Computer graphics, computer-aided design, and computer-aided manufacturing are tools that have become indispensable to a wide array of activities in contemporary society. Euclidean processing provides the basis for these computer-aided design systems although it contains elements that inevitably lead to an inaccurate, non-robust, and complex system. The primary cause of the deficiencies of Euclidean processing is the division operation, which becomes necessary if an n-space problem is to be processed in n-space. The difficulties that accompany the division operation may be avoided if processing is conducted entirely in (n+1)-space. The paradigm attained through the logical extension of this approach, totally four-dimensional processing, is the subject of this book. This book offers a new system of geometric processing techniques that attain accurate, robust, and compact computations, and allow the construction of a systematically structured CAD system.

ThIS IS an English verSIOn of the book m two volumes, entitled "KeiJo Shon Kogaku (1), (2)" (Nikkan Kogyo Shinbun Co.) written in Japanese. The purpose of the book is a umfied and systematic exposition of the wealth of research results m the field of mathematical representation of curves and surfaces for computer aided geometric design that have appeared in the last thirty years. The material for the book started hfe as a set of notes for computer aided geometnc design courses which I had at the graduate schools of both computer SCIence, the umversity of Utah m U.S.A. and Kyushu Institute of Design in Japan. The book has been used extensively as a standard text book of curves and surfaces for students, practtcal engmeers and researchers. With the aim of systematic expositIOn, the author has arranged the book in 8 chapters: Chapter 0: The sIgmficance of mathemattcal representations of curves and surfaces is explained and histoncal research developments in this field are revIewed. Chapter 1: BasIc mathematical theones of curves and surfaces are reviewed and summanzed. Chapter 2: A classical mterpolation method, the Lagrange interpolation, is discussed. Although its use is uncommon in practice, this chapter is helpful in understanding Chaps. 4 and 6. Chapter 3: This chapter dIscusses the Coons surface in detail, which is one of the most important contributions in this field. Chapter 4: The fundamentals of spline functions, spline curves and surfaces are discussed in some detail.

Computer graphics, computer-aided design, and computer-aided manufacturing are tools that have become indispensable to a wide array of activities in contemporary society. Euclidean processing provides the basis for these computer-aided design systems although it contains elements that inevitably lead to an inaccurate, non-robust, and complex system. The primary cause of the deficiencies of Euclidean processing is the division operation, which becomes necessary if an n-space problem is to be processed in n-space. The difficulties that accompany the division operation may be avoided if processing is conducted entirely in (n+1)-space. The paradigm attained through the logical extension of this approach, totally four-dimensional processing, is the subject of this book. This book offers a new system of geometric processing techniques that attain accurate, robust, and compact computations, and allow the construction of a systematically structured CAD system.