[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572285-105":53,"doc-detail-572285-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","harolds-polar-parent-functions-cheat-sheet-read-online","Harolds Polar Parent Functions Cheat Sheet - read online","","Harold’s Polar Parent Functions Cheat Sheet provides a quick reference for common polar curves, organized by type such as circle, cardioid, one-loop limaçon, inner-loop limaçon, lemniscate, rose curve, and Archimedes’ spiral. For each family, it lists the governing polar equations and the corresponding conditions on angle ranges and parameter signs that determine which form and orientation appear. The sheet also notes constraints like when expressions use cosine versus sine and when parameters are nonzero.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/harolds-polar-parent-functions-cheat-sheet-read-online/572285/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/harolds-polar-parent-functions-cheat-sheet-read-online/572285.png","ImageObject",442,249,{"name":88,"@type":89},"ahduag","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-20","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What polar parent functions are included in this cheat sheet?","Question",{"text":108,"@type":109},"It covers circle, cardioid, one-loop limaçon, inner-loop limaçon, lemniscate, rose curve, and Archimedes’ spiral, along with the relevant angle/parameter conditions.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How does the cheat sheet determine which equation form to use for a given curve?",{"text":113,"@type":109},"It selects between cosine or sine based on the listed case and applies constraints on angle intervals and signs or inequalities for parameters to identify the correct branch.",{"name":115,"@type":106,"acceptedAnswer":116},"Why are the angle and parameter restrictions important in these polar equations?",{"text":117,"@type":109},"They control which portion of the curve is drawn and which geometric variant (e.g., inner vs outer loop, orientation) occurs under different parameter ranges.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572285,1792088584,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":76,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},3985747870414,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Harold’s Polar Parent Functions Cheat Sheet  \nAKA Library of Functions  \n4 May 2026  \n\n| Circle | 0 ≤ 􀟠 \u003C 􀟨 |  |  |\n| --- | --- | --- | --- |\n| |  |  |  |\n| 􀝎 = 􀜽 cos(􀟠)\u003Cbr>􀜽 > 0 | 􀝎 = 􀜽 cos(􀟠)\u003Cbr>􀜽 \u003C 0 | 􀝎 = 􀜽 sin(􀟠)\u003Cbr>􀜽 > 0 | 􀝎 = 􀜽 sin(􀟠)\u003Cbr>􀜽 \u003C 0 |\n| Cardioid | 0 ≤ 􀟠 \u003C 2􀟨 |  |  |\n| |  |  |  |\n| 􀝎 = 􀜽 + 􀜾 cos(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽􀜾\u003Cbr>= 1 | 􀝎 = 􀜽 − 􀜾 cos(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽􀜾\u003Cbr>= 1 | 􀝎 = 􀜽 + 􀜾 sin(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽􀜾\u003Cbr>= 1 | 􀝎 = 􀜽 − 􀜾 sin(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽􀜾\u003Cbr>= 1 |\n| One-Loop Limaçon | 0 ≤ 􀟠 \u003C 2􀟨 |  |  |\n| |  |  |  |\n| 􀝎 = 􀜽 + 􀜾 cos(􀟠)􀜽, 􀜾 > 0\u003Cbr>1 \u003C 􀜽􀜾 \u003C 2 | 􀝎 = 􀜽 − 􀜾 cos(􀟠)􀜽, 􀜾 > 0\u003Cbr>1 \u003C 􀜽􀜾 \u003C 2 | 􀝎 = 􀜽 + 􀜾 sin(􀟠)􀜽, 􀜾 > 0\u003Cbr>1 \u003C 􀜽􀜾 \u003C 2 | 􀝎 = 􀜽 − 􀜾 sin(􀟠)􀜽, 􀜾 > 0\u003Cbr>1 \u003C 􀜽􀜾 \u003C 2 |\n\n\n| Inner-Loop Limaçon | 0 ≤ 􀟠 \u003C 2􀟨 |  |  |\n| --- | --- | --- | --- |\n| |  |  |  |\n| 􀝎 = 􀜽 + 􀜾 cos(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽 \u003C 􀜾 | 􀝎 = 􀜽 − 􀜾 cos(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽 \u003C 􀜾 | 􀝎 = 􀜽 + 􀜾 sin(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽 \u003C 􀜾 | 􀝎 = 􀜽 − 􀜾 sin(􀟠)􀜽, 􀜾 > 0\u003Cbr>􀜽 \u003C 􀜾 |\n| lemniscate | 0 ≤ 􀟠 \u003C 2􀟨 |  |  |\n| |  |  |  |\n| 􀝎 2 = 􀜽 2 cos (2􀟠)\u003Cbr>􀜽 ≠ 0 | 􀝎 2 = −􀜽2 cos (2􀟠)\u003Cbr>􀜽 ≠ 0 | 􀝎 2 = 􀜽 2 sin (2􀟠)\u003Cbr>􀜽 ≠ 0 | 􀝎 2 = −􀜽2 sin (2􀟠)\u003Cbr>􀜽 ≠ 0 |\n| Rose Curve 0 ≤ 􀟠 \u003C 2􀟨 | 0 ≤ 􀟠 \u003C 􀟨 | Archimedes’ Spiral |  |\n| |  | |  |\n| 􀝎 = 􀜽 cos(􀝊􀟠)\u003Cbr>􀜽 ≠ 0\u003Cbr>􀝊 even, 2􀝊 petals | 􀝎 = 􀜽 sin(􀝊􀟠)\u003Cbr>􀜽 ≠ 0\u003Cbr>􀝊 odd. 􀝊 petals | 􀝎 = 􀟠\u003Cbr>􀟠 ≥ 0\u003Cbr>0 ≤ 􀟠 ≤ 2􀟨 | 􀝎 = 􀟠\u003Cbr>􀟠 ≥ 0\u003Cbr>0 ≤ 􀟠 ≤ 4􀟨 |\n\nSources:  \n• LibreTexts Mathematics (2025) . 6. 2: Graphing Basic Polar Equations.  \nSee also:  \n•  Harold’s Algebraic, Transcendental, and Conic Parent Functions Cheat Sheet  \n•  Harold’s Trigonometry Parent Functions Cheat Sheet  \n•  Harold’s Polar Parent Functions Cheat Sheet","cbCaijjG3ZmHqK6B","https://ap.wps.com/l/cbCaijjG3ZmHqK6B","pdf",756813,"English","# Polar Parent Functions\n## Circle\n## Cardioid\n## One-Loop Limaçon\n## Inner-Loop Limaçon\n## Lemniscate\n## Rose Curve\n## Archimedes’ Spiral","[{\"question\":\"What polar parent functions are included in this cheat sheet?\",\"answer\":\"It covers circle, cardioid, one-loop limaçon, inner-loop limaçon, lemniscate, rose curve, and Archimedes’ spiral, along with the relevant angle/parameter conditions.\"},{\"question\":\"How does the cheat sheet determine which equation form to use for a given curve?\",\"answer\":\"It selects between cosine or sine based on the listed case and applies constraints on angle intervals and signs or inequalities for parameters to identify the correct branch.\"},{\"question\":\"Why are the angle and parameter restrictions important in these polar equations?\",\"answer\":\"They control which portion of the curve is drawn and which geometric variant (e.g., inner vs outer loop, orientation) occurs under different parameter ranges.\"}]","Harolds Polar Parent Functions Cheat Sheet - read online | PDF",1791926569]