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				****
				Math
				****

				.. For writing style guide, see: :doc:`/about/contribute/style_guides/writing_guide`
				If you add new entries, keep the alphabetical sorting!

				.. glossary::

				Interpolation
				Method of calculating new data between points of known value, like :term:`keyframes <keyframe>`.

				Boolean
				A type of logic dealing with binary true/false states.

				See also :doc:`Boolean Modifier </modeling/modifiers/generate/booleans>`.


				Clamp
				Clamping
				Limits a variable to a range. The values over or under the range are set
				to the constant values of the ranges minimum or maximum.


				Geometry
				========

				.. glossary::

				Axis
				A reference line which defines coordinates along one cardinal direction in n-D space.

				Axis Angle
				Rotation method where X, Y, and Z correspond to the axis definition,
				while W corresponds to the angle around that axis, in radians.

				Bevel
				The operation to chamfer or bevel edges of an object.

				Bounding Box
				The box that encloses the shape of an object. The box is aligned with the local space of the object.

				Concave face
				Face in which one vertex is inside a triangle formed by other vertices of the face.

				Convex face
				Face where, if lines were drawn from each vertex to every other vertex, all lines would remain in the face.
				Opposite of a :term:`concave face`.

				Coplanar
				Refers to any set of elements that are all aligned to the same 2D plane in 3D space.

				Edge
				Straight segment (line) that connects two :term:`vertices <vertex>`, and can be part of a :term:`face`.

				Edge Loop
				Chain of :term:`edges <edge>` belonging to consecutive :term:`quads <quad>`.
				An edge loop ends at a pole or a boundary. Otherwise, it is cyclic.

				Edge Ring
				Path of all :term:`edges <edge>` along a :term:`face loop` that share two faces belonging to that loop.

				Euler
				Euler Rotation
				Rotation method where rotations applied on each X, Y, Z axis component.

				Face
				Mesh element that defines a piece of surface. It consists of three or more :term:`edges <edge>`.

				Face Loop
				Chain of consecutive :term:`quads <quad>`.
				A face loop stops at a :term:`triangle` or :term:`N-gon` (which do not belong to the loop), or at a boundary.
				Otherwise, it is cyclic.

				Face Normal
				The normalized vector perpendicular to the plane that a :term:`face` lies in.
				Each face has its own normal.

				Geometric Center
				The mean average of the positions of all vertices making up the object.

				Gimbal
				A pivoted support that allows the rotation of an object about a single axis.

				See also `Gimbal <https://en.wikipedia.org/wiki/Gimbal>`__ on Wikipedia.

				Gimbal Lock
				The limitation where axes of rotation can become aligned,
				losing the ability to rotate on an axis (typically associated with :term:`euler rotation`).

				- See also `Gimbal lock <https://en.wikipedia.org/wiki/Gimbal_lock>`__ on Wikipedia.
				- See also `Gimbal lock <https://blender.stackexchange.com/questions/469>`__ on Stackexchange.

				Global Space
				See :term:`World Space`.

				Local Space
				A 3D coordinate system that originates (for Objects) at the :term:`Object Origin`.
				or (for Bones) at the :term:`Head` of the :term:`Bone`.

				Compare to :term:`World Space`.

				Manifold
				Manifold meshes, also called water tight meshes,
				define a closed non-self-intersecting volume (see also :term:`non-manifold`).
				A manifold mesh is a mesh in which the structure of the connected
				faces in a closed volume will always point the normals (and there
				surfaces) to the outside or to the inside of the mesh without any overlaps.
				If you recalculate those normals, they will always point at
				a predictable direction (To the outside or to the inside of the volume).
				When working with non-closed volumes, a manifold mesh is a mesh in which
				the normals will always define two different and non-consecutive surfaces.
				A manifold mesh will always define an even number of non-overlapped surfaces.

				Mesh
				Type of object consisting of :term:`vertices <vertex>`, :term:`edges <edge>` and :term:`faces <face>`.

				Micropolygons
				A polygon roughly the size of a pixel or smaller.

				N-gon
				A :term:`face` that contains more than four :term:`vertices <vertex>`.

				Non-manifold
				Non-Manifold meshes essentially define geometry which cannot exist in the real world.
				This kind of geometry is not suitable for several types of operations,
				especially those where knowing the volume (inside/outside) of the object is important
				(refraction, fluids, booleans, or 3D printing, to name a few).
				A non-manifold mesh is a mesh in which the structure of a
				non-overlapped surface (based on its connected faces) will not determine
				the inside or the outside of a volume based on its normals, defining
				a single surface for both sides, but ended with flipped normals.
				When working with non-closed volumes, a non-manifold mesh will always
				determine at least one discontinuity in the normal directions, either
				by an inversion of a connected loop, or by an odd number of surfaces.
				A non-manifold mesh will always define an odd number of surfaces.

				There are several types of non-manifold geometry:

				- Some borders and holes (edges with only a single connected face), as faces have no thickness.
				- Edges and vertices not belonging to any face (wire).
				- Edges connected to three or more faces (interior faces).
				- Vertices belonging to faces that are not adjoining (e.g. two cones sharing the vertex at the apex).

				See also: :ref:`Select Non-Manifold <mesh-select-non-manifold>` tool.

				Normal
				The normalized vector perpendicular to a surface.

				Normals can be assigned to vertices,
				faces and modulated across a surface using :term:`normal mapping`.

				Object Center
				Object Origin
				A reference point used to position, rotate, and scale an :term:`Object`
				and to define its :term:`Local Space` coordinates.

				Octahedron
				An eight-sided figure commonly used to depict the :term:`Bones <Bone>` of an :term:`Armature`.

				Pivot Point
				The pivot point is the point in space around which all rotations,
				scalings and mirror transformations are centered.

				See also the :doc:`Pivot Point </editors/3dview/object/transform/control/pivot_point/index>` docs.

				Pole
				:term:`Vertex` where three, five, or more edges meet.
				A vertex connected to one, two, or four edges is not a pole.

				Quad
				Quadrilateral
				Quadrangle
				:term:`Face` that contains exactly four :term:`vertices <vertex>`.

				Quaternion
				Quaternion Rotation
				Rotation method where rotations are defined by four values (X, Y, Z and W).
				X, Y, and Z also define an :term:`axis`, and W an angle,
				but it is quite different from :term:`Axis Angle`.

				Roll
				Roll Angle
				The orientation of the local X and Z axes of a :term:`Bone`.
				Has no effect on the local Y axis as local Y is determined by the location of the :term:`Head` and :term:`Tail`.

				Smoothing
				Defines how :term:`faces <face>` are shaded. Face can be either solid (faces are rendered flat)
				or smooth (faces are smoothed by interpolating the normal on every point of the face).

				Subdividing
				Technique for adding more geometry to a mesh.
				It creates new vertices on subdivided edges, new edges between subdivisions and new faces based on new edges.
				If new edges cross a new vertex is created at their crossing point.

				Subsurf
				Subdivision Surface
				A method of creating smooth higher poly surfaces which can take a low polygon mesh as input.

				Sometimes abbreviated to Subsurf.

				See also
				`Catmull-Clark subdivision surface <https://en.wikipedia.org/wiki/Catmull%E2%80%93Clark_subdivision_surface>`__
				on Wikipedia.

				Tessellation
				The tiling of a plane using one or more geometric shapes usually resulting in :term:`Micropolygons`.

				Topology
				The arrangement of Vertices, Edges, and Faces which define the shape of a mesh.
				See :term:`vertex`, :term:`edge`, and :term:`face`.

				Transforms
				The combined idea of location, rotation and scale.

				Triangle
				:term:`Face` with exactly three :term:`vertices <vertex>`.

				Vertex
				Vertices
				A point in 3D space containing a location. It may also have a defined color.
				Vertices are the terminating points of :term:`edges <edge>`.

				World Space
				A 3D coordinate system that originates at a point at the origin of the world.
				Compare to :term:`Local Space`.