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category-numeric-analysis category-programming untagged y2015 m07 d23 h23 slug-generating-equidistant-points-on-unit-disk"><h1 class="entry-title">Generating Equidistant Points on Unit Disk</h1><div class="entry-meta"><span class="entry-date"><abbr class="published" title="2015-07-23T14:48:53+0900">July 23, 2015</abbr></span></div><div class="entry-content"><p>Recently I needed an algorithm to generate equidistant points inside the unit disk with some of them placed on the boundary (circle). To my surprise, quick search didn’t reveal any simple method for this. Below is obvious solution & code, hopefully it will save some time for others.</p><p>MATLAB code:</p><pre class="brush: matlabkey; title: ; notranslate" title=""> function [x, y, Nb, Np] = eqdisk(Nr) %EQDISK Generates equidistant points inside the unit disk. % Nr [in] - number of radial circles % x,y [out] - coordinates of generated points % Np [out] - total number of generated points % Nb [out] - points on boundary (on r = 1 circle) dR = 1/Nr; x(1) = 0; y(1) = 0; k = 1; for r = dR:dR:1 n = round(pi/asin(1/(2*k))); theta = linspace(0, 2*pi, n+1)'; x = [x; r.*cos(theta(1:n))]; y = [y; r.*sin(theta(1:n))]; k = k+1; end; Nb = n; Np = size(x,1); end </pre><p>Algorithm is based on the idea of placing the points on concentric circles with (near-) equal arc length between them. Here is some examples:</p><p><span id="more-3453"></span></p><div><div style="width:475px; float:left; text-align:justify;"> <img src="http://www.holoborodko.com/pavel/downloads/ud1.svg" width=475 alt="Nr=1" title="Equidistant points on unit disk, Nr=1"/></div><div style="float:right; width:475px; text-align:justify;"> <img src="http://www.holoborodko.com/pavel/downloads/ud2.svg" width=475 alt="Nr=2" title="Equidistant points on unit disk, Nr=2"/></div><p><br style="clear: both;" /></div><div><div style="width:475px; float:left; text-align:justify;"> <img src="http://www.holoborodko.com/pavel/downloads/ud3.svg" width=475 alt="Nr=3" title="Equidistant points on unit disk, Nr=3"/></div><div style="float:right; width:475px; text-align:justify;"> <img src="http://www.holoborodko.com/pavel/downloads/ud4.svg" width=475 alt="Nr=4" title="Equidistant points on unit disk, Nr=4"/></div><p><br style="clear: both;" /></div><div><div style="width:475px; float:left; text-align:justify;"> <img src="http://www.holoborodko.com/pavel/downloads/ud5.svg" width=475 alt="Nr=5" title="Equidistant points on unit disk, Nr=5"/></div><div style="float:right; width:475px; text-align:justify;"> <img src="http://www.holoborodko.com/pavel/downloads/ud10.svg" width=475 alt="Nr=10" title="Equidistant points on unit disk, Nr=10"/></div><p><br style="clear: both;" /></div><p style="margin-top:22px"> Note the “equilibrium” case, when we have the same number of points on boundary and inside the disk, Nr=3 (19/19).</p><div id="post-ratings-3453" class="post-ratings" itemscope itemtype="http://schema.org/Article" data-nonce="dd6ac1f776"><img id="rating_3453_1" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="1 Star" title="1 Star" onmouseover="current_rating(3453, 1, '1 Star');" onmouseout="ratings_off(4.9, 5, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_3453_2" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="2 Stars" title="2 Stars" onmouseover="current_rating(3453, 2, '2 Stars');" onmouseout="ratings_off(4.9, 5, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_3453_3" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="3 Stars" title="3 Stars" onmouseover="current_rating(3453, 3, '3 Stars');" onmouseout="ratings_off(4.9, 5, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_3453_4" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="4 Stars" title="4 Stars" onmouseover="current_rating(3453, 4, '4 Stars');" onmouseout="ratings_off(4.9, 5, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_3453_5" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_half.gif" alt="5 Stars" title="5 Stars" onmouseover="current_rating(3453, 5, '5 Stars');" onmouseout="ratings_off(4.9, 5, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /> (<strong>13</strong> votes, average: <strong>4.92</strong>)<br /><span class="post-ratings-text" id="ratings_3453_text"></span><meta itemprop="name" content="Generating Equidistant Points on Unit Disk" /><meta itemprop="headline" content="Generating Equidistant Points on Unit Disk" /><meta itemprop="description" content="Recently I needed an algorithm to generate equidistant points inside the unit disk with some of them placed on the boundary (circle). To my surprise, quick search didn't reveal any simple method for t..." /><meta itemprop="datePublished" content="2015-07-23T14:48:53+09:00" /><meta itemprop="dateModified" content="2015-07-23T16:06:55+09:00" /><meta itemprop="url" content="http://www.holoborodko.com/pavel/2015/07/23/generating-equidistant-points-on-unit-disk/" /><meta itemprop="author" content="Pavel Holoborodko" /><meta itemprop="mainEntityOfPage" content="http://www.holoborodko.com/pavel/2015/07/23/generating-equidistant-points-on-unit-disk/" /><div style="display: none;" itemprop="publisher" itemscope itemtype="https://schema.org/Organization"><meta itemprop="name" content="Pavel Holoborodko" /><meta itemprop="url" content="http://www.holoborodko.com/pavel" /><div itemprop="logo" itemscope itemtype="https://schema.org/ImageObject"><meta itemprop="url" content="" /></div></div><div style="display: none;" itemprop="aggregateRating" itemscope itemtype="http://schema.org/AggregateRating"><meta itemprop="bestRating" content="5" /><meta itemprop="worstRating" content="1" /><meta itemprop="ratingValue" content="4.92" /><meta itemprop="ratingCount" content="13" /></div></div><div id="post-ratings-3453-loading" class="post-ratings-loading"><img src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/loading.gif" width="16" height="16" class="post-ratings-image" />Loading...</div></div></div><div id="nav-below" class="navigation"><div class="nav-previous"><a href="http://www.holoborodko.com/pavel/2014/11/04/computing-mixed-derivatives-by-finite-differences/" rel="prev"><span class="meta-nav">«</span> Computing Mixed Derivatives by Finite Differences</a></div><div class="nav-next"></div></div><div id="comments"><div id="comments-list" class="comments"><h3><span>One</span> Comment</h3><ol><li id="comment-161982" class="comment c c-y2018 c-m08 c-d11 c-h17 alt depth-1"><div class="comment-author vcard"> <span class="fn n"><a href="http://mixkino.net/" onclick="javascript:_gaq.push(['_trackEvent','outbound-commentauthor','http://mixkino.net/']);" rel='external nofollow ugc' class='url url'>Mixkino</a></span></div><div class="comment-meta">Posted August 11, 2018 at 8:40 am <span class="meta-sep">|</span> <a href="#comment-161982" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Tammes and Thomson problems can be considered as an extension for non-Platonic cases. For instance, see Math Overflow: Distributing points evenly on a sphere . Sadly, general solutions to these problems are not trivial.</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/2015/07/23/generating-equidistant-points-on-unit-disk/?replytocom=161982#respond' data-commentid="161982" data-postid="3453" data-belowelement="comment-161982" data-respondelement="respond" aria-label='Reply to Mixkino'>Reply</a></div></li></ol><div id="comments-nav-below" class="comment-navigation"><div class="paginated-comments-links"></div></div></div><div id="respond"><h3>Post a Comment</h3><div id="cancel-comment-reply"><a rel="nofollow" id="cancel-comment-reply-link" href="/pavel/2015/07/23/generating-equidistant-points-on-unit-disk/#respond" style="display:none;">Click here to cancel reply.</a></div><div class="formcontainer"><form id="commentform" action="http://www.holoborodko.com/pavel/wp-comments-post.php" method="post"><p id="comment-notes">Your email is <em>never</em> published nor shared.</p><div id="form-section-author" class="form-section"><div class="form-label"><label for="author">Name</label></div><div class="form-input"><input id="author" name="author" type="text" value="" size="30" maxlength="20" tabindex="3" /></div></div><div id="form-section-email" class="form-section"><div class="form-label"><label for="email">Email</label></div><div class="form-input"><input id="email" name="email" type="text" value="" size="30" maxlength="50" tabindex="4" /></div></div><div id="form-section-url" class="form-section"><div class="form-label"><label for="url">Website</label></div><div class="form-input"><input id="url" name="url" type="text" value="" size="30" maxlength="50" tabindex="5" /></div></div><div id="form-section-comment" class="form-section"><div class="form-label"><label for="comment">Comment</label></div><div class="form-textarea"><textarea id="comment" name="comment" cols="45" rows="8" tabindex="6"></textarea></div></div><div id="form-allowed-tags" class="form-section"><p><span>Use native LaTeX syntax to include formulas:</span> $ ... $, \[ ... \], <span>etc. Do not forget to preview comment before posting.</span></p><p><span>Also you may use these <abbr title="HyperText Markup Language">HTML</abbr> tags and attributes: </span><a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong></p></div><p style="display: none;"><input type="hidden" id="akismet_comment_nonce" name="akismet_comment_nonce" value="30bb40aed7" /></p><input name="acp-preview" type="button" id="acp-preview" tabindex="6" value="Preview" /><div id="ajax-comment-preview"></div><p style="clear: both;" class="subscribe-to-comments"> <input type="checkbox" name="subscribe" id="subscribe" value="subscribe" style="width: auto;" /> <label for="subscribe">Notify me of followup comments via e-mail</label></p><p style="display: none;"><input type="hidden" id="ak_js" name="ak_js" value="102"/></p><div class="form-submit"><input id="submit" name="submit" type="submit" value="Post Comment" tabindex="7" /><input type="hidden" name="comment_post_ID" value="3453" /></div> <input type='hidden' name='comment_post_ID' value='3453' id='comment_post_ID' /> <input type='hidden' name='comment_parent' id='comment_parent' value='0' /></form><form action="" method="post"> <input type="hidden" name="solo-comment-subscribe" value="solo-comment-subscribe" /> <input type="hidden" name="postid" value="3453" /> <input type="hidden" name="ref" value="http://www.holoborodko.com/pavel/2015/07/23/generating-equidistant-points-on-unit-disk/" /><p class="solo-subscribe-to-comments"> Subscribe without commenting <br /> <label for="solo-subscribe-email">E-Mail: <input type="text" name="email" id="solo-subscribe-email" size="22" value="" /></label> <input type="submit" name="submit" value="Subscribe" /></p></form></div></div></div></div></div></div><div id="footer"><div id="siteinfo"> Powered by <a class="wp-link" href="http://WordPress.org/" title="WordPress" rel="generator">WordPress</a> with <a href="http://www.holoborodko.com/pavel/quicklatex/">QuickLaTeX</a>. 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