{"id":842,"date":"2016-08-28T17:23:37","date_gmt":"2016-08-28T22:23:37","guid":{"rendered":"https:\/\/trigonography.com\/?p=842"},"modified":"2022-06-25T04:30:57","modified_gmt":"2022-06-25T09:30:57","slug":"involute-zig-zag-power-series-for-secant-and-tangent","status":"publish","type":"post","link":"https:\/\/trigonography.com\/?p=842","title":{"rendered":"Involute Zig-Zag: Power Series for Secant and Tangent"},"content":{"rendered":"\n

My counterpart to Chaikovsky’s Involute Pinwheel<\/a> for sine and cosine.
(See “Zig-Zag Involutes, Up-Down Permutations, and Secant and Tangent” (PDF)<\/a>.)<\/p>\n\n\n

\n
\"trigonograph-sectanzigzag\"<\/a><\/figure>\n<\/div>\n\n\n

\\(\\overset{\\displaystyle\\frown}{P_1P_2}\\) is an arc of the unit circle; thereafter, \\(\\overset{\\displaystyle\\frown}{P_iP_{i+1}}\\) is an involute<\/a> of \\(\\overset{\\displaystyle\\frown}{P_{i-1}P_i}\\).
Importantly, the involutes here unfurl in alternating directions<\/em>,
whereas those in the pinwheel all unfurl the same way.<\/p>\n\n\n\n

$$
\\begin{align}
\\sec\\theta &\\;=\\; |\\overline{P_0P_2}| \\;+\\; |\\overline{P_2 P_4}| \\;+\\; |\\overline{P_4P_6}| \\;+\\; \\cdots \\\\[4pt]
&\\;=\\; |\\overline{P_0P_1}| \\;+\\; |\\overset{\\displaystyle\\frown}{P_2 P_3}| \\;+\\; |\\overset{\\displaystyle\\frown}{P_4P_5}| \\;+\\; \\cdots \\\\[4pt]
&\\;=\\; \\sum_{k=0}^{\\infty} \\frac{U_{2k}}{(2k)!}\\;\\theta^{2k} \\\\[12pt]
\\tan\\theta &\\;=\\; |\\overline{P_1P_3}| \\;+\\; |\\overline{P_3 P_5}| \\;+\\; |\\overline{P_5P_7}| \\;+\\; \\cdots \\\\[4pt]
&\\;=\\; |\\overset{\\displaystyle\\frown}{P_1P_2}| \\;+\\; |\\overset{\\displaystyle\\frown}{P_3 P_4}| \\;+\\; |\\overset{\\displaystyle\\frown}{P_5P_6}| \\;+\\; \\cdots \\\\[4pt]
&\\;=\\; \\sum_{k=0}^{\\infty} \\frac{U_{2k+1}}{(2k+1)!}\\;\\theta^{2k+1}
\\end{align}$$
where \\(U_n\\) counts up-down permutations<\/a> of the set \\(\\{1, 2, \\ldots, n\\}\\).<\/div>\n\n\n\n

As with the pinwheel, the convergence is more obvious than the numerics.
The referenced note walks through the somewhat-involved combinatorics
of up-down permutations.<\/p>\n\n\n\n

Get the t-shirt!<\/b>
\n<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"

My counterpart to Chaikovsky’s Involute Pinwheel for sine and cosine.(See “Zig-Zag Involutes, Up-Down Permutations, and Secant and Tangent” (PDF).) \\(\\overset{\\displaystyle\\frown}{P_1P_2}\\) is an arc of the unit circle; thereafter, \\(\\overset{\\displaystyle\\frown}{P_iP_{i+1}}\\) is an involute of \\(\\overset{\\displaystyle\\frown}{P_{i-1}P_i}\\).Importantly, the involutes here unfurl in alternating directions,whereas those in the pinwheel all unfurl the same way. $$\\begin{align}\\sec\\theta &\\;=\\; |\\overline{P_0P_2}| \\;+\\; |\\overline{P_2 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-842","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/842","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=842"}],"version-history":[{"count":10,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/842\/revisions"}],"predecessor-version":[{"id":1251,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/842\/revisions\/1251"}],"wp:attachment":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=842"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=842"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=842"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}