Invalid brush

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In essence an invalid brush is one that cannot exsist within Hammer or within game. The simplest check is to use the one built into Hammer itself, Alt+P will yield an error check that finds most invalid brushes. Invalid brushes will often appear to change between saving and reloading maps due to Hammer's automatic fixing process, this process also causes a shape to appear to change when clipped. The causes of invalid brushes come down to simply three categories; a concave shape, floating vertecies, or improperly defined faces.

Further info

A concave shape means that an object does not have any faces that form an interior angle greater than or equal to 180 degrees. In other words if a line could be drawn through the object that passes through it twice or more then the object is concave. Concave objects need simply to be made into multiple brushes.

Floating vertices are basically any vertex with less that three faces adjacent. Without the three adjacent faces it is impossible for a brush to enclose an area. If the brushes shape is not continuous and has holes then it cannot be defined by the algorithms used in Hammer. Fixing it require merging vertices to cover the holes.

An improperly defined face refers to any face that is not flat or has no surface area. All points on a face should exist in the same two dimensional plane. The plane can be at any rotation, but they need to exist on the same plane. It determines what planes are possible by sampling three sequential points from the face. If one or more vertices don’t line up with the rest it will cause errors. The face ends up unable to determine which two dimensional plane it exists in as sampling different points finds different planes. Fixing it requires the definition of new edges to give each face its own plane, or manipulation of the vertices to line up.

Note:This only applies to quadrilaterals or greater as a faces existence is defined by three points, it is impossible for a triangular face to be improperly defined, hence their preference in complex situations. Faces with two or one side cannot exist via the definition of a polygon.