 voxel;

float x1, y1, z1; // start point   
float x2, y2, z2; // end point   

dx = SIGN(x2 - x1);
if (dx != 0) tDeltaX = fmin(dx / (x2 - x1), 10000000.0f); else tDeltaX = 10000000.0f;
if (dx > 0) tMaxX = tDeltaX * FRAC1(x1); else tMaxX = tDeltaX * FRAC0(x1);
voxel.x = (int) x1;

int dy = SIGN(y2 - y1);
if (dy != 0) tDeltaY = fmin(dy / (y2 - y1), 10000000.0f); else tDeltaY = 10000000.0f;
if (dy > 0) tMaxY = tDeltaY * FRAC1(y1); else tMaxY = tDeltaY * FRAC0(y1);
voxel.y = (int) y1;

int dz = SIGN(z2 - z1);
if (dz != 0) tDeltaZ = fmin(dz / (z2 - z1), 10000000.0f); else tDeltaZ = 10000000.0f;
if (dz > 0) tMaxZ = tDeltaZ * FRAC1(z1); else tMaxZ = tDeltaZ * FRAC0(z1);
voxel.z = (int) z1;

while (true) {
    if (tMaxX < tMaxY) {
        if (tMaxX < tMaxZ) {
            voxel.x += dx;
            tMaxX += tDeltaX;
        } else {
            voxel.z += dz;
            tMaxZ += tDeltaZ;
        }
    } else {
        if (tMaxY < tMaxZ) {
            voxel.y += dy;
            tMaxY += tDeltaY;
        } else {
            voxel.z += dz;
            tMaxZ += tDeltaZ;
        }
    }
    if (tMaxX > 1 && tMaxY > 1 && tMaxZ > 1) break;
    // process voxel here
}

Note, grid cell's width/height/depth are all equal to 1 in my case, but you can easily modify it for your needs.