diff --git a/doc/api/next_api_changes/behavior/24062-tb.rst b/doc/api/next_api_changes/behavior/24062-tb.rst
new file mode 100644
index 000000000000..7e5beaecba53
--- /dev/null
+++ b/doc/api/next_api_changes/behavior/24062-tb.rst
@@ -0,0 +1,8 @@
+``TrapezoidMapTriFinder`` uses different random number generator
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The random number generator used to determine the order of insertion of
+triangle edges in ``TrapezoidMapTriFinder`` has changed. This can result in a
+different triangle index being returned for a point that lies exactly on an
+edge between two triangles. This can also affect triangulation interpolation
+and refinement algorithms that use ``TrapezoidMapTriFinder``.
diff --git a/src/tri/_tri.cpp b/src/tri/_tri.cpp
index b7a87783de29..80a14201645b 100644
--- a/src/tri/_tri.cpp
+++ b/src/tri/_tri.cpp
@@ -12,6 +12,7 @@
 
 #include <algorithm>
 #include <set>
+#include <random>
 
 
 TriEdge::TriEdge()
@@ -1465,8 +1466,8 @@ TrapezoidMapTriFinder::initialize()
     _tree->assert_valid(false);
 
     // Randomly shuffle all edges other than first 2.
-    RandomNumberGenerator rng(1234);
-    std::random_shuffle(_edges.begin()+2, _edges.end(), rng);
+    std::mt19937 rng(1234);
+    std::shuffle(_edges.begin()+2, _edges.end(), rng);
 
     // Add edges, one at a time, to tree.
     size_t nedges = _edges.size();
@@ -2056,16 +2057,3 @@ TrapezoidMapTriFinder::Trapezoid::set_upper_right(Trapezoid* upper_right_)
     if (upper_right != 0)
         upper_right->upper_left = this;
 }
-
-
-
-RandomNumberGenerator::RandomNumberGenerator(unsigned long seed)
-    : _m(21870), _a(1291), _c(4621), _seed(seed % _m)
-{}
-
-unsigned long
-RandomNumberGenerator::operator()(unsigned long max_value)
-{
-    _seed = (_seed*_a + _c) % _m;
-    return (_seed*max_value) / _m;
-}
diff --git a/src/tri/_tri.h b/src/tri/_tri.h
index 28c8e07933cc..29b4ff81fb17 100644
--- a/src/tri/_tri.h
+++ b/src/tri/_tri.h
@@ -791,28 +791,4 @@ class TrapezoidMapTriFinder
     Node* _tree;    // Root node of the trapezoid map search tree.  Owned.
 };
 
-
-
-/* Linear congruential random number generator.  Edges in the triangulation are
- * randomly shuffled before being added to the trapezoid map.  Want the
- * shuffling to be identical across different operating systems and the same
- * regardless of previous random number use.  Would prefer to use a STL or
- * Boost random number generator, but support is not consistent across
- * different operating systems so implementing own here.
- *
- * This is not particularly random, but is perfectly adequate for the use here.
- * Coefficients taken from Numerical Recipes in C. */
-class RandomNumberGenerator
-{
-public:
-    RandomNumberGenerator(unsigned long seed);
-
-    // Return random integer in the range 0 to max_value-1.
-    unsigned long operator()(unsigned long max_value);
-
-private:
-    const unsigned long _m, _a, _c;
-    unsigned long _seed;
-};
-
 #endif