diff --git a/pytorch3d/csrc/iou_box3d/iou_box3d.cu b/pytorch3d/csrc/iou_box3d/iou_box3d.cu
index 270f7f18a..4da2f1e8b 100644
--- a/pytorch3d/csrc/iou_box3d/iou_box3d.cu
+++ b/pytorch3d/csrc/iou_box3d/iou_box3d.cu
@@ -88,9 +88,9 @@ __global__ void IoUBox3DKernel(
       for (int b1 = 0; b1 < box1_count; ++b1) {
         for (int b2 = 0; b2 < box2_count; ++b2) {
           const bool is_coplanar =
-              IsCoplanarFace(box1_intersect[b1], box2_intersect[b2]);
+              IsCoplanarTriTri(box1_intersect[b1], box2_intersect[b2]);
           const float area = FaceArea(box1_intersect[b1]);
-          if ((is_coplanar) && (area > kEpsilon)) {
+          if ((is_coplanar) && (area > aEpsilon)) {
             tri2_keep[b2].keep = false;
           }
         }
diff --git a/pytorch3d/csrc/iou_box3d/iou_box3d_cpu.cpp b/pytorch3d/csrc/iou_box3d/iou_box3d_cpu.cpp
index 3e5771780..3bc66de4e 100644
--- a/pytorch3d/csrc/iou_box3d/iou_box3d_cpu.cpp
+++ b/pytorch3d/csrc/iou_box3d/iou_box3d_cpu.cpp
@@ -80,9 +80,9 @@ std::tuple<at::Tensor, at::Tensor> IoUBox3DCpu(
         for (int b1 = 0; b1 < box1_intersect.size(); ++b1) {
           for (int b2 = 0; b2 < box2_intersect.size(); ++b2) {
             const bool is_coplanar =
-                IsCoplanarFace(box1_intersect[b1], box2_intersect[b2]);
+                IsCoplanarTriTri(box1_intersect[b1], box2_intersect[b2]);
             const float area = FaceArea(box1_intersect[b1]);
-            if ((is_coplanar) && (area > kEpsilon)) {
+            if ((is_coplanar) && (area > aEpsilon)) {
               tri2_keep[b2] = 0;
             }
           }
diff --git a/pytorch3d/csrc/iou_box3d/iou_utils.cuh b/pytorch3d/csrc/iou_box3d/iou_utils.cuh
index 0882c5481..b4d08563a 100644
--- a/pytorch3d/csrc/iou_box3d/iou_utils.cuh
+++ b/pytorch3d/csrc/iou_box3d/iou_utils.cuh
@@ -12,7 +12,15 @@
 #include <cstdio>
 #include "utils/float_math.cuh"
 
-__constant__ const float kEpsilon = 1e-5;
+// dEpsilon: Used in dot products and is used to assess whether two unit vectors
+// are orthogonal (or coplanar). It's an epsilon on cos(θ).
+// With dEpsilon = 0.001, two unit vectors are considered co-planar
+// if their θ = 2.5 deg.
+__constant__ const float dEpsilon = 1e-3;
+// aEpsilon: Used once in main function to check for small face areas
+__constant__ const float aEpsilon = 1e-4;
+// kEpsilon: Used only for norm(u) = u/max(||u||, kEpsilon)
+__constant__ const float kEpsilon = 1e-8;
 
 /*
 _PLANES and _TRIS define the 4- and 3-connectivity
@@ -120,10 +128,37 @@ __device__ inline void GetBoxPlanes(
   }
 }
 
-// The normal of the face defined by vertices (v0, v1, v2)
-// Define e0 to be the edge connecting (v1, v0)
-// Define e1 to be the edge connecting (v2, v0)
-// normal is the cross product of e0, e1
+// The normal of a plane spanned by vectors e0 and e1
+//
+// Args
+//    e0, e1: float3 vectors defining a plane
+//
+// Returns
+//    float3: normal of the plane
+//
+__device__ inline float3 GetNormal(const float3 e0, const float3 e1) {
+  float3 n = cross(e0, e1);
+  n = n / std::fmaxf(norm(n), kEpsilon);
+  return n;
+}
+
+// The center of a triangle defined by vertices (v0, v1, v2)
+//
+// Args
+//    v0, v1, v2: float3 coordinates of the vertices of the triangle
+//
+// Returns
+//    float3: center of the triangle
+//
+__device__ inline float3
+TriCenter(const float3 v0, const float3 v1, const float3 v2) {
+  float3 ctr = (v0 + v1 + v2) / 3.0f;
+  return ctr;
+}
+
+// The normal of the triangle defined by vertices (v0, v1, v2)
+// We find the "best" edges connecting the face center to the vertices,
+// such that the cross product between the edges is maximized.
 //
 // Args
 //    v0, v1, v2: float3 coordinates of the vertices of the face
@@ -132,9 +167,24 @@ __device__ inline void GetBoxPlanes(
 //    float3: normal for the face
 //
 __device__ inline float3
-FaceNormal(const float3 v0, const float3 v1, const float3 v2) {
-  float3 n = cross(v1 - v0, v2 - v0);
-  n = n / fmaxf(norm(n), kEpsilon);
+TriNormal(const float3 v0, const float3 v1, const float3 v2) {
+  const float3 ctr = TriCenter(v0, v1, v2);
+
+  const float d01 = norm(cross(v0 - ctr, v1 - ctr));
+  const float d02 = norm(cross(v0 - ctr, v2 - ctr));
+  const float d12 = norm(cross(v1 - ctr, v2 - ctr));
+
+  float3 n = GetNormal(v0 - ctr, v1 - ctr);
+  float max_dist = d01;
+
+  if (d02 > max_dist) {
+    max_dist = d02;
+    n = GetNormal(v0 - ctr, v2 - ctr);
+  }
+  if (d12 > max_dist) {
+    n = GetNormal(v1 - ctr, v2 - ctr);
+  }
+
   return n;
 }
 
@@ -158,8 +208,75 @@ __device__ inline float FaceArea(const FaceVerts& tri) {
   return norm(n) / 2.0;
 }
 
-// The normal of a box plane defined by the verts in `plane` with
-// the centroid of the box given by `center`.
+// The center of a plane defined by vertices (v0, v1, v2, v3)
+//
+// Args
+//    v0, v1, v2, v3: float3 coordinates of the vertices of the plane
+//
+// Returns
+//    float3: center of the plane
+//
+__device__ inline float3 PlaneCenter(
+    const float3 v0,
+    const float3 v1,
+    const float3 v2,
+    const float3 v3) {
+  float3 ctr = (v0 + v1 + v2 + v3) / 4.0f;
+  return ctr;
+}
+
+// The normal of a planar face with vertices (v0, v1, v2, v3)
+// We find the "best" edges connecting the face center to the vertices,
+// such that the cross product between the edges is maximized.
+//
+// Args
+//    e0, e1: float3 coordinates of the vertices of the plane
+//
+// Returns
+//    float3: center of the plane
+//
+__device__ inline float3 PlaneNormal(
+    const float3 v0,
+    const float3 v1,
+    const float3 v2,
+    const float3 v3) {
+  const float3 ctr = PlaneCenter(v0, v1, v2, v3);
+
+  const float d01 = norm(cross(v0 - ctr, v1 - ctr));
+  const float d02 = norm(cross(v0 - ctr, v2 - ctr));
+  const float d03 = norm(cross(v0 - ctr, v3 - ctr));
+  const float d12 = norm(cross(v1 - ctr, v2 - ctr));
+  const float d13 = norm(cross(v1 - ctr, v3 - ctr));
+  const float d23 = norm(cross(v2 - ctr, v3 - ctr));
+
+  float max_dist = d01;
+  float3 n = GetNormal(v0 - ctr, v1 - ctr);
+
+  if (d02 > max_dist) {
+    max_dist = d02;
+    n = GetNormal(v0 - ctr, v2 - ctr);
+  }
+  if (d03 > max_dist) {
+    max_dist = d03;
+    n = GetNormal(v0 - ctr, v3 - ctr);
+  }
+  if (d12 > max_dist) {
+    max_dist = d12;
+    n = GetNormal(v1 - ctr, v2 - ctr);
+  }
+  if (d13 > max_dist) {
+    max_dist = d13;
+    n = GetNormal(v1 - ctr, v3 - ctr);
+  }
+  if (d23 > max_dist) {
+    n = GetNormal(v2 - ctr, v3 - ctr);
+  }
+  return n;
+}
+
+// The normal of a box plane defined by the verts in `plane` such that it
+// points toward the centroid of the box given by `center`.
+//
 // Args
 //    plane: float3 coordinates of the vertices of the plane
 //    center: float3 coordinates of the center of the box from
@@ -173,23 +290,30 @@ template <typename FaceVertsPlane>
 __device__ inline float3 PlaneNormalDirection(
     const FaceVertsPlane& plane,
     const float3& center) {
-  // Only need the first 3 verts of the plane
+  // The plane's vertices
   const float3 v0 = plane.v0;
   const float3 v1 = plane.v1;
   const float3 v2 = plane.v2;
+  const float3 v3 = plane.v3;
+
+  // The plane's center
+  const float3 plane_center = PlaneCenter(v0, v1, v2, v3);
+
+  // The plane's normal
+  float3 n = PlaneNormal(v0, v1, v2, v3);
 
-  // We project the center on the plane defined by (v0, v1, v2)
-  // We can write center = v0 + a * e0 + b * e1 + c * n
+  // We project the center on the plane defined by (v0, v1, v2, v3)
+  // We can write center = plane_center + a * e0 + b * e1 + c * n
   // We know that <e0, n> = 0 and <e1, n> = 0 and
   // <a, b> is the dot product between a and b.
   // This means we can solve for c as:
-  // c = <center - v0 - a * e0 - b * e1, n> = <center - v0, n>
-  float3 n = FaceNormal(v0, v1, v2);
-  const float c = dot((center - v0), n);
+  // c = <center - plane_center - a * e0 - b * e1, n>
+  //   = <center - plane_center, n>
+  const float c = dot((center - plane_center), n);
 
   // If c is negative, then we revert the direction of n such that n
   // points "inside"
-  if (c < kEpsilon) {
+  if (c < 0.0f) {
     n = -1.0f * n;
   }
 
@@ -318,16 +442,22 @@ __device__ inline float3 PolyhedronCenter(
 //
 __device__ inline bool
 IsInside(const FaceVerts& plane, const float3& normal, const float3& point) {
-  // Get one vert of the plane
+  // Vertices of the plane
   const float3 v0 = plane.v0;
+  const float3 v1 = plane.v1;
+  const float3 v2 = plane.v2;
+  const float3 v3 = plane.v3;
 
-  // Every point p can be written as p = v0 + a e0 + b e1 + c n
+  // The center of the plane
+  const float3 plane_ctr = PlaneCenter(v0, v1, v2, v3);
+
+  // Every point p can be written as p = plane_ctr + a e0 + b e1 + c n
   // Solving for c:
-  // c = (point - v0 - a * e0 - b * e1).dot(n)
+  // c = (point - plane_ctr - a * e0 - b * e1).dot(n)
   // We know that <e0, n> = 0 and <e1, n> = 0
   // So the calculation can be simplified as:
-  const float c = dot((point - v0), normal);
-  const bool inside = c > -1.0f * kEpsilon;
+  const float c = dot((point - plane_ctr), normal);
+  const bool inside = c >= 0.0f;
   return inside;
 }
 
@@ -348,20 +478,150 @@ __device__ inline float3 PlaneEdgeIntersection(
     const float3& normal,
     const float3& p0,
     const float3& p1) {
-  // Get one vert of the plane
+  // Vertices of the plane
   const float3 v0 = plane.v0;
+  const float3 v1 = plane.v1;
+  const float3 v2 = plane.v2;
+  const float3 v3 = plane.v3;
+
+  // The center of the plane
+  const float3 plane_ctr = PlaneCenter(v0, v1, v2, v3);
 
   // The point of intersection can be parametrized
   // p = p0 + a (p1 - p0) where a in [0, 1]
   // We want to find a such that p is on plane
-  // <p - v0, n> = 0
-  const float top = dot(-1.0f * (p0 - v0), normal);
-  const float bot = dot(p1 - p0, normal);
-  const float a = top / bot;
-  const float3 p = p0 + a * (p1 - p0);
+  // <p - plane_ctr, n> = 0
+
+  float3 direc = p1 - p0;
+  direc = direc / fmaxf(norm(direc), kEpsilon);
+
+  float3 p = (p1 + p0) / 2.0f;
+
+  if (abs(dot(direc, normal)) >= dEpsilon) {
+    const float top = -1.0f * dot(p0 - plane_ctr, normal);
+    const float bot = dot(p1 - p0, normal);
+    const float a = top / bot;
+    p = p0 + a * (p1 - p0);
+  }
+
   return p;
 }
 
+// Compute the most distant points between two sets of vertices
+//
+// Args
+//    verts1, verts2: list of float3 defining the list of vertices
+//
+// Returns
+//    v1m, v2m: float3 vectors of the most distant points
+//          in verts1 and verts2 respectively
+//
+__device__ inline std::tuple<float3, float3> ArgMaxVerts(
+    std::initializer_list<float3> verts1,
+    std::initializer_list<float3> verts2) {
+  auto v1m = float3();
+  auto v2m = float3();
+  float maxdist = -1.0f;
+
+  for (const auto& v1 : verts1) {
+    for (const auto& v2 : verts2) {
+      if (norm(v1 - v2) > maxdist) {
+        v1m = v1;
+        v2m = v2;
+        maxdist = norm(v1 - v2);
+      }
+    }
+  }
+  return std::make_tuple(v1m, v2m);
+}
+
+// Compute a boolean indicator for whether or not two faces
+// are coplanar
+//
+// Args
+//    tri1, tri2: FaceVerts struct of the vertex coordinates of
+//       the triangle face
+//
+// Returns
+//    bool: whether or not the two faces are coplanar
+//
+__device__ inline bool IsCoplanarTriTri(
+    const FaceVerts& tri1,
+    const FaceVerts& tri2) {
+  // Get verts for face 1
+  const float3 v0_1 = tri1.v0;
+  const float3 v1_1 = tri1.v1;
+  const float3 v2_1 = tri1.v2;
+
+  const float3 tri1_ctr = TriCenter(v0_1, v1_1, v2_1);
+  const float3 tri1_n = TriNormal(v0_1, v1_1, v2_1);
+
+  // Get verts for face 2
+  const float3 v0_2 = tri2.v0;
+  const float3 v1_2 = tri2.v1;
+  const float3 v2_2 = tri2.v2;
+
+  const float3 tri2_ctr = TriCenter(v0_2, v1_2, v2_2);
+  const float3 tri2_n = TriNormal(v0_2, v1_2, v2_2);
+
+  // Check if parallel
+  const bool check1 = abs(dot(tri1_n, tri2_n)) > 1 - dEpsilon;
+
+  // Compute most distant points
+  auto argvs =
+      ArgMaxVerts({tri1.v0, tri1.v1, tri1.v2}, {tri2.v0, tri2.v1, tri2.v2});
+  const float3 v1m = std::get<0>(argvs);
+  const float3 v2m = std::get<1>(argvs);
+
+  float3 n12m = v1m - v2m;
+  n12m = n12m / fmaxf(norm(n12m), kEpsilon);
+
+  const bool check2 = (abs(dot(n12m, tri1_n)) < dEpsilon) ||
+      (abs(dot(n12m, tri2_n)) < dEpsilon);
+
+  return (check1 && check2);
+}
+
+// Compute a boolean indicator for whether or not a triangular and a planar
+// face are coplanar
+//
+// Args
+//    tri, plane: FaceVerts struct of the vertex coordinates of
+//       the triangle and planar face
+//  normal: the normal direction of the plane pointing "inside"
+//
+// Returns
+//    bool: whether or not the two faces are coplanar
+//
+__device__ inline bool IsCoplanarTriPlane(
+    const FaceVerts& tri,
+    const FaceVerts& plane,
+    const float3& normal) {
+  // Get verts for tri
+  const float3 v0t = tri.v0;
+  const float3 v1t = tri.v1;
+  const float3 v2t = tri.v2;
+
+  const float3 tri_ctr = TriCenter(v0t, v1t, v2t);
+  const float3 nt = TriNormal(v0t, v1t, v2t);
+
+  // check if parallel
+  const bool check1 = abs(dot(nt, normal)) > 1 - dEpsilon;
+
+  // Compute most distant points
+  auto argvs = ArgMaxVerts(
+      {tri.v0, tri.v1, tri.v2}, {plane.v0, plane.v1, plane.v2, plane.v3});
+  const float3 v1m = std::get<0>(argvs);
+  const float3 v2m = std::get<1>(argvs);
+
+  float3 n12m = v1m - v2m;
+  n12m = n12m / fmaxf(norm(n12m), kEpsilon);
+
+  const bool check2 = abs(dot(n12m, normal)) < dEpsilon;
+
+  return (check1 && check2);
+}
+
 // Triangle is clipped into a quadrilateral
 // based on the intersection points with the plane.
 // Then the quadrilateral is divided into two triangles.
@@ -477,6 +737,14 @@ __device__ inline int ClipTriByPlane(
   const bool isin1 = IsInside(plane, normal, v1);
   const bool isin2 = IsInside(plane, normal, v2);
 
+  // Check coplanar
+  const bool iscoplanar = IsCoplanarTriPlane(tri, plane, normal);
+  if (iscoplanar) {
+    // Return input vertices
+    face_verts_out[0] = {v0, v1, v2};
+    return 1;
+  }
+
   // All in
   if (isin0 && isin1 && isin2) {
     // Return input vertices
@@ -515,40 +783,6 @@ __device__ inline int ClipTriByPlane(
   return 0;
 }
 
-// Compute a boolean indicator for whether or not two faces
-// are coplanar
-//
-// Args
-//    tri1, tri2: FaceVerts struct of the vertex coordinates of
-//       the triangle face
-//
-// Returns
-//    bool: whether or not the two faces are coplanar
-//
-__device__ inline bool IsCoplanarFace(
-    const FaceVerts& tri1,
-    const FaceVerts& tri2) {
-  // Get verts for face 1
-  const float3 v0 = tri1.v0;
-  const float3 v1 = tri1.v1;
-  const float3 v2 = tri1.v2;
-
-  const float3 n1 = FaceNormal(v0, v1, v2);
-  int coplanar_count = 0;
-
-  // Check v0, v1, v2
-  if (abs(dot(tri2.v0 - v0, n1)) < kEpsilon) {
-    coplanar_count++;
-  }
-  if (abs(dot(tri2.v1 - v0, n1)) < kEpsilon) {
-    coplanar_count++;
-  }
-  if (abs(dot(tri2.v2 - v0, n1)) < kEpsilon) {
-    coplanar_count++;
-  }
-  return (coplanar_count == 3);
-}
-
 // Get the triangles from each box which are part of the
 // intersecting polyhedron by computing the intersection
 // points with each of the planes.
diff --git a/pytorch3d/csrc/iou_box3d/iou_utils.h b/pytorch3d/csrc/iou_box3d/iou_utils.h
index f29f4f026..e3fb58dd3 100644
--- a/pytorch3d/csrc/iou_box3d/iou_utils.h
+++ b/pytorch3d/csrc/iou_box3d/iou_utils.h
@@ -18,7 +18,15 @@
 #include <type_traits>
 #include "utils/vec3.h"
 
-const auto kEpsilon = 1e-5;
+// dEpsilon: Used in dot products and is used to assess whether two unit vectors
+// are orthogonal (or coplanar). It's an epsilon on cos(θ).
+// With dEpsilon = 0.001, two unit vectors are considered co-planar
+// if their θ = 2.5 deg.
+const auto dEpsilon = 1e-3;
+// aEpsilon: Used once in main function to check for small face areas
+const auto aEpsilon = 1e-4;
+// kEpsilon: Used only for norm(u) = u/max(||u||, kEpsilon)
+const auto kEpsilon = 1e-8;
 
 /*
 _PLANES and _TRIS define the 4- and 3-connectivity
@@ -129,23 +137,111 @@ inline face_verts GetBoxPlanes(const Box& box) {
   return box_planes;
 }
 
-// The normal of the face defined by vertices (v0, v1, v2)
-// Define e0 to be the edge connecting (v1, v0)
-// Define e1 to be the edge connecting (v2, v0)
-// normal is the cross product of e0, e1
+// The normal of a plane spanned by vectors e0 and e1
 //
 // Args
-//    v0, v1, v2: vec3 coordinates of the vertices of the face
+//    e0, e1: vec3 vectors defining a plane
 //
 // Returns
-//    vec3: normal for the face
+//    vec3: normal of the plane
 //
-inline vec3<float> FaceNormal(vec3<float> v0, vec3<float> v1, vec3<float> v2) {
-  vec3<float> n = cross(v1 - v0, v2 - v0);
+inline vec3<float> GetNormal(const vec3<float> e0, const vec3<float> e1) {
+  vec3<float> n = cross(e0, e1);
   n = n / std::fmaxf(norm(n), kEpsilon);
   return n;
 }
 
+// The center of a triangle tri
+//
+// Args
+//    tri: vec3 coordinates of the vertices of the triangle
+//
+// Returns
+//    vec3: center of the triangle
+//
+inline vec3<float> TriCenter(const std::vector<vec3<float>>& tri) {
+  // Vertices of the triangle
+  const vec3<float> v0 = tri[0];
+  const vec3<float> v1 = tri[1];
+  const vec3<float> v2 = tri[2];
+
+  return (v0 + v1 + v2) / 3.0f;
+}
+
+// The normal of the triangle defined by vertices (v0, v1, v2)
+// We find the "best" edges connecting the face center to the vertices,
+// such that the cross product between the edges is maximized.
+//
+// Args
+//    tri: vec3 coordinates of the vertices of the face
+//
+// Returns
+//    vec3: normal for the face
+//
+inline vec3<float> TriNormal(const std::vector<vec3<float>>& tri) {
+  // Get center of triangle
+  const vec3<float> ctr = TriCenter(tri);
+
+  // find the "best" normal as cross product of edges from center
+  float max_dist = -1.0f;
+  vec3<float> n = {0.0f, 0.0f, 0.0f};
+  for (int i = 0; i < 2; ++i) {
+    for (int j = i + 1; j < 3; ++j) {
+      const float dist = norm(cross(tri[i] - ctr, tri[j] - ctr));
+      if (dist > max_dist) {
+        n = GetNormal(tri[i] - ctr, tri[j] - ctr);
+      }
+    }
+  }
+  return n;
+}
+
+// The center of a plane
+//
+// Args
+//    plane: vec3 coordinates of the vertices of the plane
+//
+// Returns
+//    vec3: center of the plane
+//
+inline vec3<float> PlaneCenter(const std::vector<vec3<float>>& plane) {
+  // Vertices of the plane
+  const vec3<float> v0 = plane[0];
+  const vec3<float> v1 = plane[1];
+  const vec3<float> v2 = plane[2];
+  const vec3<float> v3 = plane[3];
+
+  return (v0 + v1 + v2 + v3) / 4.0f;
+}
+
+// The normal of a planar face with vertices (v0, v1, v2, v3)
+// We find the "best" edges connecting the face center to the vertices,
+// such that the cross product between the edges is maximized.
+//
+// Args
+//    plane: vec3 coordinates of the vertices of the planar face
+//
+// Returns
+//    vec3: normal of the planar face
+//
+inline vec3<float> PlaneNormal(const std::vector<vec3<float>>& plane) {
+  // Get center of planar face
+  vec3<float> ctr = PlaneCenter(plane);
+
+  // find the "best" normal as cross product of edges from center
+  float max_dist = -1.0f;
+  vec3<float> n = {0.0f, 0.0f, 0.0f};
+  for (int i = 0; i < 3; ++i) {
+    for (int j = i + 1; j < 4; ++j) {
+      const float dist = norm(cross(plane[i] - ctr, plane[j] - ctr));
+      if (dist > max_dist) {
+        n = GetNormal(plane[i] - ctr, plane[j] - ctr);
+      }
+    }
+  }
+  return n;
+}
+
 // The area of the face defined by vertices (v0, v1, v2)
 // Define e0 to be the edge connecting (v1, v0)
 // Define e1 to be the edge connecting (v2, v0)
@@ -166,8 +262,9 @@ inline float FaceArea(const std::vector<vec3<float>>& tri) {
   return norm(n) / 2.0;
 }
 
-// The normal of a box plane defined by the verts in `plane` with
-// the centroid of the box given by `center`.
+// The normal of a box plane defined by the verts in `plane` such that it
+// points toward the centroid of the box given by `center`.
+//
 // Args
 //    plane: vec3 coordinates of the vertices of the plane
 //    center: vec3 coordinates of the center of the box from
@@ -180,23 +277,22 @@ inline float FaceArea(const std::vector<vec3<float>>& tri) {
 inline vec3<float> PlaneNormalDirection(
     const std::vector<vec3<float>>& plane,
     const vec3<float>& center) {
-  // Only need the first 3 verts of the plane
-  const vec3<float> v0 = plane[0];
-  const vec3<float> v1 = plane[1];
-  const vec3<float> v2 = plane[2];
+  // The plane's center & normal
+  const vec3<float> plane_center = PlaneCenter(plane);
+  vec3<float> n = PlaneNormal(plane);
 
-  // We project the center on the plane defined by (v0, v1, v2)
-  // We can write center = v0 + a * e0 + b * e1 + c * n
+  // We project the center on the plane defined by (v0, v1, v2, v3)
+  // We can write center = plane_center + a * e0 + b * e1 + c * n
   // We know that <e0, n> = 0 and <e1, n> = 0 and
   // <a, b> is the dot product between a and b.
   // This means we can solve for c as:
-  // c = <center - v0 - a * e0 - b * e1, n> = <center - v0, n>
-  vec3<float> n = FaceNormal(v0, v1, v2);
-  const float c = dot((center - v0), n);
+  // c = <center - plane_center - a * e0 - b * e1, n>
+  //   = <center - plane_center, n>
+  const float c = dot((center - plane_center), n);
 
   // If c is negative, then we revert the direction of n such that n
   // points "inside"
-  if (c < kEpsilon) {
+  if (c < 0.0f) {
     n = -1.0f * n;
   }
 
@@ -308,16 +404,16 @@ inline bool IsInside(
     const std::vector<vec3<float>>& plane,
     const vec3<float>& normal,
     const vec3<float>& point) {
-  // Get one vert of the plane
-  const vec3<float> v0 = plane[0];
+  // The center of the plane
+  const vec3<float> plane_ctr = PlaneCenter(plane);
 
-  // Every point p can be written as p = v0 + a e0 + b e1 + c n
+  // Every point p can be written as p = plane_ctr + a e0 + b e1 + c n
   // Solving for c:
-  // c = (point - v0 - a * e0 - b * e1).dot(n)
+  // c = (point - plane_ctr - a * e0 - b * e1).dot(n)
   // We know that <e0, n> = 0 and <e1, n> = 0
   // So the calculation can be simplified as:
-  const float c = dot((point - v0), normal);
-  const bool inside = c > -1.0f * kEpsilon;
+  const float c = dot((point - plane_ctr), normal);
+  const bool inside = c >= 0.0f;
   return inside;
 }
 
@@ -338,20 +434,124 @@ inline vec3<float> PlaneEdgeIntersection(
     const vec3<float>& normal,
     const vec3<float>& p0,
     const vec3<float>& p1) {
-  // Get one vert of the plane
-  const vec3<float> v0 = plane[0];
+  // The center of the plane
+  const vec3<float> plane_ctr = PlaneCenter(plane);
 
   // The point of intersection can be parametrized
   // p = p0 + a (p1 - p0) where a in [0, 1]
   // We want to find a such that p is on plane
-  // <p - v0, n> = 0
-  const float top = dot(-1.0f * (p0 - v0), normal);
-  const float bot = dot(p1 - p0, normal);
-  const float a = top / bot;
-  const vec3<float> p = p0 + a * (p1 - p0);
+  // <p - ctr, n> = 0
+
+  vec3<float> direc = p1 - p0;
+  direc = direc / std::fmaxf(norm(direc), kEpsilon);
+
+  vec3<float> p = (p1 + p0) / 2.0f;
+
+  if (std::abs(dot(direc, normal)) >= dEpsilon) {
+    const float top = -1.0f * dot(p0 - plane_ctr, normal);
+    const float bot = dot(p1 - p0, normal);
+    const float a = top / bot;
+    p = p0 + a * (p1 - p0);
+  }
   return p;
 }
 
+// Compute the most distant points between two sets of vertices
+//
+// Args
+//    verts1, verts2: vec3 defining the list of vertices
+//
+// Returns
+//    v1m, v2m: vec3 vectors of the most distant points
+//          in verts1 and verts2 respectively
+//
+inline std::tuple<vec3<float>, vec3<float>> ArgMaxVerts(
+    const std::vector<vec3<float>>& verts1,
+    const std::vector<vec3<float>>& verts2) {
+  vec3<float> v1m = {0.0f, 0.0f, 0.0f};
+  vec3<float> v2m = {0.0f, 0.0f, 0.0f};
+  float maxdist = -1.0f;
+
+  for (const auto& v1 : verts1) {
+    for (const auto& v2 : verts2) {
+      if (norm(v1 - v2) > maxdist) {
+        v1m = v1;
+        v2m = v2;
+        maxdist = norm(v1 - v2);
+      }
+    }
+  }
+  return std::make_tuple(v1m, v2m);
+}
+
+// Compute a boolean indicator for whether or not two faces
+// are coplanar
+//
+// Args
+//    tri1, tri2: std:vector<vec3> of the vertex coordinates of
+//        triangle faces
+//
+// Returns
+//    bool: whether or not the two faces are coplanar
+//
+inline bool IsCoplanarTriTri(
+    const std::vector<vec3<float>>& tri1,
+    const std::vector<vec3<float>>& tri2) {
+  // Get normal for tri 1
+  const vec3<float> n1 = TriNormal(tri1);
+
+  // Get normal for tri 2
+  const vec3<float> n2 = TriNormal(tri2);
+
+  // Check if parallel
+  const bool check1 = std::abs(dot(n1, n2)) > 1 - dEpsilon;
+
+  // Compute most distant points
+  auto argvs = ArgMaxVerts(tri1, tri2);
+  const auto [v1m, v2m] = argvs;
+
+  vec3<float> n12m = v1m - v2m;
+  n12m = n12m / std::fmaxf(norm(n12m), kEpsilon);
+
+  const bool check2 = (std::abs(dot(n12m, n1)) < dEpsilon) ||
+      (std::abs(dot(n12m, n2)) < dEpsilon);
+
+  return (check1 && check2);
+}
+
+// Compute a boolean indicator for whether or not a triangular and a planar
+// face are coplanar
+//
+// Args
+//    tri, plane: std:vector<vec3> of the vertex coordinates of
+//        triangular face and planar face
+//    normal: the normal direction of the plane pointing "inside"
+//
+// Returns
+//    bool: whether or not the two faces are coplanar
+//
+inline bool IsCoplanarTriPlane(
+    const std::vector<vec3<float>>& tri,
+    const std::vector<vec3<float>>& plane,
+    const vec3<float>& normal) {
+  // Get normal for tri
+  const vec3<float> nt = TriNormal(tri);
+
+  // check if parallel
+  const bool check1 = std::abs(dot(nt, normal)) > 1 - dEpsilon;
+
+  // Compute most distant points
+  auto argvs = ArgMaxVerts(tri, plane);
+  const auto [v1m, v2m] = argvs;
+
+  vec3<float> n12m = v1m - v2m;
+  n12m = n12m / std::fmaxf(norm(n12m), kEpsilon);
+
+  const bool check2 = std::abs(dot(n12m, normal)) < dEpsilon;
+
+  return (check1 && check2);
+}
+
 // Triangle is clipped into a quadrilateral
 // based on the intersection points with the plane.
 // Then the quadrilateral is divided into two triangles.
@@ -436,6 +636,14 @@ inline face_verts ClipTriByPlane(
   const vec3<float> v1 = tri[1];
   const vec3<float> v2 = tri[2];
 
+  // Check coplanar
+  const bool iscoplanar = IsCoplanarTriPlane(tri, plane, normal);
+  if (iscoplanar) {
+    // Return input vertices
+    face_verts tris = {{v0, v1, v2}};
+    return tris;
+  }
+
   // Check each of the triangle vertices to see if it is inside the plane
   const bool isin0 = IsInside(plane, normal, v0);
   const bool isin1 = IsInside(plane, normal, v1);
@@ -480,35 +688,6 @@ inline face_verts ClipTriByPlane(
   return empty_tris;
 }
 
-// Compute a boolean indicator for whether or not two faces
-// are coplanar
-//
-// Args
-//    tri1, tri2: std:vector<vec3> of the vertex coordinates of
-//        triangle faces
-//
-// Returns
-//    bool: whether or not the two faces are coplanar
-//
-inline bool IsCoplanarFace(
-    const std::vector<vec3<float>>& tri1,
-    const std::vector<vec3<float>>& tri2) {
-  // Get verts for face 1
-  const vec3<float> v0 = tri1[0];
-  const vec3<float> v1 = tri1[1];
-  const vec3<float> v2 = tri1[2];
-
-  const vec3<float> n1 = FaceNormal(v0, v1, v2);
-  int coplanar_count = 0;
-  for (int i = 0; i < 3; ++i) {
-    float d = std::abs(dot(tri2[i] - v0, n1));
-    if (d < kEpsilon) {
-      coplanar_count = coplanar_count + 1;
-    }
-  }
-  return (coplanar_count == 3);
-}
-
 // Get the triangles from each box which are part of the
 // intersecting polyhedron by computing the intersection
 // points with each of the planes.
diff --git a/tests/test_iou_box3d.py b/tests/test_iou_box3d.py
index a1f727806..f5f2b9e44 100644
--- a/tests/test_iou_box3d.py
+++ b/tests/test_iou_box3d.py
@@ -21,7 +21,8 @@
 OBJECTRON_TO_PYTORCH3D_FACE_IDX = [0, 4, 6, 2, 1, 5, 7, 3]
 DATA_DIR = get_tests_dir() / "data"
 DEBUG = False
-EPS = 1e-5
+DOT_EPS = 1e-3
+AREA_EPS = 1e-4
 
 UNIT_BOX = [
     [0, 0, 0],
@@ -457,6 +458,207 @@ def _test_iou(self, overlap_fn, device):
         self.assertClose(
             iou, torch.tensor([[0.91]], device=vol.device, dtype=vol.dtype), atol=1e-2
         )
+        # symmetry
+        vol, iou = overlap_fn(box15b[None], box15a[None])
+        self.assertClose(
+            iou, torch.tensor([[0.91]], device=vol.device, dtype=vol.dtype), atol=1e-2
+        )
+
+        # 16th test: From GH issue 1287
+        box16a = torch.tensor(
+            [
+                [-167.5847, -70.6167, -2.7927],
+                [-166.7333, -72.4264, -2.7927],
+                [-166.7333, -72.4264, -4.5927],
+                [-167.5847, -70.6167, -4.5927],
+                [-163.0605, -68.4880, -2.7927],
+                [-162.2090, -70.2977, -2.7927],
+                [-162.2090, -70.2977, -4.5927],
+                [-163.0605, -68.4880, -4.5927],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+
+        box16b = torch.tensor(
+            [
+                [-167.5847, -70.6167, -2.7927],
+                [-166.7333, -72.4264, -2.7927],
+                [-166.7333, -72.4264, -4.5927],
+                [-167.5847, -70.6167, -4.5927],
+                [-163.0605, -68.4880, -2.7927],
+                [-162.2090, -70.2977, -2.7927],
+                [-162.2090, -70.2977, -4.5927],
+                [-163.0605, -68.4880, -4.5927],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+        vol, iou = overlap_fn(box16a[None], box16b[None])
+        self.assertClose(
+            iou, torch.tensor([[1.0]], device=vol.device, dtype=vol.dtype), atol=1e-2
+        )
+        # symmetry
+        vol, iou = overlap_fn(box16b[None], box16a[None])
+        self.assertClose(
+            iou, torch.tensor([[1.0]], device=vol.device, dtype=vol.dtype), atol=1e-2
+        )
+
+        # 17th test: From GH issue 1287
+        box17a = torch.tensor(
+            [
+                [-33.94158, -4.51639, 0.96941],
+                [-34.67156, -2.65437, 0.96941],
+                [-34.67156, -2.65437, -0.95367],
+                [-33.94158, -4.51639, -0.95367],
+                [-38.75954, -6.40521, 0.96941],
+                [-39.48952, -4.54319, 0.96941],
+                [-39.48952, -4.54319, -0.95367],
+                [-38.75954, -6.40521, -0.95367],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+
+        box17b = torch.tensor(
+            [
+                [-33.94159, -4.51638, 0.96939],
+                [-34.67158, -2.65437, 0.96939],
+                [-34.67158, -2.65437, -0.95368],
+                [-33.94159, -4.51638, -0.95368],
+                [-38.75954, -6.40523, 0.96939],
+                [-39.48953, -4.54321, 0.96939],
+                [-39.48953, -4.54321, -0.95368],
+                [-38.75954, -6.40523, -0.95368],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+        vol, iou = overlap_fn(box17a[None], box17b[None])
+        self.assertClose(
+            iou, torch.tensor([[1.0]], device=vol.device, dtype=vol.dtype), atol=1e-2
+        )
+        # symmetry
+        vol, iou = overlap_fn(box17b[None], box17a[None])
+        self.assertClose(
+            iou, torch.tensor([[1.0]], device=vol.device, dtype=vol.dtype), atol=1e-2
+        )
+
+        # 18th test: From GH issue 1287
+        box18a = torch.tensor(
+            [
+                [-105.6248, -32.7026, -1.2279],
+                [-106.4690, -30.8895, -1.2279],
+                [-106.4690, -30.8895, -3.0279],
+                [-105.6248, -32.7026, -3.0279],
+                [-110.1575, -34.8132, -1.2279],
+                [-111.0017, -33.0001, -1.2279],
+                [-111.0017, -33.0001, -3.0279],
+                [-110.1575, -34.8132, -3.0279],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+        box18b = torch.tensor(
+            [
+                [-105.5094, -32.9504, -1.0641],
+                [-106.4272, -30.9793, -1.0641],
+                [-106.4272, -30.9793, -3.1916],
+                [-105.5094, -32.9504, -3.1916],
+                [-110.0421, -35.0609, -1.0641],
+                [-110.9599, -33.0899, -1.0641],
+                [-110.9599, -33.0899, -3.1916],
+                [-110.0421, -35.0609, -3.1916],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+        # from Meshlab
+        vol_inters = 17.108501
+        vol_box1 = 18.000067
+        vol_box2 = 23.128527
+        iou_mesh = vol_inters / (vol_box1 + vol_box2 - vol_inters)
+        vol, iou = overlap_fn(box18a[None], box18b[None])
+        self.assertClose(
+            iou,
+            torch.tensor([[iou_mesh]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
+        self.assertClose(
+            vol,
+            torch.tensor([[vol_inters]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
+        # symmetry
+        vol, iou = overlap_fn(box18b[None], box18a[None])
+        self.assertClose(
+            iou,
+            torch.tensor([[iou_mesh]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
+        self.assertClose(
+            vol,
+            torch.tensor([[vol_inters]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
+
+        # 19th example: From GH issue 1287
+        box19a = torch.tensor(
+            [
+                [-59.4785, -15.6003, 0.4398],
+                [-60.2263, -13.6928, 0.4398],
+                [-60.2263, -13.6928, -1.3909],
+                [-59.4785, -15.6003, -1.3909],
+                [-64.1743, -17.4412, 0.4398],
+                [-64.9221, -15.5337, 0.4398],
+                [-64.9221, -15.5337, -1.3909],
+                [-64.1743, -17.4412, -1.3909],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+        box19b = torch.tensor(
+            [
+                [-59.4874, -15.5775, -0.1512],
+                [-60.2174, -13.7155, -0.1512],
+                [-60.2174, -13.7155, -1.9820],
+                [-59.4874, -15.5775, -1.9820],
+                [-64.1832, -17.4185, -0.1512],
+                [-64.9132, -15.5564, -0.1512],
+                [-64.9132, -15.5564, -1.9820],
+                [-64.1832, -17.4185, -1.9820],
+            ],
+            device=device,
+            dtype=torch.float32,
+        )
+        # from Meshlab
+        vol_inters = 12.505723
+        vol_box1 = 18.918238
+        vol_box2 = 18.468531
+        iou_mesh = vol_inters / (vol_box1 + vol_box2 - vol_inters)
+        vol, iou = overlap_fn(box19a[None], box19b[None])
+        self.assertClose(
+            iou,
+            torch.tensor([[iou_mesh]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
+        self.assertClose(
+            vol,
+            torch.tensor([[vol_inters]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
+        # symmetry
+        vol, iou = overlap_fn(box19b[None], box19a[None])
+        self.assertClose(
+            iou,
+            torch.tensor([[iou_mesh]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
+        self.assertClose(
+            vol,
+            torch.tensor([[vol_inters]], device=vol.device, dtype=vol.dtype),
+            atol=1e-2,
+        )
 
     def _test_real_boxes(self, overlap_fn, device):
         data_filename = "./real_boxes.pkl"
@@ -715,7 +917,86 @@ def get_plane_verts(box: torch.Tensor) -> torch.Tensor:
     return plane_verts
 
 
-def box_planar_dir(box: torch.Tensor, eps: float = 1e-4) -> torch.Tensor:
+def get_tri_center_normal(tris: torch.Tensor) -> torch.Tensor:
+    """
+    Returns the center and normal of triangles
+    Args:
+        tris: tensor of shape (T, 3, 3)
+    Returns:
+        center: tensor of shape (T, 3)
+        normal: tensor of shape (T, 3)
+    """
+    add_dim0 = False
+    if tris.ndim == 2:
+        tris = tris.unsqueeze(0)
+        add_dim0 = True
+
+    ctr = tris.mean(1)  # (T, 3)
+    normals = torch.zeros_like(ctr)
+
+    v0, v1, v2 = tris.unbind(1)  # 3 x (T, 3)
+
+    # unvectorized solution
+    T = tris.shape[0]
+    for t in range(T):
+        ns = torch.zeros((3, 3), device=tris.device)
+        ns[0] = torch.cross(v0[t] - ctr[t], v1[t] - ctr[t], dim=-1)
+        ns[1] = torch.cross(v0[t] - ctr[t], v2[t] - ctr[t], dim=-1)
+        ns[2] = torch.cross(v1[t] - ctr[t], v2[t] - ctr[t], dim=-1)
+
+        i = torch.norm(ns, dim=-1).argmax()
+        normals[t] = ns[i]
+
+    if add_dim0:
+        ctr = ctr[0]
+        normals = normals[0]
+    normals = F.normalize(normals, dim=-1)
+    return ctr, normals
+
+
+def get_plane_center_normal(planes: torch.Tensor) -> torch.Tensor:
+    """
+    Returns the center and normal of planes
+    Args:
+        planes: tensor of shape (P, 4, 3)
+    Returns:
+        center: tensor of shape (P, 3)
+        normal: tensor of shape (P, 3)
+    """
+    add_dim0 = False
+    if planes.ndim == 2:
+        planes = planes.unsqueeze(0)
+        add_dim0 = True
+
+    ctr = planes.mean(1)  # (P, 3)
+    normals = torch.zeros_like(ctr)
+
+    v0, v1, v2, v3 = planes.unbind(1)  # 4 x (P, 3)
+
+    # unvectorized solution
+    P = planes.shape[0]
+    for t in range(P):
+        ns = torch.zeros((6, 3), device=planes.device)
+        ns[0] = torch.cross(v0[t] - ctr[t], v1[t] - ctr[t], dim=-1)
+        ns[1] = torch.cross(v0[t] - ctr[t], v2[t] - ctr[t], dim=-1)
+        ns[2] = torch.cross(v0[t] - ctr[t], v3[t] - ctr[t], dim=-1)
+        ns[3] = torch.cross(v1[t] - ctr[t], v2[t] - ctr[t], dim=-1)
+        ns[4] = torch.cross(v1[t] - ctr[t], v3[t] - ctr[t], dim=-1)
+        ns[5] = torch.cross(v2[t] - ctr[t], v3[t] - ctr[t], dim=-1)
+
+        i = torch.norm(ns, dim=-1).argmax()
+        normals[t] = ns[i]
+
+    if add_dim0:
+        ctr = ctr[0]
+        normals = normals[0]
+    normals = F.normalize(normals, dim=-1)
+    return ctr, normals
+
+
+def box_planar_dir(
+    box: torch.Tensor, dot_eps: float = DOT_EPS, area_eps: float = AREA_EPS
+) -> torch.Tensor:
     """
     Finds the unit vector n which is perpendicular to each plane in the box
     and points towards the inside of the box.
@@ -731,33 +1012,33 @@ def box_planar_dir(box: torch.Tensor, eps: float = 1e-4) -> torch.Tensor:
     assert box.shape[0] == 8 and box.shape[1] == 3
 
     # center point of each box
-    ctr = box.mean(0).view(1, 3)
-
-    verts = get_plane_verts(box)  # (6, 4, 3)
+    box_ctr = box.mean(0).view(1, 3)
 
-    v0, v1, v2, v3 = verts.unbind(1)  # each v of shape (6, 3)
-
-    # We project the ctr on the plane defined by (v0, v1, v2, v3)
-    # We define e0 to be the edge connecting (v1, v0)
-    # We define e1 to be the edge connecting (v2, v0)
-    # And n is the cross product of e0, e1, either pointing "inside" or not.
-    e0 = F.normalize(v1 - v0, dim=-1)
-    e1 = F.normalize(v2 - v0, dim=-1)
-    n = F.normalize(torch.cross(e0, e1, dim=-1), dim=-1)
+    # box planes
+    plane_verts = get_plane_verts(box)  # (6, 4, 3)
+    v0, v1, v2, v3 = plane_verts.unbind(1)
+    plane_ctr, n = get_plane_center_normal(plane_verts)
 
     # Check all verts are coplanar
-    if not ((v3 - v0).unsqueeze(1).bmm(n.unsqueeze(2)).abs() < eps).all().item():
+    if (
+        not (
+            F.normalize(v3 - v0, dim=-1).unsqueeze(1).bmm(n.unsqueeze(2)).abs()
+            < dot_eps
+        )
+        .all()
+        .item()
+    ):
         msg = "Plane vertices are not coplanar"
         raise ValueError(msg)
 
     # Check all faces have non zero area
     area1 = torch.cross(v1 - v0, v2 - v0, dim=-1).norm(dim=-1) / 2
     area2 = torch.cross(v3 - v0, v2 - v0, dim=-1).norm(dim=-1) / 2
-    if (area1 < eps).any().item() or (area2 < eps).any().item():
+    if (area1 < area_eps).any().item() or (area2 < area_eps).any().item():
         msg = "Planes have zero areas"
         raise ValueError(msg)
 
-    # We can write:  `ctr = v0 + a * e0 + b * e1 + c * n`, (1).
+    # We can write:  `box_ctr = plane_ctr + a * e0 + b * e1 + c * n`, (1).
     # With <e0, n> = 0 and <e1, n> = 0, where <.,.> refers to the dot product,
     # since that e0 is orthogonal to n. Same for e1.
     """
@@ -768,16 +1049,17 @@ def box_planar_dir(box: torch.Tensor, eps: float = 1e-4) -> torch.Tensor:
     B = torch.ones((numF, 2), dtype=torch.float32, device=device)
     A[:, 0, 1] = (e0 * e1).sum(-1)
     A[:, 1, 0] = (e0 * e1).sum(-1)
-    B[:, 0] = ((ctr - v0) * e0).sum(-1)
-    B[:, 1] = ((ctr - v1) * e1).sum(-1)
+    B[:, 0] = ((box_ctr - plane_ctr) * e0).sum(-1)
+    B[:, 1] = ((box_ctr - plane_ctr) * e1).sum(-1)
     ab = torch.linalg.solve(A, B)  # (numF, 2)
     a, b = ab.unbind(1)
     # solving for c
-    c = ((ctr - v0 - a.view(numF, 1) * e0 - b.view(numF, 1) * e1) * n).sum(-1)
+    c = ((box_ctr - plane_ctr - a.view(numF, 1) * e0 - b.view(numF, 1) * e1) * n).sum(-1)
     """
     # Since we know that <e0, n> = 0 and <e1, n> = 0 (e0 and e1 are orthogonal to n),
     # the above solution is equivalent to
-    c = ((ctr - v0) * n).sum(-1)
+    direc = F.normalize(box_ctr - plane_ctr, dim=-1)  # (6, 3)
+    c = (direc * n).sum(-1)
     # If c is negative, then we revert the direction of n such that n points "inside"
     negc = c < 0.0
     n[negc] *= -1.0
@@ -848,7 +1130,7 @@ def box_volume(box: torch.Tensor) -> torch.Tensor:
     return vols
 
 
-def coplanar_tri_faces(tri1: torch.Tensor, tri2: torch.Tensor, eps: float = EPS):
+def coplanar_tri_faces(tri1: torch.Tensor, tri2: torch.Tensor, eps: float = DOT_EPS):
     """
     Determines whether two triangle faces in 3D are coplanar
     Args:
@@ -857,17 +1139,49 @@ def coplanar_tri_faces(tri1: torch.Tensor, tri2: torch.Tensor, eps: float = EPS)
     Returns:
         is_coplanar: bool
     """
-    v0, v1, v2 = tri1.unbind(0)
-    e0 = F.normalize(v1 - v0, dim=0)
-    e1 = F.normalize(v2 - v0, dim=0)
-    e2 = F.normalize(torch.cross(e0, e1), dim=0)
+    tri1_ctr, tri1_n = get_tri_center_normal(tri1)
+    tri2_ctr, tri2_n = get_tri_center_normal(tri2)
+
+    check1 = tri1_n.dot(tri2_n).abs() > 1 - eps  # checks if parallel
+
+    dist12 = torch.norm(tri1.unsqueeze(1) - tri2.unsqueeze(0), dim=-1)
+    dist12_argmax = dist12.argmax()
+    i1 = dist12_argmax // 3
+    i2 = dist12_argmax % 3
+    assert dist12[i1, i2] == dist12.max()
 
-    coplanar2 = torch.zeros((3,), dtype=torch.bool, device=tri1.device)
-    for i in range(3):
-        if (tri2[i] - v0).dot(e2).abs() < eps:
-            coplanar2[i] = 1
-    coplanar2 = coplanar2.all()
-    return coplanar2
+    check2 = (
+        F.normalize(tri1[i1] - tri2[i2], dim=0).dot(tri1_n).abs() < eps
+    ) or F.normalize(tri1[i1] - tri2[i2], dim=0).dot(tri2_n).abs() < eps
+
+    return check1 and check2
+
+
+def coplanar_tri_plane(
+    tri: torch.Tensor, plane: torch.Tensor, n: torch.Tensor, eps: float = DOT_EPS
+):
+    """
+    Determines whether two triangle faces in 3D are coplanar
+    Args:
+            tri: tensor of shape (3, 3) of the vertices of the triangle
+            plane: tensor of shape (4, 3) of the vertices of the plane
+            n: tensor of shape (3,) of the unit "inside" direction on the plane
+    Returns:
+            is_coplanar: bool
+    """
+    tri_ctr, tri_n = get_tri_center_normal(tri)
+
+    check1 = tri_n.dot(n).abs() > 1 - eps  # checks if parallel
+
+    dist12 = torch.norm(tri.unsqueeze(1) - plane.unsqueeze(0), dim=-1)
+    dist12_argmax = dist12.argmax()
+    i1 = dist12_argmax // 4
+    i2 = dist12_argmax % 4
+    assert dist12[i1, i2] == dist12.max()
+
+    check2 = F.normalize(tri[i1] - plane[i2], dim=0).dot(n).abs() < eps
+
+    return check1 and check2
 
 
 def is_inside(
@@ -875,7 +1189,6 @@ def is_inside(
     n: torch.Tensor,
     points: torch.Tensor,
     return_proj: bool = True,
-    eps: float = EPS,
 ):
     """
     Computes whether point is "inside" the plane.
@@ -900,12 +1213,13 @@ def is_inside(
       p_proj: tensor of shape (P, 3) of the projected point on plane
     """
     device = plane.device
-    v0, v1, v2, v3 = plane
-    e0 = F.normalize(v1 - v0, dim=0)
-    e1 = F.normalize(v2 - v0, dim=0)
-    if not torch.allclose(e0.dot(n), torch.zeros((1,), device=device), atol=1e-6):
+    v0, v1, v2, v3 = plane.unbind(0)
+    plane_ctr = plane.mean(0)
+    e0 = F.normalize(v0 - plane_ctr, dim=0)
+    e1 = F.normalize(v1 - plane_ctr, dim=0)
+    if not torch.allclose(e0.dot(n), torch.zeros((1,), device=device), atol=1e-2):
         raise ValueError("Input n is not perpendicular to the plane")
-    if not torch.allclose(e1.dot(n), torch.zeros((1,), device=device), atol=1e-6):
+    if not torch.allclose(e1.dot(n), torch.zeros((1,), device=device), atol=1e-2):
         raise ValueError("Input n is not perpendicular to the plane")
 
     add_dim = False
@@ -914,7 +1228,7 @@ def is_inside(
         add_dim = True
 
     assert points.shape[1] == 3
-    # Every point p can be written as p = v0 + a e0 + b e1 + c n
+    # Every point p can be written as p = ctr + a e0 + b e1 + c n
 
     # If return_proj is True, we need to solve for (a, b)
     p_proj = None
@@ -924,16 +1238,17 @@ def is_inside(
             [[1.0, e0.dot(e1)], [e0.dot(e1), 1.0]], dtype=torch.float32, device=device
         )
         B = torch.zeros((2, points.shape[0]), dtype=torch.float32, device=device)
-        B[0, :] = torch.sum((points - v0.view(1, 3)) * e0.view(1, 3), dim=-1)
-        B[1, :] = torch.sum((points - v0.view(1, 3)) * e1.view(1, 3), dim=-1)
-
+        B[0, :] = torch.sum((points - plane_ctr.view(1, 3)) * e0.view(1, 3), dim=-1)
+        B[1, :] = torch.sum((points - plane_ctr.view(1, 3)) * e1.view(1, 3), dim=-1)
         ab = A.inverse() @ B  # (2, P)
-        p_proj = v0.view(1, 3) + ab.transpose(0, 1) @ torch.stack((e0, e1), dim=0)
+        p_proj = plane_ctr.view(1, 3) + ab.transpose(0, 1) @ torch.stack(
+            (e0, e1), dim=0
+        )
 
     # solving for c
-    # c = (point - v0 - a * e0 - b * e1).dot(n)
-    c = torch.sum((points - v0.view(1, 3)) * n.view(1, 3), dim=-1)
-    ins = c > -eps
+    # c = (point - ctr - a * e0 - b * e1).dot(n)
+    direc = torch.sum((points - plane_ctr.view(1, 3)) * n.view(1, 3), dim=-1)
+    ins = direc >= 0.0
 
     if add_dim:
         assert p_proj.shape[0] == 1
@@ -942,7 +1257,7 @@ def is_inside(
     return ins, p_proj
 
 
-def plane_edge_point_of_intersection(plane, n, p0, p1):
+def plane_edge_point_of_intersection(plane, n, p0, p1, eps: float = DOT_EPS):
     """
     Finds the point of intersection between a box plane and
     a line segment connecting (p0, p1).
@@ -961,11 +1276,17 @@ def plane_edge_point_of_intersection(plane, n, p0, p1):
     # The point of intersection can be parametrized
     # p = p0 + a (p1 - p0) where a in [0, 1]
     # We want to find a such that p is on plane
-    # <p - v0, n> = 0
-    v0, v1, v2, v3 = plane
-    a = -(p0 - v0).dot(n) / (p1 - p0).dot(n)
-    p = p0 + a * (p1 - p0)
-    return p, a
+    # <p - ctr, n> = 0
+
+    # if segment (p0, p1) is parallel to plane (it can only be on it)
+    direc = F.normalize(p1 - p0, dim=0)
+    if direc.dot(n).abs() < eps:
+        return (p1 + p0) / 2.0, 0.5
+    else:
+        ctr = plane.mean(0)
+        a = -(p0 - ctr).dot(n) / ((p1 - p0).dot(n))
+        p = p0 + a * (p1 - p0)
+        return p, a
 
 
 """
@@ -983,7 +1304,6 @@ def clip_tri_by_plane_oneout(
     vout: torch.Tensor,
     vin1: torch.Tensor,
     vin2: torch.Tensor,
-    eps: float = EPS,
 ) -> Tuple[torch.Tensor, torch.Tensor]:
     """
     Case (a).
@@ -1004,10 +1324,10 @@ def clip_tri_by_plane_oneout(
     device = plane.device
     # point of intersection between plane and (vin1, vout)
     pint1, a1 = plane_edge_point_of_intersection(plane, n, vin1, vout)
-    assert a1 >= -eps and a1 <= 1.0 + eps, a1
+    assert a1 >= -0.0001 and a1 <= 1.0001, a1
     # point of intersection between plane and (vin2, vout)
     pint2, a2 = plane_edge_point_of_intersection(plane, n, vin2, vout)
-    assert a2 >= -eps and a2 <= 1.0 + eps, a2
+    assert a2 >= -0.0001 and a2 <= 1.0001, a2
 
     verts = torch.stack((vin1, pint1, pint2, vin2), dim=0)  # 4x3
     faces = torch.tensor(
@@ -1022,7 +1342,6 @@ def clip_tri_by_plane_twoout(
     vout1: torch.Tensor,
     vout2: torch.Tensor,
     vin: torch.Tensor,
-    eps: float = EPS,
 ) -> Tuple[torch.Tensor, torch.Tensor]:
     """
     Case (b).
@@ -1042,10 +1361,10 @@ def clip_tri_by_plane_twoout(
     device = plane.device
     # point of intersection between plane and (vin, vout1)
     pint1, a1 = plane_edge_point_of_intersection(plane, n, vin, vout1)
-    assert a1 >= -eps and a1 <= 1.0 + eps, a1
+    assert a1 >= -0.0001 and a1 <= 1.0001, a1
     # point of intersection between plane and (vin, vout2)
     pint2, a2 = plane_edge_point_of_intersection(plane, n, vin, vout2)
-    assert a2 >= -eps and a2 <= 1.0 + eps, a2
+    assert a2 >= -0.0001 and a2 <= 1.0001, a2
 
     verts = torch.stack((vin, pint1, pint2), dim=0)  # 3x3
     faces = torch.tensor(
@@ -1071,6 +1390,9 @@ def clip_tri_by_plane(plane, n, tri_verts) -> Union[List, torch.Tensor]:
         tri_verts: tensor of shape (K, 3, 3) of the vertex coordinates of the triangles formed
             after clipping. All K triangles are now "inside" the plane.
     """
+    if coplanar_tri_plane(tri_verts, plane, n):
+        return tri_verts.view(1, 3, 3)
+
     v0, v1, v2 = tri_verts.unbind(0)
     isin0, _ = is_inside(plane, n, v0)
     isin1, _ = is_inside(plane, n, v1)
@@ -1191,7 +1513,7 @@ def box3d_overlap_naive(box1: torch.Tensor, box2: torch.Tensor):
         for i2 in range(tri_verts2.shape[0]):
             if (
                 coplanar_tri_faces(tri_verts1[i1], tri_verts2[i2])
-                and tri_verts_area(tri_verts1[i1]) > 1e-4
+                and tri_verts_area(tri_verts1[i1]) > AREA_EPS
             ):
                 keep2[i2] = 0
     keep2 = keep2.nonzero()[:, 0]