Mathematics and Applications of Computer Graphics

This book equips readers with the fundamentals of 3DCG that support the advanced technologies. The content of this book is divided into four categories: modeling, rendering, character animation, and physical simulation, and each is explained in a separate chapter. In simpler terms, these four technologies can be described as creating shapes, creating images, creating motion, and reproducing the behavior of the natural world. The "Modeling" chapter describes the theory behind describing the shape of an object using surface meshes and then editing and manipulating them. The chapter "Rendering" focuses on the calculation of the luminance of an object's surface, and describes methods for calculating global illumination and generating fast images using precomputation to produce realistic images. Physical phenomena such as light scattering, which is necessary to improve realism, are also explained. The "Character Animation" chapter introduces the skeleton method, a standard technique in animation production, and describes various animation editing techniques for humanoid character models. Finally, the chapter on "Physical Simulation" describes methods to compute the behavior of complex natural phenomena, including not only hard objects but also flowing objects such as water and air. The four topics listed here are the fundamental elements that support 3DCG. The chapters were written by researchers who are active at the forefront of their respective fields. Each of the authors has a remarkable track record and can be said to have led the development of 3DCG technology in Japan with their wealth of experience and knowledge.

To understand the mathematical content of this book, basic knowledge of analysis and linear algebra is required. This book is suitable for upper undergraduate and graduate students in computer science, engineering, and other fields in science, and is also useful for practitioners in game production and those who are interested in 3DCG.

Jun Mitani is a professor at the University of Tsukuba in the Graduate School of Science and Technology. He received his Ph.D. in engineering from the University of Tokyo in 2004. He has held his current position since April 2015. His research interests center on computer graphics, particularly geometric modeling techniques and their application to origami design. The origami artworks he creates are characterized by three-dimensional shapes with smooth curved surfaces. In 2019, he was appointed as a Japan Cultural Envoy by the Agency for Cultural Affairs and traveled to eight Asian countries to promote cultural exchange through origami.

Kenshi Takayama is a research scientist at CyberAgent’s AI Lab, where he conducts research related to digital humans and visual content creation. He obtained his B.Sc., M.Sc., and Ph.D. from the University of Tokyo in 2007, 2009, and 2012, respectively. After a postdoctoral period at ETH Zurich funded by a JSPS fellowship until 2014, he served as an assistant professor at the National Institute of Informatics before joining CyberAgent in 2022. His research interests lie in geometric modeling and user interfaces.

Yoshinori Dobashi is a professor at Hokkaido University in the Graduate School of Information Science and Technology, Japan. His research interests span a diverse range of topics in computer graphics, including efficient rendering techniques, fluid simulation, sound simulation, image-based approaches, directable graphics for enhanced user control, haptics for tactile interaction, and digital fabrication technologies. He has published numerous papers at major international conferences and journals, including SIGGRAPH, Eurographics, Computer Graphics Forum, and Pacific Graphics, highlighting his significant contributions to advancing state-of-the-art methodologies in computer graphics.

Tomohiko Mukai is an associate professor at the Department of Industrial Art, Faculty of Systems Design, Tokyo Metropolitan University. He received his M.E. and PhD in Engineering from Toyohashi University of Technology in 2002 and 2006. Before joining Tokyo Metropolitan University, he worked at Toyohashi University of Technology as an assistant professor, Square Enix Co., Ltd as a senior researcher, and Tokai University as an associate professor.

Makoto Fujisawa is an associate professor at the Institute of Library, Information and Media Science, University of Tsukuba. He received B.Eng., M.Eng., and Ph.D. degrees in mechanical engineering from Shizuoka University in 2003, 2005, and 2008 respectively. He worked at Nara Institute of Science and Technology from 2008 to 2010 and University of Tsukuba from 2011 to 2020 as an assistant professor. His research interests include computer graphics and physics-based simulations.