Introduce an element trait

For now, this is just a thin wrapper around the old element structure,
which was renamed to `Sphere` in the previous commit. The biggest
organizational change is moving `cast` into the `DisplayItem` trait.

app-proto/src/display.rs

@@ -56,7 +56,7 @@ impl ScenePoints {
     }
 }
 
-struct Scene {
+pub struct Scene {
     spheres: SceneSpheres,
     points: ScenePoints
 }
@@ -70,8 +70,13 @@ impl Scene {
     }
 }
 
-trait DisplayItem {
+pub trait DisplayItem {
     fn show(&self, scene: &mut Scene, selected: bool);
+    
+    // the smallest positive depth, represented as a multiple of `dir`, where
+    // the line generated by `dir` hits the element. returns `None` if the line
+    // misses the element
+    fn cast(&self, dir: Vector3<f64>, assembly_to_world: &DMatrix<f64>) -> Option<f64>;
 }
 
 impl DisplayItem for Sphere {
@@ -82,6 +87,46 @@ impl DisplayItem for Sphere {
         let highlight = if selected { 1.0 } else { HIGHLIGHT };
         scene.spheres.push(representation, color, highlight);
     }
+    
+    // this method should be kept synchronized with `sphere_cast` in
+    // `spheres.frag`, which does essentially the same thing on the GPU side
+    fn cast(&self, dir: Vector3<f64>, assembly_to_world: &DMatrix<f64>) -> Option<f64> {
+        // if `a/b` is less than this threshold, we approximate
+        // `a*u^2 + b*u + c` by the linear function `b*u + c`
+        const DEG_THRESHOLD: f64 = 1e-9;
+        
+        let rep = self.representation.with_untracked(|rep| assembly_to_world * rep);
+        let a = -rep[3] * dir.norm_squared();
+        let b = rep.rows_range(..3).dot(&dir);
+        let c = -rep[4];
+        
+        let adjust = 4.0*a*c/(b*b);
+        if adjust < 1.0 {
+            // as long as `b` is non-zero, the linear approximation of
+            //
+            //   a*u^2 + b*u + c
+            //
+            // at `u = 0` will reach zero at a finite depth `u_lin`. the root of
+            // the quadratic adjacent to `u_lin` is stored in `lin_root`. if
+            // both roots have the same sign, `lin_root` will be the one closer
+            // to `u = 0`
+            let square_rect_ratio = 1.0 + (1.0 - adjust).sqrt();
+            let lin_root = -(2.0*c)/b / square_rect_ratio;
+            if a.abs() > DEG_THRESHOLD * b.abs() {
+                if lin_root > 0.0 {
+                    Some(lin_root)
+                } else {
+                    let other_root = -b/(2.*a) * square_rect_ratio;
+                    (other_root > 0.0).then_some(other_root)
+                }
+            } else {
+                (lin_root > 0.0).then_some(lin_root)
+            }
+        } else {
+            // the line through `dir` misses the sphere completely
+            None
+        }
+    }
 }
 
 // --- WebGL utilities ---
@@ -264,7 +309,7 @@ pub fn Display() -> View {
     create_effect(move || {
         state.assembly.elements.with(|elts| {
             for (_, elt) in elts {
-                elt.representation.track();
+                elt.representation().track();
             }
         });
         state.selection.track();