diff --git a/app-proto/full-interface/src/inversive.frag b/app-proto/full-interface/src/inversive.frag
index b7fb100..339be00 100644
--- a/app-proto/full-interface/src/inversive.frag
+++ b/app-proto/full-interface/src/inversive.frag
@@ -63,15 +63,13 @@ vec3 sRGB(vec3 color) {
 
 // --- shading ---
 
-struct taggedFrag {
-    int id;
-    vec4 color;
-    float highlight;
+struct Fragment {
     vec3 pt;
     vec3 normal;
+    vec4 color;
 };
 
-taggedFrag sphere_shading(vecInv v, vec3 pt, vec3 base_color, float highlight, int id) {
+Fragment sphere_shading(vecInv v, vec3 pt, vec3 base_color) {
     // the expression for normal needs to be checked. it's supposed to give the
     // negative gradient of the lorentz product between the impact point vector
     // and the sphere vector with respect to the coordinates of the impact
@@ -81,11 +79,26 @@ taggedFrag sphere_shading(vecInv v, vec3 pt, vec3 base_color, float highlight, i
     
     float incidence = dot(normal, light_dir);
     float illum = mix(0.4, 1.0, max(incidence, 0.0));
-    return taggedFrag(id, vec4(illum * base_color, opacity), highlight, pt, normal);
+    return Fragment(pt, normal, vec4(illum * base_color, opacity));
+}
+
+float intersection_dist(Fragment a, Fragment b) {
+    float intersection_sin = length(cross(a.normal, b.normal));
+    vec3 disp = a.pt - b.pt;
+    return max(
+        abs(dot(a.normal, disp)),
+        abs(dot(b.normal, disp))
+    ) / intersection_sin;
 }
 
 // --- ray-casting ---
 
+struct TaggedDepth {
+    float depth;
+    float dimming;
+    int id;
+};
+
 // if `a/b` is less than this threshold, we approximate `a*u^2 + b*u + c` by
 // the linear function `b*u + c`
 const float DEG_THRESHOLD = 1e-9;
@@ -127,36 +140,29 @@ void main() {
     
     // cast rays through the spheres
     const int LAYER_MAX = 12;
-    taggedFrag frags [LAYER_MAX];
+    TaggedDepth top_hits [LAYER_MAX];
     int layer_cnt = 0;
     for (int id = 0; id < sphere_cnt; ++id) {
         // find out where the ray hits the sphere
         vec2 hit_depths = sphere_cast(sphere_list[id], dir);
         
-        // insertion-sort the fragments we hit into the fragment list
+        // insertion-sort the points we hit into the hit list
         float dimming = 1.;
         for (int side = 0; side < 2; ++side) {
-            float hit_z = -hit_depths[side];
-            if (0. > hit_z) {
+            float depth = hit_depths[side];
+            if (depth > 0.) {
                 for (int layer = layer_cnt; layer >= 0; --layer) {
-                    if (layer < 1 || frags[layer-1].pt.z >= hit_z) {
-                        // we're not as close to the screen as the fragment
-                        // before the empty slot, so insert here
+                    if (layer < 1 || top_hits[layer-1].depth <= depth) {
+                        // we're not as close to the screen as the hit before
+                        // the empty slot, so insert here
                         if (layer < LAYER_MAX) {
-                            frags[layer] = sphere_shading(
-                                sphere_list[id],
-                                hit_depths[side] * dir,
-                                dimming * color_list[id],
-                                highlight_list[id],
-                                id
-                            );
+                            top_hits[layer] = TaggedDepth(depth, dimming, id);
                         }
                         break;
                     } else {
-                        // we're closer to the screen than the fragment before
-                        // the empty slot, so move that fragment into the empty
-                        // slot
-                        frags[layer] = frags[layer-1];
+                        // we're closer to the screen than the hit before the
+                        // empty slot, so move that hit into the empty slot
+                        top_hits[layer] = top_hits[layer-1];
                     }
                 }
                 layer_cnt = min(layer_cnt + 1, LAYER_MAX);
@@ -182,35 +188,22 @@ void main() {
         return;
     }
     
-    // highlight intersections and cusps
-    for (int i = layer_cnt-1; i >= 1; --i) {
-        // intersections
-        taggedFrag frag0 = frags[i];
-        taggedFrag frag1 = frags[i-1];
-        float ixn_sin = length(cross(frag0.normal, frag1.normal));
-        vec3 disp = frag0.pt - frag1.pt;
-        float ixn_dist = max(
-            abs(dot(frag1.normal, disp)),
-            abs(dot(frag0.normal, disp))
-        ) / ixn_sin;
-        float max_highlight = max(frags[i].highlight, frags[i-1].highlight);
-        float ixn_highlight = 0.5 * max_highlight * (1. - smoothstep(2./3.*ixn_threshold, 1.5*ixn_threshold, ixn_dist));
-        frags[i].color = mix(frags[i].color, vec4(1.), ixn_highlight);
-        frags[i-1].color = mix(frags[i-1].color, vec4(1.), ixn_highlight);
-        
-        // cusps
-        float cusp_cos = abs(dot(dir, frag0.normal));
-        float cusp_threshold = 2.*sqrt(ixn_threshold * sphere_list[frag0.id].lt.s);
-        float highlight = frags[i].highlight;
-        float cusp_highlight = highlight * (1. - smoothstep(2./3.*cusp_threshold, 1.5*cusp_threshold, cusp_cos));
-        frags[i].color = mix(frags[i].color, vec4(1.), cusp_highlight);
-    }
-    
     // composite the sphere fragments
     vec3 color = vec3(0.);
-    for (int i = layer_cnt-1; i >= layer_threshold; --i) {
-        vec4 frag_color = frags[i].color;
-        color = mix(color, frag_color.rgb, frag_color.a);
+    int layer = layer_cnt - 1;
+    TaggedDepth hit;
+    for (; layer >= layer_threshold; --layer) {
+        // shade the current fragment
+        hit = top_hits[layer];
+        Fragment frag = sphere_shading(
+            sphere_list[hit.id],
+            hit.depth * dir,
+            hit.dimming * color_list[hit.id]
+        );
+        float highlight = highlight_list[hit.id];
+        
+        // composite the current fragment
+        color = mix(color, frag.color.rgb, frag.color.a);
     }
     outColor = vec4(sRGB(color), 1.);
 }
\ No newline at end of file