Polygon

A polygonal volume is the intersection of one, two, or three polygons, each defined in a coordinate plane and extruded along the unused axis. The corresponding input card is POLYGON.

Volume

Let \(P_{xy}\), \(P_{xz}\), and \(P_{yz}\) be (possibly undefined) polygonal regions in the \(xy\), \(xz\), and \(yz\) planes. The volume is

\[V = E_z(P_{xy}) \;\cap\; E_y(P_{xz}) \;\cap\; E_x(P_{yz}),\]

where \(E_z(P_{xy})\) is the extrusion of \(P_{xy}\) along \(z\) (all \(z\)), and similarly for the other two. If a polygon is not supplied, that extrusion is taken to be all of \(\mathbb{R}^3\). Thus one polygon is an infinite prism; two polygons are the intersection of two prisms; three polygons bound a finite body aligned with the coordinate axes.

A two-vertex list in a given plane is interpreted as an axis-aligned rectangle (the bounding box of the two points). More than two vertices define a simple polygon; they must be ordered clockwise or counter-clockwise and must not repeat the first vertex. At most 100 vertices are stored per plane. The unused coordinate on each vertex is required in the input and is ignored in the in-plane test.

Point in polygon

Membership in a 2-D polygon is the even-odd ray test along \(+x\) in the in-plane coordinates \((x',y')\):

  • \(XY\) uses \((x,y)\)

  • \(XZ\) uses \((x,z)\)

  • \(YZ\) uses \((y,z)\)

An edge from \((x_i,y_i)\) to \((x_j,y_j)\) contributes a crossing when one endpoint is strictly below the query \(y'\) and the other is at or above it, and the intersection abscissa is strictly less than the query \(x'\). Horizontal edges do not contribute. The point is inside if the number of crossings is odd.

A point \(\mathbf{x}=(x,y,z)\) lies in \(V\) if and only if it lies in every defined 2-D polygon (equivalently, in every defined extrusion).

Cell mapping

CELL_CENTERS_IN_VOLUME (default)

A cell is in the region iff its center lies in \(V\). The cell body is not tested. Supported on structured, implicit unstructured, and explicit unstructured grids.

BOUNDARY_FACES_IN_VOLUME

The same point-in-volume test is applied to the centroids of faces on the domain boundary. Implemented for implicit unstructured grids only.

Because each polygon is extruded along a coordinate axis, the intersection of two or three filled polygons is a prismatic body whose generators are axis-aligned. A dipping geologic surface is not a thin sheet in this representation: its coordinate-plane projections, once extruded and intersected, occupy a volume whose thickness is set by the size of those projections, not by a prescribed aperture.