feat(universe_utils): add SAT implementation for 2D convex polygon collision check

common/autoware_universe_utils/CMakeLists.txt

@@ -16,6 +16,7 @@ ament_auto_add_library(autoware_universe_utils SHARED
   src/geometry/boost_polygon_utils.cpp
   src/geometry/random_convex_polygon.cpp
   src/geometry/gjk_2d.cpp
+  src/geometry/sat_2d.cpp
   src/math/sin_table.cpp
   src/math/trigonometry.cpp
   src/ros/msg_operation.cpp

common/autoware_universe_utils/include/autoware/universe_utils/geometry/sat_2d.hpp

@@ -0,0 +1,30 @@
+// Copyright 2024 Tier IV, Inc.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+#ifndef AUTOWARE__UNIVERSE_UTILS__GEOMETRY__SAT_2D_HPP_
+#define AUTOWARE__UNIVERSE_UTILS__GEOMETRY__SAT_2D_HPP_
+
+#include "autoware/universe_utils/geometry/boost_geometry.hpp"
+
+namespace autoware::universe_utils::sat
+{
+/**
+ * @brief Check if 2 convex polygons intersect using the SAT algorithm
+ * @details faster than boost::geometry::overlap() but speed decline sharply as vertices increase
+ */
+bool intersects(const Polygon2d & convex_polygon1, const Polygon2d & convex_polygon2);
+
+}  // namespace autoware::universe_utils::sat
+
+#endif  // AUTOWARE__UNIVERSE_UTILS__GEOMETRY__SAT_2D_HPP_

common/autoware_universe_utils/src/geometry/sat_2d.cpp

@@ -0,0 +1,82 @@
+// Copyright 2024 Tier IV, Inc.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+#include "autoware/universe_utils/geometry/sat_2d.hpp"
+
+#include <boost/geometry/algorithms/for_each.hpp>
+
+namespace autoware::universe_utils::sat
+{
+
+namespace
+{
+/// @brief calculate the edge normal of two points
+Point2d edge_normal(const Point2d & p1, const Point2d & p2)
+{
+  return {p2.y() - p1.y(), p1.x() - p2.x()};
+}
+
+/// @brief project a polygon onto an axis and return the minimum and maximum values
+std::pair<double, double> project_polygon(const Polygon2d & poly, const Point2d & axis)
+{
+  double min = poly.outer()[0].dot(axis);
+  double max = min;
+  for (const auto & point : poly.outer()) {
+    double projection = point.dot(axis);
+    if (projection < min) {
+      min = projection;
+    }
+    if (projection > max) {
+      max = projection;
+    }
+  }
+  return {min, max};
+}
+
+/// @brief check if two projections overlap
+bool projections_overlap(
+  const std::pair<double, double> & proj1, const std::pair<double, double> & proj2)
+{
+  return proj1.second >= proj2.first && proj2.second >= proj1.first;
+}
+
+/// @brief check is all edges of a polygon can be separated from the other polygon with a separating
+/// axis
+bool has_separating_axis(const Polygon2d & polygon, const Polygon2d & other)
+{
+  for (size_t i = 0; i < polygon.outer().size(); ++i) {
+    const size_t next_i = (i + 1) % polygon.outer().size();
+    const Point2d edge = edge_normal(polygon.outer()[i], polygon.outer()[next_i]);
+    const auto projection1 = project_polygon(polygon, edge);
+    const auto projection2 = project_polygon(other, edge);
+    if (!projections_overlap(projection1, projection2)) {
+      return false;
+    }
+  }
+  return true;
+}
+}  // namespace
+
+/// @brief check if two convex polygons intersect using the SAT algorithm
+/// @details this function uses the Separating Axis Theorem (SAT) to determine if two convex
+/// polygons intersect. If projections overlap on all tested axes, the function returns `true`;
+/// otherwise, it returns `false`. Note that touching polygons (e.g., at a point or along an edge)
+/// will be considered as not intersecting.
+bool intersects(const Polygon2d & convex_polygon1, const Polygon2d & convex_polygon2)
+{
+  return has_separating_axis(convex_polygon1, convex_polygon2) &&
+         has_separating_axis(convex_polygon2, convex_polygon1);
+}
+
+}  // namespace autoware::universe_utils::sat

common/autoware_universe_utils/test/src/geometry/test_geometry.cpp

@@ -1,4 +1,4 @@
 // Copyright 2020-2024 Tier IV, Inc.
 //
 // Licensed under the Apache License, Version 2.0 (the "License");
 // you may not use this file except in compliance with the License.
@@ -15,6 +15,7 @@
 #include "autoware/universe_utils/geometry/boost_geometry.hpp"
 #include "autoware/universe_utils/geometry/geometry.hpp"
 #include "autoware/universe_utils/geometry/random_convex_polygon.hpp"
+#include "autoware/universe_utils/geometry/sat_2d.hpp"
 #include "autoware/universe_utils/math/unit_conversion.hpp"
 #include "autoware/universe_utils/system/stop_watch.hpp"
 
@@ -1933,54 +1934,82 @@
   constexpr auto max_values = 1000;
 
   autoware::universe_utils::StopWatch<std::chrono::nanoseconds, std::chrono::nanoseconds> sw;
+
   for (auto vertices = 3UL; vertices < max_vertices; ++vertices) {
     double ground_truth_intersect_ns = 0.0;
     double ground_truth_no_intersect_ns = 0.0;
     double gjk_intersect_ns = 0.0;
     double gjk_no_intersect_ns = 0.0;
+    double sat_intersect_ns = 0.0;
+    double sat_no_intersect_ns = 0.0;
     int intersect_count = 0;
     polygons.clear();
+
     for (auto i = 0; i < polygons_nb; ++i) {
       polygons.push_back(autoware::universe_utils::random_convex_polygon(vertices, max_values));
     }
+
     for (auto i = 0UL; i < polygons.size(); ++i) {
       for (auto j = 0UL; j < polygons.size(); ++j) {
         sw.tic();
         const auto ground_truth = boost::geometry::intersects(polygons[i], polygons[j]);
         if (ground_truth) {
           ++intersect_count;
           ground_truth_intersect_ns += sw.toc();
         } else {
           ground_truth_no_intersect_ns += sw.toc();
         }
+
         sw.tic();
         const auto gjk = autoware::universe_utils::intersects_convex(polygons[i], polygons[j]);
         if (gjk) {
           gjk_intersect_ns += sw.toc();
         } else {
           gjk_no_intersect_ns += sw.toc();
         }
+
+        sw.tic();
+        const auto sat = autoware::universe_utils::sat::intersects(polygons[i], polygons[j]);
+        if (sat) {
+          sat_intersect_ns += sw.toc();
+        } else {
+          sat_no_intersect_ns += sw.toc();
+        }
+
+        EXPECT_EQ(ground_truth, gjk);
+        EXPECT_EQ(ground_truth, sat);
+
         if (ground_truth != gjk) {
-          std::cout << "Failed for the 2 polygons: ";
+          std::cout << "Failed for the 2 polygons with GJK: ";
+          std::cout << boost::geometry::wkt(polygons[i]) << boost::geometry::wkt(polygons[j])
+                    << std::endl;
+        }
+
+        if (ground_truth != sat) {
+          std::cout << "Failed for the 2 polygons with SAT: ";
           std::cout << boost::geometry::wkt(polygons[i]) << boost::geometry::wkt(polygons[j])
                     << std::endl;
         }
-        EXPECT_EQ(ground_truth, gjk);
       }
     }
+
     std::printf(
       "polygons_nb = %d, vertices = %ld, %d / %d pairs with intersects\n", polygons_nb, vertices,
       intersect_count, polygons_nb * polygons_nb);
+
     std::printf(
-      "\tIntersect:\n\t\tBoost::geometry = %2.2f ms\n\t\tGJK = %2.2f ms\n",
-      ground_truth_intersect_ns / 1e6, gjk_intersect_ns / 1e6);
+      "\tIntersect:\n\t\tBoost::geometry = %2.2f ms\n\t\tGJK = %2.2f ms\n\t\tSAT = %2.2f ms\n",
+      ground_truth_intersect_ns / 1e6, gjk_intersect_ns / 1e6, sat_intersect_ns / 1e6);
+
     std::printf(
-      "\tNo Intersect:\n\t\tBoost::geometry = %2.2f ms\n\t\tGJK = %2.2f ms\n",
-      ground_truth_no_intersect_ns / 1e6, gjk_no_intersect_ns / 1e6);
+      "\tNo Intersect:\n\t\tBoost::geometry = %2.2f ms\n\t\tGJK = %2.2f ms\n\t\tSAT = %2.2f ms\n",
+      ground_truth_no_intersect_ns / 1e6, gjk_no_intersect_ns / 1e6, sat_no_intersect_ns / 1e6);
+
     std::printf(
-      "\tTotal:\n\t\tBoost::geometry = %2.2f ms\n\t\tGJK = %2.2f ms\n",
+      "\tTotal:\n\t\tBoost::geometry = %2.2f ms\n\t\tGJK = %2.2f ms\n\t\tSAT = %2.2f ms\n",
       (ground_truth_no_intersect_ns + ground_truth_intersect_ns) / 1e6,
-      (gjk_no_intersect_ns + gjk_intersect_ns) / 1e6);
+      (gjk_no_intersect_ns + gjk_intersect_ns) / 1e6,
+      (sat_no_intersect_ns + sat_intersect_ns) / 1e6);
   }
 }