Degrees of Freedom

Each type of geometry has a number of degrees of freedom equal to the minimum number of coordinates required to specify the state of the geometric object. For example, a 2D point has two degrees of freedom. A circle has three degrees of freedom, as it is completely defined by three parameters <X, Y, R>, namely the center coordinates and radius. According to Item S.1.7. Representation of Geometric Objects, a line is represented by a position point and a normal vector. This presentation is convenient, but it is redundant; minimum sufficient line presentation can be a pair of values, such as offset value and slope angle, so for a line the degrees of freedom are 2. For an ellipse, the degrees of freedom are 5. For a spline, the degrees of freedom are the sum of the degrees of freedom of its control points.

A parametric curve is fully determined by the application, i.e., it has zero degrees of freedom. Table 1 lists the degrees of freedom for all types supported by the solver.

Table 1 - Degrees of freedom for geometric objects

Geometry type	Degrees of freedom
Point	2
Line	2
Circle	3
Ellipse	5
Spline	2 \(\cdot\) Number of control points
Parametric curve	0
Bounded curve	2 + Degrees of freedom of the base curve

Every geometric object included in the system adds a number of its degrees of freedom to the overall degrees of freedom for the sketch. From the other side, every added constraint takes away one or more degrees of freedom. To define the state of all sketch geometric objects, it is required to add some number of constraints that take away all degrees of freedom for the object.

Most constraints take away one degree of freedom. These are constraints like parallelism, perpendicularity, horizontality/verticality, equality of radii, equality of lengths, a point on curve, tangency, and most dimensional constraints. Other constraints take away two degrees of freedom: middle point, collinearity, symmetry, and bisector.

It can be said that the task of parametric drawing is to completely determine geometric objects in the sketch. The number of geometric constraints required to completely determine the sketch is the sum of degrees of freedom of all the sketch objects.