diff --git a/src/OpenFOAM/meshes/meshShapes/edge/edge.H b/src/OpenFOAM/meshes/meshShapes/edge/edge.H
index 5557ce4045..c479af2217 100644
--- a/src/OpenFOAM/meshes/meshShapes/edge/edge.H
+++ b/src/OpenFOAM/meshes/meshShapes/edge/edge.H
@@ -28,13 +28,13 @@ Class
     Foam::edge
 
 Description
-    An edge is a list of two point labels. The functionality it provides
-    supports the discretisation on a 2-D flat mesh.
+    An edge is a list of two vertex labels.
+    This can correspond to a directed graph edge or an edge on a mesh.
 
     The edge is implemented as a Pair/FixedList of labels.
     As well as geometrically relevant methods, it also provides methods
     similar to HashSet for additional convenience.
-    Valid point labels are always non-negative (since they correspond to
+    Valid vertex labels are always non-negative (eg, since they correspond to
     addressing within the mesh). The value '-1' is used to tag invalid
     point labels that correspond conceptually to open 'slots', which
     can be filled with a HashSet-like functionality.
@@ -69,30 +69,31 @@ public:
 
     // Static Data Members
 
+        //- The typeName ("edge")
         static const char* const typeName;
 
 
     // Constructors
 
-        //- Default construct, with invalid point labels (-1)
+        //- Default construct, with invalid vertex labels (-1)
         inline edge();
 
-        //- Construct from two point labels
+        //- Construct from two vertex labels
         inline edge(const label from, const label to);
 
-        //- Construct from pair of point labels
+        //- Construct from pair of vertex labels
         inline edge(const labelPair& pair);
 
-        //- Construct from list of point labels
+        //- Construct from list of vertex labels
         inline edge(const FixedList<label, 2>& list);
 
-        //- Construct from two point labels, sorted with first less-than second
+        //- Construct from two vertex labels, sorted with first less-than second
         inline edge(const label from, const label to, const bool doSort);
 
         //- Construct from list, sorted with first less-than second
         inline edge(const FixedList<label, 2>& list, const bool doSort);
 
-        //- Copy construct from a subset of point labels
+        //- Copy construct from a subset of vertex labels
         inline edge
         (
             const UList<label>& list,
@@ -133,53 +134,50 @@ public:
 
 
         //- Return reverse edge as copy.
-        //  No special handling of negative point labels.
+        //  No special handling of negative vertex labels.
         inline edge reverseEdge() const;
 
 
     // Queries
 
-        //- Return the smallest point label used by the edge
-        //  No special handling of negative point labels.
-        inline label minVertex() const;
+        //- Return the smallest vertex label used by the edge
+        //  No special handling of negative vertex labels.
+        inline label minVertex() const noexcept;
 
-        //- Return the largest point label used by the edge
-        //  No special handling of negative point labels.
-        inline label maxVertex() const;
+        //- Return the largest vertex label used by the edge
+        //  No special handling of negative vertex labels.
+        inline label maxVertex() const noexcept;
 
         //- True if the vertices are unique and non-negative.
         inline bool good() const noexcept;
 
-        //- True if the vertices are unique and non-negative.
-        bool valid() const noexcept { return good(); }
+        //- Return true if the vertex label is contained in the edge.
+        //  Always false for a negative vertex label.
+        inline bool contains(const label vertex) const noexcept;
 
-        //- Return true if the point label is contained in the edge.
-        //  Always false for a negative label.
-        inline bool contains(const label pointLabel) const;
-
-        //- Return local index (0,1) of point label in edge -1 on failure
-        //  Always return -1 for a negative label.
-        inline label which(const label pointLabel) const;
+        //- Return local index (0,1) of vertex label in edge -1 on failure
+        //  Always return -1 for a negative vertex label.
+        inline label which(const label vertex) const;
 
         //- True if the edge has at least one vertex in common with other
         inline bool connected(const edge& other) const;
 
         //- Return vertex common with other edge or -1 on failure
-        //  Negative point labels are never considered common between edges.
+        //  Negative vertex labels are never considered common between edges.
         inline label commonVertex(const edge& other) const;
 
-        //- Given the point label for one vertex, return the other one.
-        //  No special treatment for negative point labels.
-        inline label otherVertex(const label pointLabel) const;
+        //- Given one vertex label, return the other one.
+        //  No special treatment for negative vertex labels.
+        inline label otherVertex(const label vertex) const;
 
         //- Do the edges share a common vertex index?
-        //  Negative point labels never connect.
+        //  Negative vertex labels never connect.
         bool connects(const edge& other) const { return connected(other); }
 
 
     // Editing
 
-        //- 'Collapse' edge by marking duplicate point labels as '-1',
+        //- 'Collapse' edge by marking duplicate vertex labels as '-1',
         //- the lower vertex is retained.
         //  Return the effective size after collapsing.
         inline label collapse();
@@ -187,21 +185,22 @@ public:
 
     // Hash-like Functions
 
-        //- Return the number of unique, valid (non -1) point labels.
+        //- Return the number of unique, valid (non -1) vertex labels.
         //  Similar to a HashTable::size().
         inline label count() const;
 
-        //- Return true if edge has no valid point labels.
-        inline bool empty() const;
+        //- Return true if edge has no valid vertex labels.
+        inline bool empty() const noexcept;
 
-        //- 'Clears' edge by setting both ends to invalid point labels.
+        //- 'Clears' edge by setting both ends to invalid vertex labels.
         inline void clear();
 
 
-        //- Fill any open slot with the index if it did not previously exist.
-        //  Returns true on success. A negative label never inserts.
+        //- Fill any open slot with the vertex label
+        //- (if not previously contained in the edge).
+        //  Returns true on success. A negative vertex label never inserts.
         //  Similar to a HashTable::insert().
-        inline bool insert(const label index);
+        inline bool insert(const label vertex);
 
         //- Insert values, using begin/end iterators.
         template<class InputIterator>
@@ -227,10 +226,11 @@ public:
         inline label insert(const labelUList& list);
 
 
-        //- Remove an existing index from the edge and set its location to '-1'.
-        //  Returns the number of changes. A negative label never removes.
+        //- Remove an existing vertex from the edge and set its location
+        //- to '-1'. A negative vertex label never removes.
+        //  Returns the number of changes.
         //  Similar to a HashTable::erase().
-        inline label erase(const label index);
+        inline label erase(const label vertex);
 
         //- Remove values, using begin/end iterators.
         template<class InputIterator>
@@ -253,26 +253,30 @@ public:
     // Geometric Functions
 
         //- Return centre point (centroid) of the edge.
-        //  No special handling of negative point labels.
+        //  No special handling of negative vertex labels.
         inline point centre(const UList<point>& pts) const;
 
-        //- Return the vector (end - start)
-        //  No special handling of negative point labels.
+        //- Return the vector (from first to second).
+        //  No special handling of negative vertex labels.
         inline vector vec(const UList<point>& pts) const;
 
-        //- Return the unit vector (end - start)
-        //  No special handling of negative point labels.
+        //- Return the unit vector (from first to second).
+        //  No special handling of negative vertex labels.
         inline vector unitVec(const UList<point>& pts) const;
 
-        //- Return scalar magnitude of the edge.
-        //  No special handling of negative point labels.
+        //- The length (L2-norm) of the edge vector.
+        //  No special handling of negative vertex labels.
         inline scalar mag(const UList<point>& pts) const;
 
+        //- The length (L2-norm) squared of the edge vector.
+        //  No special handling of negative vertex labels.
+        inline scalar magSqr(const UList<point>& pts) const;
+
         //- The enclosing (bounding) box for the edge
         inline Pair<point> box(const UList<point>& pts) const;
 
         //- Return edge line
-        //  No special handling of negative point labels.
+        //  No special handling of negative vertex labels.
         inline linePointRef line(const UList<point>& pts) const;
 
 
@@ -327,6 +331,9 @@ public:
         //- Same as contains()
         bool found(label vertex) const { return contains(vertex); }
 
+        //- Same as good()
+        bool valid() const noexcept { return good(); }
+
         //- Deprecated(2021-04) hashing functor. Use hasher()
         // \deprecated(2021-04) - use hasher() functor
         template<class Unused=bool>
diff --git a/src/OpenFOAM/meshes/meshShapes/edge/edgeI.H b/src/OpenFOAM/meshes/meshShapes/edge/edgeI.H
index 879a413de1..a3b6c7f4e3 100644
--- a/src/OpenFOAM/meshes/meshShapes/edge/edgeI.H
+++ b/src/OpenFOAM/meshes/meshShapes/edge/edgeI.H
@@ -92,15 +92,15 @@ inline Foam::edge::edge(Istream& is)
 
 // * * * * * * * * * * * * * * * Member Functions  * * * * * * * * * * * * * //
 
-inline Foam::label Foam::edge::minVertex() const
+inline Foam::label Foam::edge::minVertex() const noexcept
 {
     return (first() < second() ? first() : second());
 }
 
 
-inline Foam::label Foam::edge::maxVertex() const
+inline Foam::label Foam::edge::maxVertex() const noexcept
 {
-    return (first() > second() ? first() : second());
+    return (second() < first() ? first() : second());
 }
 
 
@@ -110,27 +110,27 @@ inline bool Foam::edge::good() const noexcept
 }
 
 
-inline bool Foam::edge::contains(const label pointLabel) const
+inline bool Foam::edge::contains(const label vertex) const noexcept
 {
     // -1: always false
     return
     (
-        pointLabel >= 0
-     && (pointLabel == first() || pointLabel == second())
+        vertex >= 0
+     && (vertex == first() || vertex == second())
     );
 }
 
 
-inline Foam::label Foam::edge::which(const label pointLabel) const
+inline Foam::label Foam::edge::which(const label vertex) const
 {
     // -1: always false
-    if (pointLabel >= 0)
+    if (vertex >= 0)
     {
-        if (pointLabel == first())
+        if (vertex == first())
         {
             return 0;
         }
-        if (pointLabel == second())
+        if (vertex == second())
         {
             return 1;
         }
@@ -162,13 +162,13 @@ inline Foam::label Foam::edge::commonVertex(const edge& other) const
 }
 
 
-inline Foam::label Foam::edge::otherVertex(const label pointLabel) const
+inline Foam::label Foam::edge::otherVertex(const label vertex) const
 {
-    if (pointLabel == first())
+    if (vertex == first())
     {
         return second();
     }
-    if (pointLabel == second())
+    if (vertex == second())
     {
         return first();
     }
@@ -228,35 +228,35 @@ inline Foam::label Foam::edge::count() const
 }
 
 
-inline bool Foam::edge::empty() const
+inline bool Foam::edge::empty() const noexcept
 {
     return (first() < 0 && second() < 0);
 }
 
 
-inline bool Foam::edge::insert(const label index)
+inline bool Foam::edge::insert(const label vertex)
 {
-    if (index < 0)
+    if (vertex < 0)
     {
-        // Cannot insert invalid point labels (use direct assignment for that)
+        // Cannot insert invalid vertex labels (use direct assignment for that)
         return false;
     }
 
     if (first() < 0)
     {
         // Store at first, if not duplicate of second
-        if (index != second())
+        if (vertex != second())
         {
-            first() = index;
+            first() = vertex;
             return true;
         }
     }
     else if (second() < 0)
     {
         // Store at second, if not duplicate of first
-        if (index != first())
+        if (vertex != first())
         {
-            second() = index;
+            second() = vertex;
             return true;
         }
     }
@@ -308,23 +308,23 @@ inline Foam::label Foam::edge::insert(const labelUList& list)
 }
 
 
-inline Foam::label Foam::edge::erase(const label index)
+inline Foam::label Foam::edge::erase(const label vertex)
 {
-    if (index < 0)
+    if (vertex < 0)
     {
         // Can never remove invalid point labels!
         return 0;
     }
 
     label n = 0;
-    if (index == first())
+    if (vertex == first())
     {
         first() = -1;
         ++n;
     }
 
     // Automatically handle duplicates, which should not have been there anyhow
-    if (index == second())
+    if (vertex == second())
     {
         second() = -1;
         ++n;
@@ -426,7 +426,13 @@ inline Foam::vector Foam::edge::unitVec(const UList<point>& pts) const
 
 inline Foam::scalar Foam::edge::mag(const UList<point>& pts) const
 {
-    return ::Foam::mag(vec(pts));
+    return pts[first()].dist(pts[second()]);
+}
+
+
+inline Foam::scalar Foam::edge::magSqr(const UList<point>& pts) const
+{
+    return pts[first()].distSqr(pts[second()]);
 }
 
 
diff --git a/src/OpenFOAM/meshes/meshShapes/face/face.C b/src/OpenFOAM/meshes/meshShapes/face/face.C
index bf646864a7..4e4d745a63 100644
--- a/src/OpenFOAM/meshes/meshShapes/face/face.C
+++ b/src/OpenFOAM/meshes/meshShapes/face/face.C
@@ -776,8 +776,8 @@ Foam::edgeList Foam::face::edges() const
     edgeList theEdges(nVerts);
 
     // Last edge closes the polygon
-    theEdges.last().first() = verts.last();
-    theEdges.last().second() = verts[0];
+    theEdges.back().first() = verts.back();
+    theEdges.back().second() = verts.front();
 
     for (label verti = 0; verti < nVerts - 1; ++verti)
     {
@@ -797,8 +797,8 @@ Foam::edgeList Foam::face::rcEdges() const
     edgeList theEdges(nVerts);
 
     // First edge closes the polygon
-    theEdges.first().first() = verts[0];
-    theEdges.first().second() = verts.last();
+    theEdges.front().first() = verts.front();
+    theEdges.front().second() = verts.back();
 
     for (label verti = 1; verti < nVerts; ++verti)
     {
@@ -878,12 +878,12 @@ Foam::label Foam::face::longestEdge(const UList<point>& pts) const
 
     // Last edge closes the polygon. Use it to initialize loop
     label longest = nVerts - 1;
-    scalar longestLen = Foam::edge(verts.first(), verts.last()).mag(pts);
+    scalar longestLen = pts[verts.front()].distSqr(pts[verts.back()]);
 
     // Examine other edges
     for (label edgei = 0; edgei < nVerts - 1; ++edgei)
     {
-        scalar edgeLen = Foam::edge(verts[edgei], verts[edgei + 1]).mag(pts);
+        scalar edgeLen = pts[verts[edgei]].distSqr(pts[verts[edgei+1]]);
 
         if (longestLen < edgeLen)
         {
diff --git a/src/OpenFOAM/meshes/meshShapes/face/face.H b/src/OpenFOAM/meshes/meshShapes/face/face.H
index 3257afc7f6..96167660c0 100644
--- a/src/OpenFOAM/meshes/meshShapes/face/face.H
+++ b/src/OpenFOAM/meshes/meshShapes/face/face.H
@@ -136,26 +136,26 @@ public:
         //- Default construct
         constexpr face() noexcept = default;
 
-        //- Construct given size, with invalid point labels (-1)
+        //- Construct given size, with invalid vertex labels (-1)
         inline explicit face(const label sz);
 
-        //- Copy construct from list of point labels
+        //- Copy construct from list of vertex labels
         inline explicit face(const labelUList& list);
 
-        //- Move construct from list of point labels
+        //- Move construct from list of vertex labels
         inline explicit face(labelList&& list);
 
-        //- Copy construct from an initializer list of point labels
+        //- Copy construct from an initializer list of vertex labels
         inline explicit face(std::initializer_list<label> list);
 
-        //- Copy construct from list of point labels
+        //- Copy construct from list of vertex labels
         template<unsigned N>
         inline explicit face(const FixedList<label, N>& list);
 
-        //- Copy construct from subset of point labels
+        //- Copy construct from subset of vertex labels
         inline face(const labelUList& list, const labelUList& indices);
 
-        //- Copy construct from subset of point labels
+        //- Copy construct from subset of vertex labels
         template<unsigned N>
         inline face(const labelUList& list, const FixedList<label, N>& indices);
 
@@ -168,12 +168,12 @@ public:
 
     // Member Functions
 
-        //- Collapse face by removing duplicate point labels
+        //- Collapse face by removing duplicate vertex labels
         //  \return the collapsed size
         label collapse();
 
         //- Flip the face in-place.
-        //  The starting points of the original and flipped face are identical.
+        //  The starting vertex of the original and flipped face are identical.
         void flip();
 
         //- Return the points corresponding to this face
@@ -211,15 +211,15 @@ public:
         inline Pair<point> box(const UList<point>& pts) const;
 
         //- Return face with reverse direction
-        //  The starting points of the original and reverse face are identical.
+        //  The starting vertex of the original and reverse face are identical.
         face reverseFace() const;
 
 
         // Navigation through face vertices
 
-        //- Find local index on face for the point label, same as find()
+        //- Find local vertex on face for the vertex label, same as find()
         //  \return position in face (0,1,2,...) or -1 if not found.
-        inline label which(const label pointLabel) const;
+        inline label which(const label vertex) const;
 
         //- The vertex on face - identical to operator[], but with naming
         //- similar to nextLabel(), prevLabel()
diff --git a/src/OpenFOAM/meshes/meshShapes/face/faceI.H b/src/OpenFOAM/meshes/meshShapes/face/faceI.H
index fee4a2049c..c54c33f3cd 100644
--- a/src/OpenFOAM/meshes/meshShapes/face/faceI.H
+++ b/src/OpenFOAM/meshes/meshShapes/face/faceI.H
@@ -175,9 +175,9 @@ inline Foam::vector Foam::face::rcEdge
 }
 
 
-inline Foam::label Foam::face::which(const label pointLabel) const
+inline Foam::label Foam::face::which(const label vertex) const
 {
-    return labelList::find(pointLabel);
+    return labelList::find(vertex);
 }
 
 
diff --git a/src/OpenFOAM/meshes/meshShapes/face/faceIntersection.C b/src/OpenFOAM/meshes/meshShapes/face/faceIntersection.C
index 624a319e7e..1213813dcf 100644
--- a/src/OpenFOAM/meshes/meshShapes/face/faceIntersection.C
+++ b/src/OpenFOAM/meshes/meshShapes/face/faceIntersection.C
@@ -107,12 +107,7 @@ Foam::pointHit Foam::face::ray
 
                 // Miss distance is the distance between the plane intersection
                 // point and the nearest point of the triangle
-                scalar missDist =
-                    Foam::mag
-                    (
-                        p + curHit.distance()*n
-                      - curHit.point()
-                    );
+                scalar missDist = curHit.point().dist(p + curHit.distance()*n);
 
                 if (missDist < nearestMissDist)
                 {
diff --git a/src/OpenFOAM/meshes/meshShapes/triFace/triFace.H b/src/OpenFOAM/meshes/meshShapes/triFace/triFace.H
index 9fbe3e4905..c3753be895 100644
--- a/src/OpenFOAM/meshes/meshShapes/triFace/triFace.H
+++ b/src/OpenFOAM/meshes/meshShapes/triFace/triFace.H
@@ -84,19 +84,19 @@ public:
 
     // Constructors
 
-        //- Default construct, with invalid point labels (-1)
+        //- Default construct, with invalid vertex labels (-1)
         inline triFace();
 
-        //- Construct from three point labels
+        //- Construct from three vertex labels
         inline triFace(const label p0, const label p1, const label p2) noexcept;
 
-        //- Construct from an initializer list of three point labels
+        //- Construct from an initializer list of three vertex labels
         inline explicit triFace(std::initializer_list<label> list);
 
-        //- Copy construct from a list of three point labels.
+        //- Copy construct from a list of three  vertex labels.
         inline explicit triFace(const labelUList& list);
 
-        //- Copy construct from a subset of point labels
+        //- Copy construct from a subset of vertex labels
         inline triFace
         (
             const labelUList& list,
@@ -135,20 +135,17 @@ public:
         //- True if vertices are unique and non-negative.
         inline bool good() const noexcept;
 
-        //- True if vertices are unique and non-negative.
-        bool valid() const noexcept { return good(); }
-
 
     // Other
 
-        //- 'Collapse' face by marking duplicate point labels.
-        //  Duplicates point labels are marked with '-1'
+        //- 'Collapse' face by marking duplicate vertex labels.
+        //  Duplicates vertex labels are marked with '-1'
         //  (the lower vertex is retained).
         //  Return the collapsed size.
         inline label collapse();
 
         //- Flip the face in-place.
-        //  The starting points of the original and flipped face are identical.
+        //  The starting vertex of the original and flipped face are identical.
         inline void flip();
 
         //- Return the points corresponding to this face
@@ -191,12 +188,12 @@ public:
         inline label nTriangles() const noexcept;
 
         //- Return face with reverse direction
-        //  The starting points of the original and reverse face are identical.
+        //  The starting vertex of the original and reverse face are identical.
         inline triFace reverseFace() const;
 
-        //- Find local index on face for the point label, same as find()
+        //- Find local vertex on face for the vertex label, same as find()
         //  \return position in face (0,1,2) or -1 if not found.
-        inline label which(const label pointLabel) const;
+        inline label which(const label vertex) const;
 
         //- Next vertex on face
         inline label nextLabel(const label i) const;
@@ -378,6 +375,9 @@ public:
 
     // Housekeeping
 
+        //- Same as good()
+        bool valid() const noexcept { return good(); }
+
         //- Identical to edge()
         Foam::edge faceEdge(label edgei) const { return this->edge(edgei); }
 };
diff --git a/src/OpenFOAM/meshes/meshShapes/triFace/triFaceI.H b/src/OpenFOAM/meshes/meshShapes/triFace/triFaceI.H
index ea0698f070..d839354690 100644
--- a/src/OpenFOAM/meshes/meshShapes/triFace/triFaceI.H
+++ b/src/OpenFOAM/meshes/meshShapes/triFace/triFaceI.H
@@ -221,9 +221,9 @@ inline Foam::triFace Foam::triFace::reverseFace() const
 }
 
 
-inline Foam::label Foam::triFace::which(const label pointLabel) const
+inline Foam::label Foam::triFace::which(const label vertex) const
 {
-    return FixedList<label, 3>::find(pointLabel);
+    return FixedList<label, 3>::find(vertex);
 }