MediaWiki API result

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        "*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Line 1:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Line 1:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">&lt;strong&gt;MediaWiki has been installed.&lt;/strong&gt;</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">__NOTOC__</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Conformal Geometric Algebra ==</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br/></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br/></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Consult </del>the <del class=\"diffchange diffchange-inline\">[https://www</del>.<del class=\"diffchange diffchange-inline\">mediawiki.org/</del>wiki<del class=\"diffchange diffchange-inline\">/Special:MyLanguage/Help</del>:<del class=\"diffchange diffchange-inline\">Contents User's Guide] for information on using the wiki software.</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">This wiki is a repository of information about Conformal Geometric Algebra, and specifically </ins>the <ins class=\"diffchange diffchange-inline\">five-dimensional Clifford algebra $$\\mathcal G_{4,1}$$</ins>. <ins class=\"diffchange diffchange-inline\">This </ins>wiki <ins class=\"diffchange diffchange-inline\">is associated with the following websites</ins>:</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br/></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br/></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">== Getting started ==</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [<ins class=\"diffchange diffchange-inline\">http</ins>://<ins class=\"diffchange diffchange-inline\">projectivegeometricalgebra</ins>.org <ins class=\"diffchange diffchange-inline\">Projective Geometric Algebra overview site</ins>]</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www.mediawiki</del>.org<del class=\"diffchange diffchange-inline\">/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [<ins class=\"diffchange diffchange-inline\">http</ins>://<ins class=\"diffchange diffchange-inline\">rigidgeometricalgebra</ins>.org/wiki/<ins class=\"diffchange diffchange-inline\">index.php?title=Main_Page Rigid Geometric Algebra companion site</ins>]</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www.mediawiki</del>.org/wiki/<del class=\"diffchange diffchange-inline\">Special:MyLanguage/Manual:FAQ MediaWiki FAQ</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https://lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.org<del class=\"diffchange diffchange-inline\">/postorius/lists/mediawiki</del>-<del class=\"diffchange diffchange-inline\">announce</del>.<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org/ MediaWiki release mailing list</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Conformal geometric algebra is an area of active research, and new information is frequently being added to this wiki</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www.mediawiki</del>.org/wiki/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Localisation#Translation_resources Localise MediaWiki for your language]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org/wiki/Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage/Manual</del>:<del class=\"diffchange diffchange-inline\">Combating_spam Learn how to combat spam on your wiki</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''If you are experiencing problems with the LaTeX on this site, please clear the cookies for conformalgeometricalgebra</ins>.org <ins class=\"diffchange diffchange-inline\">and reload.'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Introduction ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Conformal geometric algebra is constructed by adding two projective basis vectors called $$\\mathbf e_-$$ and $$\\mathbf e_+$$ to the set of ordinary basis vectors $$\\mathbf e_1, \\mathbf e_2, \\ldots, \\mathbf e_n$$ of ''n''-dimensional Euclidean space. The new vectors are named this way because their squares under the dot product are</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:$$\\mathbf e_- \\cdot \\mathbf e_- = </ins>-<ins class=\"diffchange diffchange-inline\">1 \\qquad \\mathrm{and} \\qquad \\mathbf e_+ \\cdot \\mathbf e_+ = +1$$ </ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">This by itself is enough to build the entire algebra and observe all of its emergent properties</ins>. <ins class=\"diffchange diffchange-inline\">What follows is the standard way to interpret what's going on when we talk about specific vectors, bivectors, etc</ins>.<ins class=\"diffchange diffchange-inline\">, and start multiplying them together with the [[exterior product]] and [[geometric product]</ins>]<ins class=\"diffchange diffchange-inline\">. Using 3D Euclidean space as an example, we begin by considering a homogeneous point</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:$$\\mathbf p = x \\mathbf e_1 + y \\mathbf e_2 + z \\mathbf e_3 + w \\mathbf e_-$$ .</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">As in </ins>[<ins class=\"diffchange diffchange-inline\">http</ins>://<ins class=\"diffchange diffchange-inline\">rigidgeometricalgebra</ins>.org/wiki/<ins class=\"diffchange diffchange-inline\">index.php?title=Main_Page rigid geometric algebra], any nonzero scalar multiple of this point belongs to the same equivalence class, so we can just assume $$w = 1$$. Now we perform a stereographic projection onto the four-dimensional unit hypersphere centered at $$\\mathbf e_-$$ and extending into the $${\\mathbf e_1, \\mathbf e_2, \\mathbf e_3, \\mathbf e_+}$$ subspace. The north pole of this hypersphere toward which points are projected is $$\\mathbf e_- + \\mathbf e_+$$. This projection transforms the point $$\\mathbf p$$ into</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\">$$\\mathbf p = \\dfrac{2x}{x^2 + y^2 + z^2 + 1} \\mathbf e_1 + \\dfrac{2y}{x^2 + y^2 + z^2 + 1} \\mathbf e_2 + \\dfrac{2z}{x^2 + y^2 + z^2 + 1} \\mathbf e_3 + \\mathbf e_- + \\dfrac{x^2 + y^2 + z^2 - 1}{x^2 + y^2 + z^2 + 1} \\mathbf e_+$$ .</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">We can homogeneously multiply this by $$(x^2 + y^2 + z^2 + 1)</ins>/<ins class=\"diffchange diffchange-inline\">2$$ without changing the meaning of $$\\mathbf p$$ to obtain</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:$$\\mathbf p = x \\mathbf e_1 + y \\mathbf e_2 + z \\mathbf e_3 + \\dfrac{1}{2}(x^2 + y^2 + z^2 + 1) \\mathbf e_- + \\dfrac{1}{2}(x^2 + y^2 + z^2 - 1) \\mathbf e_+$$ .</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Regrouping the coefficients of the $$\\mathbf e_-$$ and $$\\mathbf e_+$$ terms lets us write</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:$$\\mathbf p = x \\mathbf e_1 + y \\mathbf e_2 + z \\mathbf e_3 + \\dfrac{1}{2}(\\mathbf e_- - \\mathbf e_+) + \\dfrac{1}{2}(x^2 + y^2 + z^2)(\\mathbf e_- + \\mathbf e_+)$$ .</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">It is convenient to define two new vectors $$\\mathbf e_4$$ and $$\\mathbf e_5$$ as</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:$$\\mathbf e_4 = \\dfrac{1}{2}(\\mathbf e_- - \\mathbf e_+) \\qquad \\mathrm{and} \\qquad \\mathbf e_5 = \\mathbf e_- + \\mathbf e_+$$ .</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Using these two vectors, the point $$\\mathbf p$$ is finally expressed as</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\">$$\\mathbf p = x \\mathbf e_1 + y \\mathbf e_2 + z \\mathbf e_3 + \\mathbf e_4 + \\dfrac{1}{2}(x^2 + y^2 + z^2) \\mathbf e_5$$ .</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Numerous alternate definitions for $$\\mathbf e_4$$ and $$\\mathbf e_5$$ in which $$\\mathbf e_4 = a(\\mathbf e_- - \\mathbf e_+)$$ and $$\\mathbf e_5 = b(\\mathbf e_- + \\mathbf e_+)$$ are possible, but the definitions above where $$a = 1</ins>/<ins class=\"diffchange diffchange-inline\">2$$ and $$b = 1$$ tend to produce the cleanest formulation of the entire algebra. In particular, any definitions in which $$ab = 1</ins>/<ins class=\"diffchange diffchange-inline\">2$$ make the bivectors $$\\mathbf e_- \\wedge \\mathbf e_+$$ and $$\\mathbf e_4 \\wedge \\mathbf e_5$$ equal to each other</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">When we consider the case that $$(x, y, z) = (0, 0, 0)$$, we can clearly identify the vector $$\\mathbf e_4$$ as the origin</ins>. <ins class=\"diffchange diffchange-inline\">If we homogeneously rescale $$\\mathbf p$$ as</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\">$$\\mathbf p = \\dfrac{2x}{x^2 + y^2 + z^2} \\mathbf e_1 + \\dfrac{2y}{x^2 + y^2 + z^2} \\mathbf e_2 + \\dfrac{2z}{x^2 + y^2 + z^2} \\mathbf e_3 + \\dfrac{2}{x^2 + y^2 + z^2} \\mathbf e_4 + \\mathbf e_5$$ ,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">and allow the magnitude of $$(x, y, z)$$ to become arbitrarily large, then it's apparent that $$\\mathbf e_5$$ represents the point at infinity.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The following diagram illustrates the image of the ''x'' axis under the homogeneous stereographic projection that we defined $$\\mathbf p$$ with. Euclidean space becomes a parabolic surface called the ''horosphere''. The ''x''-''y'' plane is mapped to a paraboloid, and the full three-dimensional Euclidean space is mapped to a higher-dimensional parabolic volume.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Image</ins>:<ins class=\"diffchange diffchange-inline\">horosphere.svg|512px]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Pages ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== The seven types of geometric objects ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Flat point]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Line]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Plane]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Round point]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Dipole]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Circle]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Sphere]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Various properties and unary operations ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Duals]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Carriers]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Attitude]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Centers]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Containers]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Partners]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Products and other binary operations ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Geometric products]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Exterior products]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Join and meet]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Connect]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Projections]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Conformal Transformations ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Translation]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Rotation]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Dilation]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Reflection]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Inversion]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Transversion]</ins>]</div></td></tr>\n"
    }
}