File:Academ Reflections with parallel axis on wallpaper.svg
Summary
A given <a href="https://en.wikipedia.org/wiki/Translation_(geometry)" class="extiw" title="w:Translation (geometry)">translation</a> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a137981d17306dd1514f144fd838043daf8fdf9f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.209ex; width:1.884ex; height:2.176ex;" alt="{\displaystyle {\mathit {T}}}"> may be decomposed into two reflections with parallel axis: two lines here named Δ 1 and Δ 2 . If one of the reflections <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e19122bb1751fb9eb31b2332be4522162a520f" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:3.154ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,1}}"> or <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ac871480ef2115d3daf895bc67cd485cb1ab86" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:3.154ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,2}}"> is given, the axis of which is perpendicular to the direction of <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/603dc836657f5c46a74a6748738930f78d5ad43d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.332ex; height:2.509ex;" alt="{\displaystyle {\mathit {T}},}"> the other reflection is unique which fulfills the equality: <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e00ec6be02fc99a99c7fb1ea7b555a5aba229ce" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:16.343ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,2}~\circ ~{\mathit {R}}_{\,1}~=~{\mathit {T.}}}"> There is a unique translation <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b047feeb7dc2e5628923eefa42344135b48a21" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.165ex; width:1.482ex; height:2.176ex;" alt="{\displaystyle {\mathit {S}}}"> that transforms Δ 1 into Δ 2 , then <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf082d96874cbf3d494bdf3ac738932426311242" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:9.368ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle {\mathit {S}}^{\,2}~=~{\mathit {T}},}"> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad26a5b7c166df297ae2c42ed55bffba599bdf01" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:34.208ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle \mathrm {and\quad } {\mathit {S}}^{\,-\,2}~=~{\mathit {T}}^{\,-\,1}~=~{\mathit {R}}_{\,1}~\circ ~{\mathit {R}}_{\,2}.}"> The <a href="https://en.wikipedia.org/wiki/Trapezoid" class="extiw" title="w:Trapezoid">trapezoid</a> ABCD is part of the tiling of the top of the drawing, where the triangular elements are green and yellow. Across the line Δ 1, the tiling and ABCD are transformed by the reflection <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ccd5e66551dd6c38a62c8c79898fcb1fd10471" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:3.811ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,1},}"> the trapezoid ABCD becomes A 1 B 1 C 1 D 1 , and the elements of the tiling become blue and gray as in <a href="//commons.wikimedia.org/wiki/File:Academ_Repeated_translations_on_tiling.svg" title="File:Academ Repeated translations on tiling.svg">this image</a> or this <a href="//commons.wikimedia.org/wiki/File:Academ_Composing_translations_on_tiling.svg" title="File:Academ Composing translations on tiling.svg">other one.</a> The image of ABCD through the translation <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a137981d17306dd1514f144fd838043daf8fdf9f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.209ex; width:1.884ex; height:2.176ex;" alt="{\displaystyle {\mathit {T}}}"> is the trapezoid A 2 B 2 C 2 D 2 , that is filled by three elements of the first tiling, one green triangle and two yellow.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 23:55, 4 January 2017 | | 750 × 750 (4 KB) | 127.0.0.1 (talk) | <br>A given <a href="https://en.wikipedia.org/wiki/Translation_(geometry)" class="extiw" title="w:Translation (geometry)">translation</a> <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">T</mi></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {T}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a137981d17306dd1514f144fd838043daf8fdf9f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.209ex; width:1.884ex; height:2.176ex;" alt="{\displaystyle {\mathit {T}}}"></span> may be decomposed into two reflections with parallel axis: two lines here named <span style="white-space:nowrap"><i>Δ</i><sub> 1</sub> and <i>Δ</i><sub> 2 </sub>. </span> If one of the reflections <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>1</mn></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {R}}_{\,1}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e19122bb1751fb9eb31b2332be4522162a520f" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:3.154ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,1}}"></span> <span style="white-space:nowrap">or <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {R}}_{\,2}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ac871480ef2115d3daf895bc67cd485cb1ab86" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:3.154ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,2}}"></span> </span> is given, the axis of which is perpendicular to the direction of <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">T</mi></mrow></mrow><mo>,</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {T}},}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/603dc836657f5c46a74a6748738930f78d5ad43d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.332ex; height:2.509ex;" alt="{\displaystyle {\mathit {T}},}"></span> the other reflection is unique which fulfills the equality: <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msub><mtext> </mtext><mo>∘<!-- ∘ --></mo><mtext> </mtext><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>1</mn></mrow></msub><mtext> </mtext><mo>=</mo><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">T</mi><mo class="MJX-tex-mathit" mathvariant="italic">.</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {R}}_{\,2}~\circ ~{\mathit {R}}_{\,1}~=~{\mathit {T.}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e00ec6be02fc99a99c7fb1ea7b555a5aba229ce" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:16.343ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,2}~\circ ~{\mathit {R}}_{\,1}~=~{\mathit {T.}}}"></span> There is a unique translation <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">S</mi></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {S}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b047feeb7dc2e5628923eefa42344135b48a21" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.165ex; width:1.482ex; height:2.176ex;" alt="{\displaystyle {\mathit {S}}}"></span> that transforms <span style="white-space:nowrap"><i>Δ</i><sub> 1</sub> into <i>Δ</i><sub> 2 </sub>, </span> then <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">S</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msup><mtext> </mtext><mo>=</mo><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">T</mi></mrow></mrow><mo>,</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {S}}^{\,2}~=~{\mathit {T}},}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf082d96874cbf3d494bdf3ac738932426311242" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:9.368ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle {\mathit {S}}^{\,2}~=~{\mathit {T}},}"></span> <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mspace width="1em"></mspace></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">S</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mo>−<!-- − --></mo><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msup><mtext> </mtext><mo>=</mo><mtext> </mtext><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">T</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mo>−<!-- − --></mo><mspace width="thinmathspace"></mspace><mn>1</mn></mrow></msup><mtext> </mtext><mo>=</mo><mtext> </mtext><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>1</mn></mrow></msub><mtext> </mtext><mo>∘<!-- ∘ --></mo><mtext> </mtext><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msub><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \mathrm {and\quad } {\mathit {S}}^{\,-\,2}~=~{\mathit {T}}^{\,-\,1}~=~{\mathit {R}}_{\,1}~\circ ~{\mathit {R}}_{\,2}.}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad26a5b7c166df297ae2c42ed55bffba599bdf01" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:34.208ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle \mathrm {and\quad } {\mathit {S}}^{\,-\,2}~=~{\mathit {T}}^{\,-\,1}~=~{\mathit {R}}_{\,1}~\circ ~{\mathit {R}}_{\,2}.}"></span> The <a href="https://en.wikipedia.org/wiki/Trapezoid" class="extiw" title="w:Trapezoid">trapezoid</a> <i>ABCD </i> is part of the tiling of the top of the drawing, where the triangular elements are green and yellow. <span style="white-space:nowrap">Across the line <i>Δ</i><sub> 1</sub>, </span> the tiling and <i>ABCD </i> are transformed by the reflection <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">R</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mn>1</mn></mrow></msub><mo>,</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {R}}_{\,1},}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ccd5e66551dd6c38a62c8c79898fcb1fd10471" class="mwe-math-fallback-image-inline" style="vertical-align: -0.671ex; width:3.811ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle {\mathit {R}}_{\,1},}"></span> the trapezoid <i>ABCD </i> <span style="white-space:nowrap">becomes <i>A</i><sub> 1 </sub><i>B</i><sub> 1 </sub><i>C</i><sub> 1 </sub><i>D</i><sub> 1 </sub>, </span> and the elements of the tiling become blue and gray as in <a href="//commons.wikimedia.org/wiki/File:Academ_Repeated_translations_on_tiling.svg" title="File:Academ Repeated translations on tiling.svg">this image</a> or this <a href="//commons.wikimedia.org/wiki/File:Academ_Composing_translations_on_tiling.svg" title="File:Academ Composing translations on tiling.svg">other one.</a> The image of <i>ABCD </i> through the translation <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-mathit" mathvariant="italic">T</mi></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\mathit {T}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a137981d17306dd1514f144fd838043daf8fdf9f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.209ex; width:1.884ex; height:2.176ex;" alt="{\displaystyle {\mathit {T}}}"></span> is the trapezoid <span style="white-space:nowrap"><i>A</i><sub> 2 </sub><i>B</i><sub> 2 </sub><i>C</i><sub> 2 </sub><i>D</i><sub> 2 </sub>, </span> that is filled by three elements of the first tiling, one green triangle and two yellow.<br> |
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