The examples in Figure 3.5 span the main qualitative differences between various two-by-two matrices . Two of them were rotation matrices: the identity matrix, which is 0 degrees of rotation, and the -degree rotation matrix. Among the set of all possible , which ones are valid rotations? We must ensure that the model does not become distorted. This is achieved by ensuring that satisfies the following rules:

- No stretching of axes.
- No shearing.
- No mirror images.

To satisfy the first rule, the columns of must have unit length:

The scaling and shearing transformations in Figure 3.5 violated this.

To satisfy the second rule, the coordinate axes must remain perpendicular. Otherwise, shearing occurs. Since the columns of indicate how axes are transformed, the rule implies that their inner (dot) product is zero:

The shearing transformations in Figure 3.5 violate this rule, which clearly causes right angles in the model to be destroyed.

Satisfying the third rule requires that the determinant of is positive. After satisfying the first two rules, the only possible remaining determinants are (the normal case) and (the mirror-image case). Thus, the rule implies that:

The mirror image example in Figure 3.5 results in .

The first constraint (3.9) indicates that each column must be chosen so that its components lie on a unit circle, centered at the origin. In standard planar coordinates, we commonly write the equation of this circle as . Recall the common parameterization of the unit circle in terms of an angle that ranges from 0 to radians (see Figure 3.6):

Instead of and , we use the notation of the matrix components. Let and . Substituting this into from (3.4) yields

in which and were uniquely determined by applying (3.10) and (3.11). By allowing to range from 0 to , the full range of all allowable rotations is generated.

Think about degrees of freedom. Originally, we could chose all four components of independently, resulting in DOFs. The constraints in (3.9) each removed a DOF. Another DOF was removed by (3.10). Note that (3.11) does not reduce the DOFs; it instead eliminates exactly half of the possible transformations: The ones that are mirror flips and rotations together. The result is one DOF, which was nicely parameterized by the angle . Furthermore, we were lucky that set of all possible 2D rotations can be nicely interpreted as points along a unit circle.

Steven M LaValle 2020-11-11