## Terms and Definitions

**Local coordinate system (LCS)** -- a coordinate system that determines
a 3D object position with respect to the *World Coordinate System*. LCS
is specified by the **origin point** and three vectors -- Z, X, Y axes
that are always orthonormal for C3D Solver tasks. The LCS definition is
related to **Transformation**.

**Geometric object** -- the main subject of the C3D Solver calculations.
It is a 3D object, such as a point, curve, surface, or LCS. C3D Solver
supports a specific set of 3D geometry types listed in section
[Supported geometry types](./Supported_geometry_types.md). Note that
solid body abstraction for C3D Solver is presented as a group of
geometric objects combined in a **Cluster** with a common LCS rather
than a topological structure (C3D Modeler). See section [Geometric scene
clustering, assembly
modeling](./Geometric_scene_clustering_assembly_modeling.md).

**Standard position** -- LCS which has the same origin point and axes as
*the World Coordinate System*.

**Transformation** -- with respect to C3D solver tasks, transformation
of a geometric object using a combination of translations and rotations
only. Mathematically, the transformation is presented in the form of a
square 4x4 matrix -- the [Transformation
Matrix](https://en.wikipedia.org/wiki/Transformation_matrix). It is
convenient to use the transformation matrix for LCS representation,
bearing in mind that any LCS can be the result of transforming from the
*standard position* (see **Standard position**). Note that C3D Solver is
used only for transformations that [retain the
distance](https://en.wikipedia.org/wiki/Isometry) between any points of
a geometric object. It means that C3D Solver does not support any
transformations that distort the orthogonality or normality of LCS
vectors, for example, scaling.

**Rigid set or Cluster** -- a subset of geometric objects within a
common LCS that C3D Solver sees as a geometric object being calculated.
Clusters are used in a geometric model to represent solids and
geometrically rigid subassemblies. Any geometric object of the LCS type
can be a cluster in the Solver.

**Geometric constraint** -- a relation that determines a link between
two, three or more geometric objects that are called *constraint
arguments*. Examples of geometric constraints: \"line L is parallel to
plane P\", \"cylinder C is tangent to plane P\", \"distance between
planes P1 and P2 is 10.0\", etc.

**Constraint argument** -- a geometric object or a numeric parameter
connected by a geometric constraint or dimension.

**Geometric constraint system** -- a set of mutually related geometric
objects. The geometric constraint system formulates a GCSP (geometric
constraint satisfaction problem) the purpose of which is to find
positions of 3D geometric objects that satisfy all the given
constraints.

**Initial approximation** -- coordinate values of all geometric objects
and constraint system variables that the solver considers a state close
to the target solution. The initial approximation generally satisfies
the majority of constraints and is suggested as the starting condition
of GCSP. The initial approximation also defines the solution which C3D
Solver selects for tasks with a non-numerable set of solutions (see
Under-defined constraint systems). C3D Solver may offer different
solutions for the same system of constraints with different initial
approximations.

**Geometric constraint satisfaction problem (GCSP)** -- a task to
transform a set of geometric objects with preliminary evaluations
(Initial approximation) into a position that satisfies all the given
constraints. In addition to the list of objects and constraints, the
GCSP definition also includes the Initial approximation that affects the
solution result.

**Three-dimensional geometric solver** -- a software component that
solves the problem of satisfying constraints for 3D objects. As part of
the C3D Solver module, this component is technically referred to as GCM
(***G**eometric **C**onstraint **M**anager*). Another component, GCE
calculates planar objects (see [Two-dimensional geometric
solver](../../../TWO-DIMENSIONAL_GEOMETRIC_SOLVER/Two-Dimensional_Geometric_Solver_en.md)).

**Well-defined constraint system** -- a geometric constraint system that
has a single possible solution or a finite range of solutions.

**Inconsistent constraint system** -- a constraint system with no
solution. A constraint system that has at least one solution is,
accordingly, referred to as **Consistent**.

**Under-defined constraint systems** -- a geometric constraint system
with infinite solutions.

**Over-defined constraint system** -- a constraint system that includes
constraints which, if removed, do not affect the set of solutions or
transform an inconsistent system into a consistent system. Redefined
constraint systems usually include unnecessary constraints, the removal
of which either does not affect anything or eliminates inconsistencies.

**Dimensional constraint or Dimensions** -- a geometric constraint that
includes a numeric parameter argument that measures distance or angle
between two geometric objects. The numeric value of this parameter is
called the **Dimension value**.

**Distance dimension** -- a dimension constraint that measures the
distance between points belonging to two geometric objects. For
instance, the distance between a point and a plane is measured as the
distance between the point and its nearest projection on the plane.
Distance dimension value is always set using distance measurement units
(meter, millimeter, inch, etc.) and may have both positive and negative
values, including zero.

**Angular dimension** -- a dimension constraint that measures the angle
between two directions taken from a pair of geometric objects. In other
words, the angular dimension value is the angle of rotating the first
object vector to align it with the second object vector. For example,
the angle between a line and a plane is determined by the angle of the
line rotation to be aligned with the plane using the shortest path.
Angular dimension values are set in radians.

**Logical constraint** -- a geometric constraint that measures a boolean
value: **true** if the constraint is satisfied and **false** if the
constraint is not satisfied. Examples of logical constraints:
parallelism of two lines, tangency between a cylinder and a plane,
coaxiality of two cones, etc. Note that dimensions are not logical
constraints.

**Parametric model** -- a *geometric model* making it possible to create
to different instances of this model by modifying independent numeric
parameters (control parameters). The geometric constraint system may be
a parametric model if at least one *driving dimension* is set for it
(see [Driving dimensions](./Driving_dimensions.md)).