Constraint Assignment

When the Regenerate button is activated, the dimensioned sketch currently displayed in the viewing screen is sent to the constraint solver component of the profile editor. There are several possible outcomes. The sketch may be well-constrained, and the display changed to display an object which is a solution to the given constraint problem. The sketch may be overconstrained, that is, too many constraint assignments were made and the sketch is therefore overdetermined. The sketch may be underconstrained, with insufficient information to determine a solution. Or the sketch may be improperly constrained. This outcome indicates that either the sketch is overconstrained in some areas and underconstrained in others, or that the values of the constraints as given are inconsistent. Any of the last three situations results in a popup message being given, and the sketch being unaltered.

It is clear that an important aspect of using the constraint solver is to be able to assign only as many constraints as are necessary and sufficient to make the problem well-constrained. In this section, we discuss what makes a sketch well-constrained or not. We then let you decide if certain sketches are over, under or well-constrained.

The number of constraints necessary for a well-constrained sketch is easy to count. The basic geometric elements available are lines, points, and fixed radius circles (that is, the radius is not allowed to vary in order to satisfy a constraint). Thus two constraints are necessary per geometric element. However, when only unfixed geometries are used, the configuration need only be specified up to a rigid body transformation. Since there are two translational and one rotational degree of freedom in two-space, the total number of constraints necessary is 2*n-3, where n is the number of geometric elements in the sketch.

Unfortunately, a blind count of geometric elements and constraints is not sufficient to decide whether a sketch is well-constrained or not. There may be some elements which are overconstrained, while other elements are underconstrained. In this case, the overall count may come out correctly, but the problem still cannot be solved. In terms of the constraint graph, a sketch is overconstrained if its constraint graph contains any subgraph with m vertices and more than 2*m-3 edges. It is underconstrained if it is not overconstrained, and the number of constraints is less than 2*n-3, where n is the number of vertices in the graph. It is well-constrained if it is not overconstrained and the number of constraints is equal to 2*n-3.

Keeping these relationships in mind helps one decide if a sketch is properly constrained. However, practice in correct constraint assignment is the best way to learn when a sketch is well-constrained, or to figure out what is wrong with an incorrectly constrained sketch. Below are five examples sketches, some of which are well-constrained, and others of which are not. The non-dimensional constraints between elements are listed near each sketch. See if you can decide which are well-constrained, and if not, why. Select your answer from the three choices beside each picture. An audio response of applause will indicate the correct selection. For an explanation of each solution, select Explanation.

Example 1

                                     Well-constrained 

                                     Overconstrained   

                                     Underconstrained


                                     Explanation

Example 2

                                     Well-constrained 

                                     Overconstrained   

                                     Underconstrained


                                     Explanation

Example 3

                                     Well-constrained 

                                     Overconstrained   

                                     Underconstrained

Non-dimensioned constraints

                                     Explanation

Example 4

                                     Well-constrained 

                                     Overconstrained   

                                     Underconstrained

Non-dimensioned constraints

                                     Explanation

Example 5

                                     Well-constrained 

                                     Overconstrained   

                                     Underconstrained

Non-dimensioned constraints

                                     Explanation