https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_norm&feed=atom&action=history
Geometric norm - Revision history
2024-03-29T13:22:18Z
Revision history for this page on the wiki
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https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_norm&diff=294&oldid=prev
Eric Lengyel: /* See Also */
2024-02-09T01:00:58Z
<p><span dir="auto"><span class="autocomment">See Also</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:00, 9 February 2024</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See Also ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See Also ==</div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Geometric <del style="font-weight: bold; text-decoration: none;">property</del>]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[Geometric <ins style="font-weight: bold; text-decoration: none;">constraint</ins>]]</div></td></tr>
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Eric Lengyel
https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_norm&diff=237&oldid=prev
Eric Lengyel at 07:23, 23 October 2023
2023-10-23T07:23:09Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:23, 23 October 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Bulk Norm ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Bulk Norm ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ''bulk norm'' of an element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$, denoted $$\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ with itself:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ''bulk norm'' of an element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$, denoted $$\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ with itself:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CF} = \sqrt{\mathbf <del style="font-weight: bold; text-decoration: none;">x </del>\mathbin{\unicode{x25CF}} \mathbf <del style="font-weight: bold; text-decoration: none;">x</del>}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CF} = \sqrt{\mathbf <ins style="font-weight: bold; text-decoration: none;">u </ins>\mathbin{\unicode{x25CF}} \mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l43">Line 43:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Weight Norm ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Weight Norm ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ''weight norm'' of an element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$, denoted $$\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ with itself:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ''weight norm'' of an element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$, denoted $$\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ with itself:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CB} = \sqrt{\mathbf <del style="font-weight: bold; text-decoration: none;">x </del>\mathbin{\unicode{x25CB}} \mathbf <del style="font-weight: bold; text-decoration: none;">x</del>}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CB} = \sqrt{\mathbf <ins style="font-weight: bold; text-decoration: none;">u </ins>\mathbin{\unicode{x25CB}} \mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(Note that the square root in this case is taken with respect to the geometric antiproduct.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(Note that the square root in this case is taken with respect to the geometric antiproduct.)</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l85">Line 85:</td>
<td colspan="2" class="diff-lineno">Line 85:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The bulk norm and weight norm are summed to construct the ''geometric norm'' given by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The bulk norm and weight norm are summed to construct the ''geometric norm'' given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert = \left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CF} + \left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CB} = \sqrt{\mathbf <del style="font-weight: bold; text-decoration: none;">x </del>\mathbin{\unicode{x25CF}} \mathbf <del style="font-weight: bold; text-decoration: none;">x</del>} + \sqrt{\mathbf <del style="font-weight: bold; text-decoration: none;">x </del>\mathbin{\unicode{x25CB}} \mathbf <del style="font-weight: bold; text-decoration: none;">x</del>}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert = \left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CF} + \left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CB} = \sqrt{\mathbf <ins style="font-weight: bold; text-decoration: none;">u </ins>\mathbin{\unicode{x25CF}} \mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>} + \sqrt{\mathbf <ins style="font-weight: bold; text-decoration: none;">u </ins>\mathbin{\unicode{x25CB}} \mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l91">Line 91:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert a\mathbf 1 + b{\large\unicode{x1D7D9}}\right\Vert = |a|\mathbf 1 + |b|{\large\unicode{x1D7D9}}$$ .</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert a\mathbf 1 + b{\large\unicode{x1D7D9}}\right\Vert = |a|\mathbf 1 + |b|{\large\unicode{x1D7D9}}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ is given by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ is given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\widehat{\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert} = \dfrac{\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert}{\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf <del style="font-weight: bold; text-decoration: none;">x </del>\mathbin{\unicode{x25CF}} \mathbf <del style="font-weight: bold; text-decoration: none;">x</del>}}{\sqrt{\mathbf <del style="font-weight: bold; text-decoration: none;">x </del>\mathbin{\unicode{x25CB}} \mathbf <del style="font-weight: bold; text-decoration: none;">x</del>}} + {\large\unicode{x1D7D9}}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\widehat{\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert} = \dfrac{\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert}{\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf <ins style="font-weight: bold; text-decoration: none;">u </ins>\mathbin{\unicode{x25CF}} \mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>}}{\sqrt{\mathbf <ins style="font-weight: bold; text-decoration: none;">u </ins>\mathbin{\unicode{x25CB}} \mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>}} + {\large\unicode{x1D7D9}}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.</div></td></tr>
</table>
Eric Lengyel
https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_norm&diff=177&oldid=prev
Eric Lengyel at 19:20, 25 August 2023
2023-08-25T19:20:26Z
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<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:20, 25 August 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Bulk Norm ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Bulk Norm ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ''bulk norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf x$$ with <del style="font-weight: bold; text-decoration: none;">its own [[reverse]]</del>:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ''bulk norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf x$$ with <ins style="font-weight: bold; text-decoration: none;">itself</ins>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf<del style="font-weight: bold; text-decoration: none;">{\tilde </del>x<del style="font-weight: bold; text-decoration: none;">}</del>}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l43">Line 43:</td>
<td colspan="2" class="diff-lineno">Line 43:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Weight Norm ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Weight Norm ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ''weight norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf x$$ with <del style="font-weight: bold; text-decoration: none;">its own [[antireverse]]</del>:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ''weight norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf x$$ with <ins style="font-weight: bold; text-decoration: none;">itself</ins>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CB}} <del style="font-weight: bold; text-decoration: none;">\smash{</del>\mathbf<del style="font-weight: bold; text-decoration: none;">{\underset{\Large\unicode{x7E}}{</del>x<del style="font-weight: bold; text-decoration: none;">}}}\vphantom{\mathbf{\tilde x}}</del>}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(Note that the square root in this case is taken with respect to the geometric antiproduct.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(Note that the square root in this case is taken with respect to the geometric antiproduct.)</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l83">Line 83:</td>
<td colspan="2" class="diff-lineno">Line 83:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Geometric Norm ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Geometric Norm ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The bulk norm and weight norm are summed to construct <del style="font-weight: bold; text-decoration: none;">a </del>''geometric norm'' given by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The bulk norm and weight norm are summed to construct <ins style="font-weight: bold; text-decoration: none;">the </ins>''geometric norm'' given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf<del style="font-weight: bold; text-decoration: none;">{\tilde </del>x<del style="font-weight: bold; text-decoration: none;">}</del>} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} <del style="font-weight: bold; text-decoration: none;">\smash{</del>\mathbf<del style="font-weight: bold; text-decoration: none;">{\underset{\Large\unicode{x7E}}{</del>x<del style="font-weight: bold; text-decoration: none;">}}}\vphantom{\mathbf{\tilde x}}</del>}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l93">Line 93:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf x$$ is given by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf x$$ is given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf<del style="font-weight: bold; text-decoration: none;">{\tilde </del>x<del style="font-weight: bold; text-decoration: none;">}</del>}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} <del style="font-weight: bold; text-decoration: none;">\smash{</del>\mathbf<del style="font-weight: bold; text-decoration: none;">{\underset{\Large\unicode{x7E}}{</del>x<del style="font-weight: bold; text-decoration: none;">}}}\vphantom{\mathbf{\tilde x}}</del>}} + {\large\unicode{x1D7D9}}$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}} + {\large\unicode{x1D7D9}}$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.</div></td></tr>
</table>
Eric Lengyel
https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_norm&diff=40&oldid=prev
Eric Lengyel: Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d..."
2023-07-15T06:02:43Z
<p>Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For <a href="/wiki/index.php?title=Points" class="mw-redirect" title="Points">points</a>, <a href="/wiki/index.php?title=Lines" class="mw-redirect" title="Lines">lines</a>, and <a href="/wiki/index.php?title=Planes" class="mw-redirect" title="Planes">planes</a>, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For <a href="/wiki/index.php?title=Motors" class="mw-redirect" title="Motors">motors</a> and <a href="/wiki/index.php?title=Flectors" class="mw-redirect" title="Flectors">flectors</a>, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d..."</p>
<p><b>New page</b></p><div>The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm.<br />
<br />
For [[points]], [[lines]], and [[planes]], the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For [[motors]] and [[flectors]], the geometric norm is equal to half the distance that the origin is moved by the isometry operator.<br />
<br />
== Bulk Norm ==<br />
<br />
The ''bulk norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf x$$ with its own [[reverse]]:<br />
<br />
:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf{\tilde x}}$$ .<br />
<br />
An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.<br />
<br />
The following table lists the bulk norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.<br />
<br />
{| class="wikitable"<br />
! Type !! Definition !! Bulk Norm<br />
|-<br />
| style="padding: 12px;" | [[Magnitude]]<br />
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CF} = |x|$$<br />
|-<br />
| style="padding: 12px;" | [[Point]]<br />
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CF} = \sqrt{p_x^2 + p_y^2 + p_z^2}$$<br />
|-<br />
| style="padding: 12px;" | [[Line]]<br />
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$<br />
| style="padding: 12px;" | $$\left\Vert\boldsymbol l\right\Vert_\unicode{x25CF} = \sqrt{l_{mx}^2 + l_{my}^2 + l_{mz}^2}$$<br />
|-<br />
| style="padding: 12px;" | [[Plane]]<br />
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf g\right\Vert_\unicode{x25CF} = |g_w|$$<br />
|-<br />
| style="padding: 12px;" | [[Motor]]<br />
| style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$<br />
|-<br />
| style="padding: 12px;" | [[Flector]]<br />
| style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf F\right\Vert_\unicode{x25CF} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$<br />
|}<br />
<br />
== Weight Norm ==<br />
<br />
The ''weight norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf x$$ with its own [[antireverse]]:<br />
<br />
:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}\vphantom{\mathbf{\tilde x}}}$$ .<br />
<br />
(Note that the square root in this case is taken with respect to the geometric antiproduct.)<br />
<br />
An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''weight normalized'' or ''[[unitized]]''.<br />
<br />
The following table lists the weight norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.<br />
<br />
{| class="wikitable"<br />
! Type !! Definition !! Weight Norm<br />
|-<br />
| style="padding: 12px;" | [[Magnitude]]<br />
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CB} = |y|{\large\unicode{x1D7D9}}$$<br />
|-<br />
| style="padding: 12px;" | [[Point]]<br />
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CB} = |p_w|{\large\unicode{x1D7D9}}$$<br />
|-<br />
| style="padding: 12px;" | [[Line]]<br />
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$<br />
| style="padding: 12px;" | $$\left\Vert\boldsymbol l\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{l_{vx}^2 + l_{vy}^2 + l_{vz}^2}$$<br />
|-<br />
| style="padding: 12px;" | [[Plane]]<br />
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf g\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{g_x^2 + g_y^2 + g_z^2}$$<br />
|-<br />
| style="padding: 12px;" | [[Motor]]<br />
| style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$<br />
|-<br />
| style="padding: 12px;" | [[Flector]]<br />
| style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$<br />
| style="padding: 12px;" | $$\left\Vert\mathbf F\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$<br />
|}<br />
<br />
== Geometric Norm ==<br />
<br />
The bulk norm and weight norm are summed to construct a ''geometric norm'' given by<br />
<br />
:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf{\tilde x}} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}\vphantom{\mathbf{\tilde x}}}$$ .<br />
<br />
This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because<br />
<br />
:$$\left\Vert a\mathbf 1 + b{\large\unicode{x1D7D9}}\right\Vert = |a|\mathbf 1 + |b|{\large\unicode{x1D7D9}}$$ .<br />
<br />
Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf x$$ is given by<br />
<br />
:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf{\tilde x}}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}\vphantom{\mathbf{\tilde x}}}} + {\large\unicode{x1D7D9}}$$ .<br />
<br />
The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.<br />
<br />
{| class="wikitable"<br />
! Type !! Definition !! Geometric Norm !! Interpretation<br />
|-<br />
| style="padding: 12px;" | [[Magnitude]]<br />
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$<br />
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf z\right\Vert} = \dfrac{|x|}{|y|}$$<br />
| style="padding: 12px;" | A Euclidean distance.<br />
|-<br />
| style="padding: 12px;" | [[Point]]<br />
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$<br />
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf p\right\Vert} = \dfrac{\sqrt{p_x^2 + p_y^2 + p_z^2}}{|p_w|}$$<br />
| style="padding: 12px;" | Distance from the origin to the point $$\mathbf p$$.<br />
<br />
Half the distance that the origin is moved by the [[flector]] $$\mathbf p$$.<br />
|-<br />
| style="padding: 12px;" | [[Line]]<br />
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$<br />
| style="padding: 12px;" | $$\widehat{\left\Vert\boldsymbol l\right\Vert} = \sqrt{\dfrac{l_{mx}^2 + l_{my}^2 + l_{mz}^2}{l_{vx}^2 + l_{vy}^2 + l_{vz}^2}}$$<br />
| style="padding: 12px;" | Perpendicular distance from the origin to the line $$\boldsymbol l$$.<br />
<br />
Half the distance that the origin is moved by the [[motor]] $$\boldsymbol l$$.<br />
|-<br />
| style="padding: 12px;" | [[Plane]]<br />
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$<br />
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf g\right\Vert} = \dfrac{|g_w|}{\sqrt{g_x^2 + g_y^2 + g_z^2}}$$<br />
| style="padding: 12px;" | Perpendicular distance from the origin to the plane $$\mathbf g$$.<br />
<br />
Half the distance that the origin is moved by the [[flector]] $$\mathbf g$$.<br />
|-<br />
| style="padding: 12px;" | [[Motor]]<br />
| style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$<br />
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{\dfrac{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}}$$<br />
| style="padding: 12px;" | Half the distance that the origin is moved by the [[motor]] $$\mathbf Q$$.<br />
|-<br />
| style="padding: 12px;" | [[Flector]]<br />
| style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$<br />
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf F\right\Vert} = \sqrt{\dfrac{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}}$$<br />
| style="padding: 12px;" | Half the distance that the origin is moved by the [[flector]] $$\mathbf F$$.<br />
|}<br />
<br />
== See Also ==<br />
<br />
* [[Geometric property]]</div>
Eric Lengyel