{"id":1809,"date":"2019-03-13T08:19:33","date_gmt":"2019-03-13T12:19:33","guid":{"rendered":"http:\/\/brickisland.net\/DDGSpring2019\/?p=1809"},"modified":"2019-03-14T04:37:24","modified_gmt":"2019-03-14T08:37:24","slug":"steiner-formula-for-surfaces-in-mathbbr3","status":"publish","type":"post","link":"https:\/\/brickisland.net\/DDGSpring2019\/2019\/03\/13\/steiner-formula-for-surfaces-in-mathbbr3\/","title":{"rendered":"Steiner Formula for Surfaces in \\(\\mathbb{R}^3\\)"},"content":{"rendered":"

\"\"<\/a><\/p>\n

In the slides<\/a> we derived a Steiner formula for polyhedral surfaces in \\(\\mathbb{R}^3\\), by considering the Minkowski sum with a ball and working out expressions for the areas and curvatures associated with vertices, edges, and faces. But we can also get a Steiner formula for smooth surfaces, using the expressions already derived in class. In particular, recall that for a closed surface \\(f: M \\to \\mathbb{R}^3\\) with Gauss map \\(N\\), we can obtain the basic curvatures by just wedging together \\(df\\) and \\(dN\\) in all possible ways:
\n\\[\\begin{array}{rcl}
\ndf \\wedge df &=& 2NdA \\\\
\ndf \\wedge dN &=& 2HNdA \\\\
\ndN \\wedge dN &=& 2KNdA \\\\
\n\\end{array}
\n\\]
\nHere \\(H\\) and \\(K\\) denote the mean and Gauss curvature (resp.), and dA is the area form induced by \\(f\\). For sufficiently small \\(t\\), taking a Minkowski sum with a ball is the same as pushing the surface in the normal direction a distance \\(t\\). In other words, the surface
\n\\[
\n f_t := f + tN
\n\\]
\nwill describe the “outer” boundary of the Minkowski sum; this surface has the same Gauss map \\(N\\) as the original one. To get its area element, we can take the wedge product
\n\\[
\n\\begin{array}{rcl}
\ndf_t \\wedge df_t
\n&=& (df + t dN) \\wedge (df + t dN) \\\\
\n&=& df \\wedge df + 2t df \\wedge dN + t^2 dN \\wedge dN,
\n\\end{array}
\n\\]
\nwhere we have used the fact that \\(\\alpha \\wedge \\beta = \\beta \\wedge \\alpha\\) when \\(\\alpha,\\beta\\) are both \\(\\mathbb{R}^3\\)-valued 1-forms. The list of identities above then yields
\n\\[
\n df_t \\wedge df_t = (2 + 4tH + 2t^2 K)NdA,
\n\\]
\nor equivalently,
\n\\[
\n \\fbox{\\(dA_t = (1+2tH+t^2K)dA.\\)}
\n\\]
\nIn other words, just as in the polyhedral case, the rate at which the area is growing is a polynomial in the ball radius \\(t\\); the coefficients of this polynomial are given by the basic curvatures of the surface (also known as quermassintegrals!<\/em>).<\/p>\n","protected":false},"excerpt":{"rendered":"

In the slides we derived a Steiner formula for polyhedral surfaces in \\(\\mathbb{R}^3\\), by considering the Minkowski sum with a ball and working out expressions for the areas and curvatures associated with vertices, edges, and faces. But we can also get a Steiner formula for smooth surfaces, using the expressions already derived in class. In … Continue reading “Steiner Formula for Surfaces in \\(\\mathbb{R}^3\\)”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1809","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/posts\/1809","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/comments?post=1809"}],"version-history":[{"count":14,"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/posts\/1809\/revisions"}],"predecessor-version":[{"id":1825,"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/posts\/1809\/revisions\/1825"}],"wp:attachment":[{"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/media?parent=1809"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/categories?post=1809"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2019\/wp-json\/wp\/v2\/tags?post=1809"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}