const float theta_spacing = 0.07;
const float phi_spacing = 0.02;

const float R1 = 1;
const float R2 = 2;
const float K2 = 5;
// Calculate K1 based on screen size: the maximum x-distance occurs
// roughly at the edge of the torus, which is at x=R1+R2, z=0. we
// want that to be displaced 3/8ths of the width of the screen, which
// is 3/4th of the way from the center to the side of the screen.
// screen_width*3/8 = K1*(R1+R2)/(K2+0)
// screen_width*K2*3/(8*(R1+R2)) = K1
const float K1 = screen_width*K2*3/(8*(R1+R2));

render_frame(float A, float B) {
// precompute sines and cosines of A and B
float cosA = cos(A), sinA = sin(A);
float cosB = cos(B), sinB = sin(B);

char output[0..screen_width, 0..screen_height] = ' ';
float zbuffer[0..screen_width, 0..screen_height] = 0;

// theta goes around the cross-sectional circle of a torus
for (float theta=0; theta < 2*pi; theta += theta_spacing) {
// precompute sines and cosines of theta
float costheta = cos(theta), sintheta = sin(theta);

// phi goes around the center of revolution of a torus
for(float phi=0; phi < 2*pi; phi += phi_spacing) {
// precompute sines and cosines of phi
float cosphi = cos(phi), sinphi = sin(phi);

// the x,y coordinate of the circle, before revolving (factored
// out of the above equations)
float circlex = R2 + R1*costheta;
float circley = R1*sintheta;

// final 3D (x,y,z) coordinate after rotations, directly from
// our math above
float x = circlex*(cosB*cosphi + sinA*sinB*sinphi)
- circley*cosA*sinB;
float y = circlex*(sinB*cosphi - sinA*cosB*sinphi)
+ circley*cosA*cosB;
float z = K2 + cosA*circlex*sinphi + circley*sinA;
float ooz = 1/z; // "one over z"

// x and y projection. note that y is negated here, because y
// goes up in 3D space but down on 2D displays.
int xp = (int) (screen_width/2 + K1*ooz*x);
int yp = (int) (screen_height/2 - K1*ooz*y);

// calculate luminance. ugly, but correct.
float L = cosphi*costheta*sinB - cosA*costheta*sinphi -
sinA*sintheta + cosB*(cosA*sintheta - costheta*sinA*sinphi);
// L ranges from -sqrt(2) to +sqrt(2). If it's < 0, the surface
// is pointing away from us, so we won't bother trying to plot it.
if (L > 0) {
// test against the z-buffer. larger 1/z means the pixel is
// closer to the viewer than what's already plotted.
if(ooz > zbuffer[xp,yp]) {
zbuffer[xp, yp] = ooz;
int luminance_index = L*8;
// luminance_index is now in the range 0..11 (8*sqrt(2) = 11.3)
// now we lookup the character corresponding to the
// luminance and plot it in our output:
output[xp, yp] = ".,-~:;=!*#$@"[luminance_index];
}
}
}
}

// now, dump output[] to the screen.
// bring cursor to "home" location, in just about any currently-used
// terminal emulation mode
printf("\x1b[H");
for (int j = 0; j < screen_height; j++) {
for (int i = 0; i < screen_width; i++) {
putchar(output[i,j]);
}
putchar('\n');
}

}