This paper addresses the representation of rigid solids in computer-based systems for modeling geometry, which are increasingly important in mechanical and civil engineering, architecture, computer graphics, and computer vision. The paper provides a coherent view of the current state of knowledge on solid representations, based on sound theoretical principles. It is divided into three parts: (1) a mathematical framework for characterizing representation schemes, (2) a description and comparison of major schemes for representing solids, and (3) a survey of existing geometric modeling systems and a design for a multiple-representation system that offers superior reliability and versatility. The paper discusses the challenges in assessing current designs, specifying future system characteristics, and designing systems to meet these specifications. It emphasizes the importance of semantic (geometric) integrity in representation schemes and the need for a framework to characterize them. The paper also explores the role of mathematical models in representing solids, defining abstract solids with properties such as rigidity, three-dimensionality, finiteness, and closure under certain operations. The paper describes various schemes for representing solids, including ambiguous schemes, pure primitive instancing schemes, spatial occupancy enumeration, cell decompositions, constructive solid geometry (CSG), sweep representations, and boundary representations. It discusses the characteristics of these schemes, their efficacy in applications, and the challenges in ensuring validity and uniqueness. The paper also addresses the importance of consistency and equivalence between representations, and the need for multiple representations in practical systems. The paper concludes that no single representation scheme is uniformly "best" for all applications, and that multiple representations are often necessary to achieve efficiency and versatility. It emphasizes the importance of designing systems that can handle a wide range of applications and that can ensure the validity and uniqueness of representations. The paper also highlights the need for further research into the theoretical foundations of representation schemes and their practical applications in geometric modeling.This paper addresses the representation of rigid solids in computer-based systems for modeling geometry, which are increasingly important in mechanical and civil engineering, architecture, computer graphics, and computer vision. The paper provides a coherent view of the current state of knowledge on solid representations, based on sound theoretical principles. It is divided into three parts: (1) a mathematical framework for characterizing representation schemes, (2) a description and comparison of major schemes for representing solids, and (3) a survey of existing geometric modeling systems and a design for a multiple-representation system that offers superior reliability and versatility. The paper discusses the challenges in assessing current designs, specifying future system characteristics, and designing systems to meet these specifications. It emphasizes the importance of semantic (geometric) integrity in representation schemes and the need for a framework to characterize them. The paper also explores the role of mathematical models in representing solids, defining abstract solids with properties such as rigidity, three-dimensionality, finiteness, and closure under certain operations. The paper describes various schemes for representing solids, including ambiguous schemes, pure primitive instancing schemes, spatial occupancy enumeration, cell decompositions, constructive solid geometry (CSG), sweep representations, and boundary representations. It discusses the characteristics of these schemes, their efficacy in applications, and the challenges in ensuring validity and uniqueness. The paper also addresses the importance of consistency and equivalence between representations, and the need for multiple representations in practical systems. The paper concludes that no single representation scheme is uniformly "best" for all applications, and that multiple representations are often necessary to achieve efficiency and versatility. It emphasizes the importance of designing systems that can handle a wide range of applications and that can ensure the validity and uniqueness of representations. The paper also highlights the need for further research into the theoretical foundations of representation schemes and their practical applications in geometric modeling.