[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81957-en":3,"doc-seo-81957-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81957,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Generalized Altitudes and Their Bounds","Introduces generalized altitudes for an n-dimensional simplex by extending the usual vertex-to-opposite-face altitude to arbitrary pairs of opposite faces. These quantities capture the relative position of the affine spans and yield a uniform closed-form expression for the angle between the faces. An equivalent algebraic representation is derived using generalized cross products and Gram determinants, connecting geometry to determinant-based analysis. The paper proves a lower bound on every generalized altitude in terms of the simplex height, so classical height or thickness conditions control this broader quality family, with applications to triangulation criteria for Riemannian manifolds.","arXiv :2607 .06 187v 1 [ cs .CG] 7 Jul 2026  \nGeneralized altitudes and their bounds  \nHana Dal Poz Kouˇrimsk´a 1* and Mathijs Wintraecken2 1* Institut f¨ur Mathematik der Universit¨at Potsdam, University of  \nPotsdam, Campus Golm, Golm, D-14476, Germany, [https://orcid.org/0000-0001-7841-0091](https://orcid.org/0000-0001-7841-0091) .  \n2 Centre Inria d’Universit´e Cˆote d’Azur, Sophia-Antipolis, 06902, France,  \n[https://orcid.org/0000-0002-7472-2220](https://orcid.org/0000-0002-7472-2220) .  \n*Corresponding author(s) . E-mail(s): [hana.dal.poz.kourimska@uni-potsdam.de](hana.dal.poz.kourimska@uni-potsdam.de) ; Contributing [authors: mathijs.wintraecken@inria.fr](authors: mathijs.wintraecken@inria.fr);  \nAbstract  \nWe introduce generalized altitudes of a simplex, extending the usual vertex-toopposite-face altitude to arbitrary pairs of opposite faces. These quantities encode the relative position of the affine spans of such faces and yield a uniform formula for the angle between them. We also derive an equivalent algebraic expression in terms of generalized cross products and Gram determinants, linking the construction to standard determinant-based tools. Finally, we prove that every generalized altitude is bounded below by a quantity controlled by the ordinary height of the simplex. Thus, classical height or thickness assumptions imply control over this broader family of geometric quantities. The results provide a compact framework for studying simplex quality and are motivated by applications to triangulation criteria for Riemannian manifolds.  \n1 Introduction  \nWell-shaped simplices are essential for numerical accuracy in several areas of computational mathematics, including the numerical solution of partial differential equations by finite element methods [1–6] and manifold meshing [7, 8] . In finite element methods, poorly shaped simplices can lead to large discretization errors and ill-conditioned stiffness matrices. In manifold meshing, simplex quality controls, among other things,  \n1  \nthe angle between a simplex whose vertices lie on the manifold and the tangent space of the manifold at, or near, those vertices [9] .  \nSeveral quality measures are used to quantify how well shaped a simplex is. The choice of measure often depends on the field, and sometimes on the specific application. Common examples include the thickness [10, 11], defined as the ratio between the height of the simplex—that is, its smallest altitude—and its largest edge length; the fatness [9], defined as the ratio between the volume of the simplex and the largest edge length raised to the dimension of the simplex; and the inradius–circumradius ratio [12] . These measures are weakly equivalent: upper and lower bounds on one of them yield corresponding upper and lower bounds on the others. Bounds on dihedral angles are also frequently used as a measure of simplex quality. Thickness, in particular, has proved especially convenient in the context of manifold meshing [13, 14], because it can be related directly to the eigenvalues associated with the edge vectors of the simplex.  \nThe main goal of this paper is to extend the classical notion of altitude, which is defined from a vertex to the opposite face, to arbitrary pairs of opposite faces of a simplex (Definition 1) . We call the resulting quantities generalized altitudes.  \nWe first show that generalized altitudes provide a natural way to describe the angle between (the affine hulls of) a face of a simplex and its opposite face (Lemma 2) . More precisely, they lead to an explicit formula for this angle. The formula is simple, geometrically transparent, and applies uniformly to faces of arbitrary dimension.  \nWe then relate generalized altitudes to standard tools from multilinear algebra and analysis, in particular the (generalized) cross product and the Gram determinant (Corollary 5) . This connection gives a second, algebraic formula for generalized altitudes, and provides a bridge between the geome","cbCaifEedYHejCG3","https://ap.wps.com/l/cbCaifEedYHejCG3","pdf",605968,7,1,9,"English","en",105,"# Introduction\n# Generalized altitude","[{\"question\":\"What are generalized altitudes of a simplex?\",\"answer\":\"Generalized altitudes extend the classical altitude of a vertex by defining the smallest distance between the affine hulls of two opposite faces of a simplex.\"},{\"question\":\"How do generalized altitudes relate to angles between faces?\",\"answer\":\"They provide an explicit, uniform formula for the angle between the affine spans of a face and its opposite face, valid across dimensions.\"},{\"question\":\"What bound is proved for generalized altitudes?\",\"answer\":\"Every generalized altitude has a lower bound controlled by the ordinary height of the simplex, implying that standard height or thickness assumptions also bound this broader family of geometric quantities.\"}]","Generalized Altitudes and Their Bounds | PDF",1784177281,23,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"generalized-altitudes-and-their-bounds","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/generalized-altitudes-and-their-bounds/81957/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-03","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What are generalized altitudes of a simplex?","Question",{"text":77,"@type":78},"Generalized altitudes extend the classical altitude of a vertex by defining the smallest distance between the affine hulls of two opposite faces of a simplex.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How do generalized altitudes relate to angles between faces?",{"text":82,"@type":78},"They provide an explicit, uniform formula for the angle between the affine spans of a face and its opposite face, valid across dimensions.",{"name":84,"@type":75,"acceptedAnswer":85},"What bound is proved for generalized altitudes?",{"text":86,"@type":78},"Every generalized altitude has a lower bound controlled by the ordinary height of the simplex, implying that standard height or thickness assumptions also bound this broader family of geometric quantities.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,117,121,124,128,131,135],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":113,"doc_module":4,"doc_module_name":47,"category_name":114,"show_sort_weight":115,"slug":116},6,"Technology",50,"technology",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":22,"doc_module":4,"doc_module_name":47,"category_name":125,"show_sort_weight":126,"slug":127},"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":47,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":47,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":47,"category_name":137,"show_sort_weight":108,"slug":138},19,"General","general"]