Orientation and Transformations

We consider the problem of the coordinate transformation about a rotation axis.

Euler's theorem:

Every spatial rotation leaves a line of fixed points: the rotation axis.

Rodrigues formula:

Given point x, decompose into components parallel and perpendicular to the rotation axis;

Figure 1

Only the perpendicular component is affected by the rotation, yielding Rodrigues formula:

Substituting the cross product matrix n into Rodrigues formula:

Factoring out x we obtain:

That is Rodrigues formula in matrix form:

with .

Transformation matrix for a rotation around an arbitrary vector

A rotation in n-dimensional space leaves invariant a n-2 dimensional subspace. In two dimensions an invariant is a point; in three dimensional space a line is invariant after an arbitrary rotation etc. If we assume that the n-2 dimensional invariant space can be described by the n-2 basis vectors e₁,…,e_n-2 then a rotation around an angle a can be described by a matrix:

where P is a skew symmetric matrix

Given an axis u = (u₁,u₂,u₃) then a rotation of a vector w around this axis by an angle a can be described by the transformation matrix R, see Fig 2(a),

where Ä denotes the tensor product. Eq. (I.1) is equivalent with Eq. (1), with matrix elements

P is a skew symmetric matrix formed by the elements of u, called the dual to u matrix.

d_ij is the Kronecker delta symbol and e_ijk the anti-symmetric permutation tensor.

The angle a and the rotation axis u can be obtained from the trace of R and its transposed matrix R^T using the Equations (I.5) and (I.6).

Fig. 2(b) shows the derivation of Equation (I.1). We consider the components of w parallel and perpendicular to u, w_|| and w_T respectively. The term uÄu = uu^T describes the projection w_|| of w onto u which remains unchanged after a rotation around u. (1-uu^T) describes the projection of w onto the plane on which u is perpendicular and P describes a projection of w onto the plane where u’s is normal. This operation is followed by an additional rotation by an angle p/2.

Given a transformation matrix R we can determine the corresponding rotation axis u from P using Eq. (I.6), whereas the corresponding rotation angle is given by Equation (I.5). If R-R^T = 0 then the rotation angle is 0 or p and the rotation axis can be obtained from R. In this case it follows that R has as diagonal matrix elements the following:

Figure 2 (a) Rotation of a vector w around an arbitrary axis u by an angle a. (b) Components of w in the plane which is perpendicular to the rotation axis u after the rotation.

Optimal Orientation

We consider the problem of the determination of a direction vector u which optimizes some utility function f(u, ...), where f can depend on additional parameters some of which are direction dependent.

In three-dimensional space the invariant can be viewed as a rotation axis u = (u₁, u₂, u₃). Using spherical coordinates this vector u can be specified using the spherical angles (j,q) i.e. u = (u₁,u₂,u₃) = (sin(q)cos(j), sin(q)sin(j), cos(q)) and the utility function f depends on the three parameters (j,q,a). Given a direction u the angle which optimizes the utility function along this direction can be determined. The search in the q direction actually is a search in cosq which ensures that the angular space is uniformly searched. In order to escape from local minima we introduced a grid search method using a multi-scale approach. From the current search position new directions specified by a grid of a specified size and a given grid size are searched for a possible better solution. If a better solution is found on one of these grid points, then the search is repeated on a grid in a smaller part of space with a smaller grid constant around the previously found minimum. This process is repeated a few times.

Such a problem is the minimum bounding box problem. Find the orientation of a bounding box of a set of points with a minimum volume. That is the orientation of a bounding box that as tight as possible encloses a set of points.

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