Further Readings
We have presented several mathematical basics that give the theoretical foundation for the deformation/animation techniques in computer graphics. As for your more visual understanding of this book, we would also recommend you to refer to our SIGGRAPH 2016 course notes [Ochiai2016a] and the video assocated with it [Ochiai2016b].
The graphics topics covered in this book are, however, not exhaustive. The mathematics employed in this book then deal only with affine transformations, quaternions, dual quaternions, and their Lie theoretic aspects.
In the following sections, we briefly describe some more mathematical aspects that should further be explored for better understanding of geometric objects and their deformation and animation.
DIFFERENTIAL GEOMETRIC APPROACHES
As shown in Chapters 6 and 7, we need various differential geometric concepts, such as curvature, geodesics, and the fundamental forms, for dealing with more complex structure of deforming geometric objects.
As a good introduction to such differential geometric approaches from the computer scientific viewpoint, we would recommend the following two books: [Kimmel2004] describes differential geometric computational methods and algorithms in image processing and analysis. [Bronstein2009] gives a differential geometric framework for computational models of 2D and 3D non-rigid objects.
Discrete Differential Geometry, DDG, is a rapidly growing new field in mathematics, having natural and fruitful applications in computer graphics. [Hoffmann2009] gives a good introduction to DDG from the mathematician’s viewpoint, which we believe gives a nice overview of the theory of discrete curves and surfaces. As for more advanced topics, [Bobenko2008] will give you a nice guidepost, including recent results in DDG.
LIE GROUP APPLICATIONS
Quaternion is a useful tool for controlling rotation. This also suggests that quaternion is powerful in camera control. [Shoemake1994a] proposed a solution of the camera twist problem using quaternions, where S3 is treated as a fiber bundle, which means that S3 is locally homeomorphic to a direct product: (an open subset of 2d sphere)× (1d circle).
A special class of Lie groups and Lie algebras is realized as (generalized) numbers, such as quaternions. This class of numbers, called geometric algebra or Clifford algebra, has a long history going back to 19th-century mathematician Clifford. A graphical approach is found in [Hanson2006, Dorst2007], and [Dorst2011].
There is another interesting Lie group application, which gives a keyframing method for crowd control [Takahashi2009]. In our context, we regard the method as an interpolation technique in the orthogonal group O(n) of size n ≫ 4.
Recently the Lie group integrators have been applied to computer animation: [Kobilarov2009] provides a holonomic system for vehicle animations, and [Tournier2012] presents a control system providing trade-off between physics-based simulation and kinematics control using a metric interpolation and real-time animation. The latter one uses PGA, as described next.
Principal geodesic analysis (PGA [Fletcher2004]) provides a new statistical shape analysis method for data manifold. The data manifold typically means the collection of shape data, such as of hippocampus in medical imaging [Fletcher2004], or the pose manifold of a motion capture sequence in computer animation [Tournier2012]. As is well known, Principal Component Analysis (PCA) is a standard technique for dimension reduction of statistical data lying on a Euclidian space. PGA is a generalization of this technique for curved data manifold. In PGA the concept of Lie group plays a fundamental role and it acts on the data manifold as a symmetric Riemannian manifold.
TOWARD LIE THEORY
How do we describe motion/deformation of objects in Rn? In this book we have shown the following examples:
• Local: Affine transformation well describes a rigid or non-rigid transformation.
• Global: The set of affine maps or Poisson editing approaches well approximate deformation and/or animation.
For global deformation, introducing a diffeomorphism (smooth bijective map whose inverse map is also smooth) may also give us a natural and wider framework of Lie group approaches. We then note that the set of all diffeomorphisms for a given surface (or manifold) constitutes an infinite-dimensional Lie group, where the multiplication is defined as composite of the diffeomorphisms. Unfortunately an infinite-dimensional Lie group is still difficult to understand with the current mathematics, although many attempts have been done. For example, [Mumford2010] gives a good introduction to this general framework, focusing on morphing between two 2D images as input.
Finally, for a more mathematical aspect of Lie groups and Lie algebras, we would like to refer to the classical literatures: [Helgason1978] for Lie groups with differential geometry, [Knapp1996] for Lie groups with representation theory, [Hochschild1965, Gorbatsevich1993], and [Duistermaat1999] for Lie groups with structure theory. For abstract Lie algebra, we see [Serre1992], and its representation theory with physics application is found in [Georgi1982]. Though those are a bit far from graphics applications, you can consult them to know more about the basic ideas in Lie theory, which we believe will be quite useful for further graphics research.