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Operations</a></li><liclass="breadcrumb-nav-list-item"><span>Catmull-Clark Subdivision</span></li></ol></nav><divid="main-content"class="main-content"role="main"><h1id="catmull-clark-subdivision"><ahref="#catmull-clark-subdivision"class="anchor-heading"aria-labelledby="catmull-clark-subdivision"><svgviewBox="0 0 16 16"aria-hidden="true"><usexlink:href="#svg-link"></use></svg></a> Catmull-Clark Subdivision </h1><p>For an in-practice example, see the <ahref="/Scotty3D/guide/model">User Guide</a>.</p><p>The only difference between Catmull-Clark and <ahref="../linear">linear</a> subdivision is the choice of positions for new vertices. Whereas linear subdivision simply takes a uniform average of the old vertex positions, Catmull-Clark uses a very carefully-designed <em>weighted</em> average to ensure that the surface converges to a nice, round surface as the number of subdivision steps increases. The original scheme is described in the paper <em>“Recursively generated B-spline surfaces on arbitrary topological meshes”</em> by (Pixar co-founder) Ed Catmull and James Clark. Since then, the scheme has been thoroughly discussed, extended, and analyzed; more modern descriptions of the algorithm may be easier to read, including those from the <ahref="https://en.wikipedia.org/wiki/Catmull-Clark_subdivision_surface">Wikipedia</a> and <ahref="http://www.rorydriscoll.com/2008/08/01/catmull-clark-subdivision-the-basics/">this webpage</a>. In short, the new vertex positions can be calculated by:</p><ol><li>setting the new vertex position at each face f to the average of all its original vertices (exactly as in linear subdivision),</li><li>setting the new vertex position at each edge e to the average of the new face positions (from step 1) and the original endpoint positions, and</li><li>setting the new vertex position at each vertex v to the weighted sum</li></ol><center><imgsrc="catmull_clark_positions.png"style="height:80px"/></center><p>where <em>n</em> is the degree of vertex <em>v</em> (i.e., the number of faces containing <em>v</em>), and</p><ul><li><em>Q</em> is the average of all new face position for faces containing <em>v</em>,</li><li><em>R</em> is the average of all original edge midpoints for edges containing <em>v</em>, and</li><li><em>S</em> is the original vertex position for vertex <em>v</em>.</li></ul><p>In other words, the new vertex positions are an “average of averages.” (Note that you <em>will</em> need to divide by <em>n</em><em>both</em> when computing <em>Q</em> and <em>R</em>, <em>and</em> when computing the final, weighted value—this is not a typo!)</p><p>Apart from changing the way vertex positions are computed, there should be no difference in your implementation of linear and Catmull-Clark subdivision.</p><p>This step should be implemented in the method <codeclass="language-plaintext highlighter-rouge">HalfedgeMesh::catmullclark_subdivide_positions</code> in <codeclass="language-plaintext highlighter-rouge">student/meshedit.cpp</code>.</p><p>This subdivision rule <strong>is not</strong> required to support meshes with boundary, unless the implementer wishes to go above and beyond.</p></div></div><divclass="search-overlay"></div></div></body></html>
