diff --git a/crates/three-d/src/solid.rs b/crates/three-d/src/solid.rs
index 27f44e60..f12fc2d7 100644
--- a/crates/three-d/src/solid.rs
+++ b/crates/three-d/src/solid.rs
@@ -701,14 +701,18 @@ impl Edges {
         Self { edges, origin: bounds.origin(), size, n: N, buckets }
     }
 
+    /// The bucket row or column of coordinate `v`, from the grid's origin `o` on that axis.
+    fn cell(&self, v: f64, o: f64) -> isize {
+        (((v - o) / self.size).floor() as isize).clamp(0, self.n as isize - 1)
+    }
+
     /// Is `p` within `tolerance` of an edge? Only the buckets `p` could share an edge with are
     /// looked at.
     fn near(&self, p: Point, tolerance: f64) -> bool {
         self.touch(Rect::new(p.x - tolerance, p.y - tolerance, p.x + tolerance, p.y + tolerance)) && {
             let n = self.n as isize;
-            let index = |v: f64, o: f64| (((v - o) / self.size).floor() as isize).clamp(0, n - 1);
-            let (i0, i1) = (index(p.x - tolerance, self.origin.x), index(p.x + tolerance, self.origin.x));
-            let (j0, j1) = (index(p.y - tolerance, self.origin.y), index(p.y + tolerance, self.origin.y));
+            let (i0, i1) = (self.cell(p.x - tolerance, self.origin.x), self.cell(p.x + tolerance, self.origin.x));
+            let (j0, j1) = (self.cell(p.y - tolerance, self.origin.y), self.cell(p.y + tolerance, self.origin.y));
             (j0..=j1).any(|j| {
                 (i0..=i1).any(|i| {
                     self.buckets[(j * n + i) as usize]
@@ -724,9 +728,8 @@ impl Edges {
     /// buckets `r` covers comes more than once).
     fn each(&self, r: Rect, mut f: impl FnMut(Point, Point)) {
         let n = self.n as isize;
-        let index = |v: f64, o: f64| (((v - o) / self.size).floor() as isize).clamp(0, n - 1);
-        for j in index(r.y0, self.origin.y)..=index(r.y1, self.origin.y) {
-            for i in index(r.x0, self.origin.x)..=index(r.x1, self.origin.x) {
+        for j in self.cell(r.y0, self.origin.y)..=self.cell(r.y1, self.origin.y) {
+            for i in self.cell(r.x0, self.origin.x)..=self.cell(r.x1, self.origin.x) {
                 for (a, b) in self.buckets[(j * n + i) as usize].iter().filter_map(|k| self.edges.get(*k as usize)) {
                     if a.x.min(b.x) <= r.x1 && a.x.max(b.x) >= r.x0 && a.y.min(b.y) <= r.y1 && a.y.max(b.y) >= r.y0 {
                         f(*a, *b);
@@ -739,8 +742,8 @@ impl Edges {
     /// Does any edge pass through `r` (its bounding box, at least, overlapping it)?
     fn touch(&self, r: Rect) -> bool {
         let n = self.n as isize;
-        let index = |v: f64, o: f64| (((v - o) / self.size).floor() as isize).clamp(0, n - 1);
-        let (i0, i1, j0, j1) = (index(r.x0, self.origin.x), index(r.x1, self.origin.x), index(r.y0, self.origin.y), index(r.y1, self.origin.y));
+        let (i0, i1) = (self.cell(r.x0, self.origin.x), self.cell(r.x1, self.origin.x));
+        let (j0, j1) = (self.cell(r.y0, self.origin.y), self.cell(r.y1, self.origin.y));
         (j0..=j1).any(|j| {
             (i0..=i1).any(|i| {
                 self.buckets[(j * n + i) as usize]
@@ -760,7 +763,7 @@ impl Edges {
         }
         let n = self.n as isize;
         let (ci, cj) = ((fx.floor() as isize).clamp(0, n - 1), (fy.floor() as isize).clamp(0, n - 1));
-        // How far `p` is outside the grid, which every bucket's distance bound must add.
+        // How far `p` is outside the grid: the nearest point of the grid to it lies in `p`'s bucket.
         let outside = (fx.clamp(0.0, n as f64) - fx).hypot(fy.clamp(0.0, n as f64) - fy) * self.size;
         let mut best = f64::INFINITY;
         for r in 0..n {
@@ -776,8 +779,9 @@ impl Edges {
                     }
                 }
             }
-            // Buckets r + 1 away are at least r buckets from any point of `p`'s own.
-            if best <= outside + r as f64 * self.size {
+            // Buckets r + 1 away are at least r buckets from any point of `p`'s own, and that
+            // far across the grid from a `p` outside it (at right angles to its way in, at worst).
+            if best <= outside.hypot(r as f64 * self.size) {
                 break;
             }
         }
diff --git a/crates/three-d/src/tests_solid.rs b/crates/three-d/src/tests_solid.rs
index 6dc94565..424303ba 100644
--- a/crates/three-d/src/tests_solid.rs
+++ b/crates/three-d/src/tests_solid.rs
@@ -349,3 +349,16 @@ fn the_edge_index_finds_the_nearest_edge() {
         assert!((fast - slow).abs() < 1e-9, "{q:?}: {fast} against {slow}");
     }
 }
+
+/// From a point outside the grid, an edge further along the grid's side can be nearer than one
+/// straight ahead: the search doesn't stop before it has looked there.
+#[test]
+fn the_edge_index_looks_along_the_side_from_outside() {
+    let ring = |b: BezPath| rings(&shape(&b).0, 64.0, MAX_SOLID_POINTS).unwrap().remove(0);
+    // Straight ahead of the point, a little further up its side, and the far corner (a 64 pt grid).
+    let rings = vec![ring(square(2.5, 0.0, 1.0)), ring(square(0.0, 4.2, 0.8)), ring(square(63.0, 63.0, 1.0))];
+    let edges = Edges::new(&rings);
+    let q = Point::new(-10.0, 0.5);
+    let (fast, slow) = (edges.distance(q), edge_distance(&rings, q));
+    assert!((fast - slow).abs() < 1e-9, "{fast} against {slow}");
+}