build based on 8d5f81a

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2024-09-22T19:16:11","documenter_version":"1.7.0"}}
+{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2024-09-23T23:05:46","documenter_version":"1.7.0"}}

dev/triangle/index.html

@@ -4,4 +4,4 @@
     = 2A \int_0^1 \int_0^{1-v} f\left( \bar{r}(u,v) \right) \, \text{d}u \, \text{d}v\]</p><p>Since the integral domain is a right-triangle in the Barycentric domain, a nested application of Gauss-Kronrod quadrature rules is capable of computing the result, albeit inefficiently. However, many numerical integration methods that require rectangular bounds can not be directly applied.</p><p>In order to enable integration methods that operate over rectangular bounds, two coordinate system transformations are applied: the first maps from Barycentric coordinates <span>$(u, v)$</span> to polar coordinates <span>$(r, \phi)$</span>, and the second is a non-linear map from polar coordinates to a new curvilinear basis <span>$(R, \phi)$</span>.</p><p>For the first transformation, let <span>$u = r~\cos\phi$</span> and <span>$v = r~\sin\phi$</span> where <span>$\text{d}u~\text{d}v = r~\text{d}r~\text{d}\phi$</span>. The Barycentric triangle&#39;s hypotenuse boundary line is described by the function <span>$v(u) = 1 - u$</span>. Substituting in the previous definitions leads to a new boundary line function in polar coordinate space <span>$r(\phi) = 1 / (\sin\phi + \cos\phi)$</span>.</p><p class="math-container">\[\int_0^1 \int_0^{1-v} f\left( \bar{r}(u,v) \right) \, \text{d}u \, \text{d}v =
     \int_0^{\pi/2} \int_0^{1/(\sin\phi+\cos\phi)} f\left( \bar{r}(r,\phi) \right) \, r \, \text{d}r \, \text{d}\phi\]</p><p>These integral boundaries remain non-rectangular, so an additional transformation will be applied to a curvilinear <span>$(R, \phi)$</span> space that normalizes all of the hypotenuse boundary line points to <span>$R=1$</span>. To achieve this, a function <span>$R(r,\phi)$</span> is required such that <span>$R(r_0, \phi) = 1$</span> where <span>$r_0 = 1 / (\sin\phi + \cos\phi)$</span></p><p>To achieve this, let <span>$R(r, \phi) = r~(\sin\phi + \cos\phi)$</span>. Now, substituting some terms leads to</p><p class="math-container">\[\int_0^{\pi/2} \int_0^{1/(\sin\phi+\cos\phi)} f\left( \bar{r}(r,\phi) \right) \, r \, \text{d}r \, \text{d}\phi
     = \int_0^{\pi/2} \int_0^{r_0} f\left( \bar{r}(r,\phi) \right) \, \left(\frac{R}{\sin\phi + \cos\phi}\right) \, \text{d}r \, \text{d}\phi\]</p><p>Since <span>$\text{d}R/\text{d}r = \sin\phi + \cos\phi$</span>, a change of integral domain leads to</p><p class="math-container">\[\int_0^{\pi/2} \int_0^{r_0} f\left( \bar{r}(r,\phi) \right) \, \left(\frac{R}{\sin\phi + \cos\phi}\right) \, \text{d}r \, \text{d}\phi
-    = \int_0^{\pi/2} \int_0^1 f\left( \bar{r}(R,\phi) \right) \, \left(\frac{R}{\left(\sin\phi + \cos\phi\right)^2}\right) \, \text{d}R \, \text{d}\phi\]</p><p>The second term in this new integrand function serves as a correction factor that corrects for the impact of the non-linear domain transformation. Since all of the integration bounds are now constants, specialized integration methods can be defined for triangles that performs these domain transformations and then solve the new rectangular integration problem using a wider range of solver options.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../usage/">« Example Usage</a><a class="docs-footer-nextpage" href="../api/">Public API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Sunday 22 September 2024 19:16">Sunday 22 September 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+    = \int_0^{\pi/2} \int_0^1 f\left( \bar{r}(R,\phi) \right) \, \left(\frac{R}{\left(\sin\phi + \cos\phi\right)^2}\right) \, \text{d}R \, \text{d}\phi\]</p><p>The second term in this new integrand function serves as a correction factor that corrects for the impact of the non-linear domain transformation. Since all of the integration bounds are now constants, specialized integration methods can be defined for triangles that performs these domain transformations and then solve the new rectangular integration problem using a wider range of solver options.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../usage/">« Example Usage</a><a class="docs-footer-nextpage" href="../api/">Public API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Monday 23 September 2024 23:05">Monday 23 September 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/usage/index.html

@@ -23,4 +23,4 @@
 
 integral(f, unit_circle_bz, GaussKronrod())
     # 0.017122 seconds (18.93 k allocations: 78.402 MiB)
-    # ans = 5.551055333711397 m^2</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../supportmatrix/">« Support Matrix</a><a class="docs-footer-nextpage" href="../triangle/">Integrating a Triangle »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Sunday 22 September 2024 19:16">Sunday 22 September 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+    # ans = 5.551055333711397 m^2</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../supportmatrix/">« Support Matrix</a><a class="docs-footer-nextpage" href="../triangle/">Integrating a Triangle »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Monday 23 September 2024 23:05">Monday 23 September 2024</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>