Rigidity through a Projective Lens

Applied Sciences

Anthony Nixon¹  Bernd Schulze¹  Walter Whiteley² 

[1] Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK;Deptartment of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada;

关键词: projective geometry;    projective statics;    projective infinitesimal motions;    bar–joint framework;    spherical framework;    body-bar framework;   

DOI  :  10.3390/app112411946

来源: DOAJ

【 摘 要 】

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

【 授权许可】

Unknown