@@ -2249,6 +2249,101 @@ inline std::vector<Vec2> sampleCurveWireByU(const CurveWire& wire, int segmentsP
   return samples;
 }
 
+/// Ear-clipping of a simple parametric wire that clips the ear with the shortest 3D diagonal, so a trimmed strip is not fanned across.
+inline std::vector<std::array<int, 3>> triangulateWireByShortestChord(const SurfaceWire& wire,
+                                                                      const std::vector<Vec3>& points)
+{
+  std::vector<int> remainingIndices;
+  remainingIndices.reserve(wire.vertices.size());
+  if (wire.signedArea() >= 0.) {
+    for (size_t vertexIndex = 0; vertexIndex < wire.vertices.size(); ++vertexIndex) {
+      remainingIndices.push_back(static_cast<int>(vertexIndex));
+    }
+  } else {
+    for (size_t reverseIndex = wire.vertices.size(); reverseIndex > 0; --reverseIndex) {
+      remainingIndices.push_back(static_cast<int>(reverseIndex - 1));
+    }
+  }
+
+  // the 3D diagonal of the ear at a position, or infinity when it is not an ear (same tests as triangulateSimpleWire)
+  const double notAnEar = std::numeric_limits<double>::infinity();
+  std::vector<char> blocked(remainingIndices.size(), 0); // convex, but another vertex lies in its triangle
+  auto earChord = [&](size_t indexPosition) {
+    blocked[indexPosition] = 0;
+    const int previousIndex = remainingIndices[(indexPosition + remainingIndices.size() - 1) % remainingIndices.size()];
+    const int currentIndex = remainingIndices[indexPosition];
+    const int nextIndex = remainingIndices[(indexPosition + 1) % remainingIndices.size()];
+    const auto& previousVertex = wire.vertices[previousIndex];
+    const auto& currentVertex = wire.vertices[currentIndex];
+    const auto& nextVertex = wire.vertices[nextIndex];
+    if (cross2D(currentVertex - previousVertex, nextVertex - currentVertex) <= kTolerance) {
+      return notAnEar;
+    }
+    for (int candidateIndex : remainingIndices) {
+      if (candidateIndex == previousIndex || candidateIndex == currentIndex || candidateIndex == nextIndex) {
+        continue;
+      }
+      if (pointInTriangle(wire.vertices[candidateIndex], previousVertex, currentVertex, nextVertex)) {
+        blocked[indexPosition] = 1;
+        return notAnEar;
+      }
+    }
+    return norm(points[nextIndex] - points[previousIndex]);
+  };
+
+  std::vector<double> chords(remainingIndices.size());
+  for (size_t indexPosition = 0; indexPosition < remainingIndices.size(); ++indexPosition) {
+    chords[indexPosition] = earChord(indexPosition);
+  }
+
+  std::vector<std::array<int, 3>> triangles;
+  while (remainingIndices.size() > 3) {
+    // the shortest diagonal wins; ties go to the lowest position
+    size_t bestPosition = remainingIndices.size();
+    double bestChord = notAnEar;
+    for (size_t indexPosition = 0; indexPosition < remainingIndices.size(); ++indexPosition) {
+      if (chords[indexPosition] < bestChord) {
+        bestChord = chords[indexPosition];
+        bestPosition = indexPosition;
+      }
+    }
+    if (bestPosition == remainingIndices.size()) {
+      break;
+    }
+
+    const size_t count = remainingIndices.size();
+    const int previousIndex = remainingIndices[(bestPosition + count - 1) % count];
+    const int clippedIndex = remainingIndices[bestPosition];
+    const int nextIndex = remainingIndices[(bestPosition + 1) % count];
+    triangles.push_back({previousIndex, clippedIndex, nextIndex});
+    remainingIndices.erase(remainingIndices.begin() + bestPosition);
+    chords.erase(chords.begin() + bestPosition);
+    blocked.erase(blocked.begin() + bestPosition);
+
+    // the two neighbours change shape; any other non-ear can only change if the clipped vertex was what blocked it
+    const size_t newCount = remainingIndices.size();
+    const size_t beforePosition = (bestPosition + newCount - 1) % newCount;
+    const size_t afterPosition = bestPosition % newCount;
+    for (size_t indexPosition = 0; indexPosition < newCount; ++indexPosition) {
+      if (indexPosition == beforePosition || indexPosition == afterPosition) {
+        chords[indexPosition] = earChord(indexPosition);
+      } else if (blocked[indexPosition]) {
+        const auto& previousVertex = wire.vertices[remainingIndices[(indexPosition + newCount - 1) % newCount]];
+        const auto& currentVertex = wire.vertices[remainingIndices[indexPosition]];
+        const auto& nextVertex = wire.vertices[remainingIndices[(indexPosition + 1) % newCount]];
+        if (pointInTriangle(wire.vertices[clippedIndex], previousVertex, currentVertex, nextVertex)) {
+          chords[indexPosition] = earChord(indexPosition);
+        }
+      }
+    }
+  }
+
+  if (remainingIndices.size() == 3) {
+    triangles.push_back({remainingIndices[0], remainingIndices[1], remainingIndices[2]});
+  }
+  return triangles;
+}
+
 /// Append the display triangulation of a wire-trimmed quadric patch: the sampled outer loop, ear-clipped; holes are omitted.
 template <typename MapUV>
 void appendCurveTrimMesh(const CurveWire& outerWire, const MapUV& mapUV, std::vector<Vec3>& vertices,
@@ -2260,10 +2355,13 @@ void appendCurveTrimMesh(const CurveWire& outerWire, const MapUV& mapUV, std::ve
     return;
   }
   const int firstVertexIndex = static_cast<int>(vertices.size());
+  std::vector<Vec3> points;
+  points.reserve(sampledWire.vertices.size());
   for (const auto& sample : sampledWire.vertices) {
-    vertices.push_back(mapUV(sample.uCoord, sample.vCoord));
+    points.push_back(mapUV(sample.uCoord, sample.vCoord));
   }
-  for (const auto& triangle : triangulateSimpleWire(sampledWire)) {
+  vertices.insert(vertices.end(), points.begin(), points.end());
+  for (const auto& triangle : triangulateWireByShortestChord(sampledWire, points)) {
     triangles.push_back(
       {firstVertexIndex + triangle[0], firstVertexIndex + triangle[1], firstVertexIndex + triangle[2]});
   }
 
@@ -2186,6 +2186,64 @@ BOOST_AUTO_TEST_CASE(WireTrimmedConeMatchesCone)
   checkClose(solid.Capacity(), reference.Capacity(), 1.e-9);
 }
 
+namespace
+{
+// Check that a patch's display mesh keeps its vertex count and stays within the chord sag of radius on the patch;
+// boundarySag adds the sag of a trim side that the wire sampling leaves as one chord.
+void checkDisplayMeshFollowsPatch(const surf::BoundedSurface& patch, double radius, size_t expectedVertices,
+                                  double boundarySag = 0.)
+{
+  std::vector<surf::Vec3> vertices;
+  std::vector<std::array<int, 3>> triangles;
+  patch.appendDisplayMesh(vertices, triangles);
+  BOOST_CHECK_EQUAL(vertices.size(), expectedVertices);
+  BOOST_CHECK_EQUAL(triangles.size(), expectedVertices - 2);
+  const double maxSag = 1.05 * (radius * (1. - std::cos(surf::kPi / surf::kArcSamples)) + boundarySag);
+  double worstSag = 0.;
+  for (const auto& triangle : triangles) {
+    const surf::Vec3& first = vertices[triangle[0]];
+    const surf::Vec3& second = vertices[triangle[1]];
+    const surf::Vec3& third = vertices[triangle[2]];
+    const double longestSq = std::max({normSq(second - first), normSq(third - second), normSq(first - third)});
+    BOOST_CHECK_GT(norm(cross(second - first, third - first)), 1.e-6 * longestSq);
+    for (const surf::Vec3& probe : {(first + second + third) * (1. / 3.), (first + second) * 0.5,
+                                    (second + third) * 0.5, (third + first) * 0.5}) {
+      worstSag = std::max(worstSag, std::sqrt(patch.distanceSqToPatch(probe)));
+    }
+  }
+  BOOST_CHECK_LE(worstSag, maxSag);
+}
+} // namespace
+
+// The display triangles of a wire-trimmed quadric follow the patch instead of fanning across it (O2-7284).
+BOOST_AUTO_TEST_CASE(WireTrimmedDisplayMeshFollowsPatch)
+{
+  std::string error;
+  constexpr double phiMin = 1.6;
+  constexpr double phiMax = 4.683;
+
+  // half-cylinder of the ALICE 3 tracker shell: 12 chords on each long side plus the two corners
+  surf::CylindricalBoundedSurface halfCylinder;
+  BOOST_REQUIRE(halfCylinder.initialize({0., 0., 0.}, {0., 0., 1.}, {1., 0., 0.}, 10., -20., 20., phiMin,
+                                        phiMax - phiMin, false, paramRectWireCurves(phiMin, phiMax, -20., 20.), {},
+                                        error));
+  checkDisplayMeshFollowsPatch(halfCylinder, 10., 26);
+
+  // quarter-torus: 6 chords on each ring side; each tube side is sampled as a single chord
+  surf::TorusBoundedSurface quarterTorus;
+  BOOST_REQUIRE(quarterTorus.initialize({0., 0., 0.}, {0., 0., 1.}, {1., 0., 0.}, 10., 0.3, 0., surf::kHalfPi, 0.,
+                                        surf::kPi / 3., false,
+                                        paramRectWireCurves(0., surf::kHalfPi, 0., surf::kPi / 3.), {}, error));
+  checkDisplayMeshFollowsPatch(quarterTorus, 10.3, 14, 0.3 * (1. - std::cos(surf::kPi / 6.)));
+
+  // half-cone
+  surf::ConicalBoundedSurface halfCone;
+  BOOST_REQUIRE(halfCone.initialize({0., 0., 0.}, {0., 0., 1.}, {1., 0., 0.}, 8., 12., -10., 10., phiMin,
+                                    phiMax - phiMin, false, paramRectWireCurves(phiMin, phiMax, -10., 10.), {},
+                                    error));
+  checkDisplayMeshFollowsPatch(halfCone, 12., 26);
+}
+
 // Green's theorem for wire-trimmed quadrics.
 //
 // The integrand does not depend on the second parameter for a cylinder, and depends on the first