to introduce these techniques and additional background is provided in appendices. The ? rst chapter presents an overview of the articulated systems that we will be considering in this book. The generic mobility of a linkage is de? ned, and we separate them into the primary classes of planar, spherical, and spatial chains. The second chapter presents the analysis of planar chains and details their movement and classi? cation. Chapter three develops the graphical design theory for planar linkages and introduces many of the geometric principlesthatappearintheremainderofthebook. Inparticular,geometric derivations of the pole triangle and the center-point theorem anticipate analytical results for the spherical and spatial cases. Chapter four presents the theory of planar displacements, and Chapter ? ve presents the algebraic design theory. The bilinear structure of the - sign equations provides a solution strategy that emphasizes the geometry underlying linear algebra. The ? ve-position solutionincludes an elimi- tion step that is probably new to most students, though it is understood and well-received in the classroom. Chapters six and seven introduce the properties of spherical linkages and detail the geometric theory of spatial rotations. Chapter eight presents the design theory for these linkages, which is analogous to the planar theory. This material exercises the student’s use of vector methods to represent geometry in three dimensions. Perpendicular bisectors in the planar design theory become perpendicular bisecting planes that intersect to de? The analogue provides students with a geometric perspective of the linear equations that they are solving.