How a 3D brush lands on a 2D screen

The exact arithmetic behind this lesson’s controls and worked example.

export function cameraSample(x,z,f,scale=0.2){
 const u=64+f*x/z,j=[f/z,-f*x/(z*z)];
 return {u,variance:scale**2*(j[0]**2+j[1]**2)+0.3,width:Math.sqrt(scale**2*(j[0]**2+j[1]**2)+0.3)};
}

The WGSL helper differentiates the clip-space quotient.

fn projected_axis(center_clip: vec4<f32>, axis: vec3<f32>) -> vec2<f32> {
  // Exact first-order derivative of ndc = clip.xy / clip.w along a world-space
  // covariance axis. This is the homogeneous form of the perspective Jacobian.
  let axis_clip = camera.view_proj * vec4<f32>(axis, 0.0);
  let w2 = max(center_clip.w * center_clip.w, 0.00000001);
  let ndc_derivative = (
    axis_clip.xy * center_clip.w - center_clip.xy * axis_clip.w
  ) / w2;
  return ndc_derivative * camera.viewport.xy * 0.5;

Projected axes produce the three independent entries of a 2 × 2 covariance.

  let d0 = projected_axis(center_clip, axis0);
  let d1 = projected_axis(center_clip, axis1);
  let d2 = projected_axis(center_clip, axis2);

  // Project J W Sigma W^T J^T by summing the projected covariance axes.
  let cov_a = dot(vec3<f32>(d0.x, d1.x, d2.x), vec3<f32>(d0.x, d1.x, d2.x)) + 0.3;
  let cov_b = d0.x * d0.y + d1.x * d1.y + d2.x * d2.y;
  let cov_c = dot(vec3<f32>(d0.y, d1.y, d2.y), vec3<f32>(d0.y, d1.y, d2.y)) + 0.3;