**5. Conclusion**

In this chapter, we have presented closed-form solutions for computing and updating the principal components of (dynamic) discrete and continuous point sets. The new principal components can be computed in constant time, when a constant number of points are added or deleted from the point set. This is a significant improvement of the commonly used approach, when the new principal components are computed from scratch, which takes linear time. The advantages of some of the theoretical results were verified and presented in the context of computing dynamic PCA bounding boxes in Dimitrov, Holst, Knauer & Kriegel (2009); Dimitrov et al. (2011).

An interesting open problem is to find a closed-form solutions for dynamical point sets different from convex polyhedra, for example, implicit surfaces or B-splines. An implementation of computing principal components in a dynamic, continuous setting could be a useful practical extension of the results presented here regarding continuous point sets. Applications of the results presented here in other fields, like computer vision or visualization, are of high interest.
