Revision #1758 → #2229 · back to history
modifiedCircle of radius 2 as an equation4ab53c152155
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| mathlib.match_kind | invocation | special_case |
| provenance | ai | ai-moderated |
addedSigned distance from the originee2108e5fb57
modifiedAffine transformations via matricesb6d6fb5f73cd
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| anchors | [{"section":"General matrix form of the transformations","snippet":"All affine transformations of the plane can be described in a uniform way by using matrices."},{"type":"math_alttext","value":"{\\displaystyle {\\begin{pmatrix}x'\\\\y'\\end{pmatrix}}=A{\\begin{pmatrix}x\\\\y\\end{pmatrix}}+b,}"},{"type":"math_alttext","value":"{\\displaystyle A={\\begin{pmatrix}A_{1,1}&A_{1,2}\\\\A_{2,1}&A_{2,2}\\end{pmatrix}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}x'&=xA_{1,1}+yA_{1,1}+b_{1}\\\\y'&=xA_{2,1}+yA_{2,2}+b_{2}.\\end{aligned}}}"}] | — |
modifiedEuclidean transformations have orthogonal matrix2eed0558efe1
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|---|
| anchors | [{"section":"General matrix form of the transformations","snippet":"the Euclidean transformations are characterized by the fact that the matrix"},{"type":"math_alttext","value":"{\\displaystyle A_{1,1}A_{1,2}+A_{2,1}A_{2,2}=0}"},{"type":"math_alttext","value":"{\\displaystyle A_{1,1}^{2}+A_{2,1}^{2}=A_{1,2}^{2}+A_{2,2}^{2}=1.}"}] | — |
modifiedTranslation iff A is identityaa29d0647bdc
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|---|
| anchors | [{"section":"General matrix form of the transformations","snippet":"The transformation is a translation if and only if A is the identity matrix ."},{"type":"math_alttext","value":"{\\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}"}] | — |
modifiedRotation iff A is rotation matrix1b995507cb78
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| anchors | [{"section":"General matrix form of the transformations","snippet":"The transformation is a rotation around some point if and only if A is a rotation matrix"},{"type":"math_alttext","value":"{\\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}"}] | — |
modifiedReflection or glide reflection condition565b391d382d
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|---|
| anchors | [{"section":"General matrix form of the transformations","snippet":"A reflection or glide reflection is obtained when"},{"type":"math_alttext","value":"{\\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=-1.}"}] | — |
modifiedComposition by multiplying matrices191908516801
| Field | From #1758 | To #2229 |
|---|
| anchors | [{"section":"General matrix form of the transformations","snippet":"transformations can be composed by simply multiplying the associated transformation matrices"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{pmatrix}x'\\\\y'\\\\1\\end{pmatrix}}=A'{\\begin{pmatrix}x\\\\y\\\\1\\end{pmatrix}},}"},{"type":"math_alttext","value":"{\\displaystyle A'={\\begin{pmatrix}A_{1,1}&A_{1,2}&b_{1}\\\\A_{2,1}&A_{2,2}&b_{2}\\\\0&0&1\\end{pmatrix}}.}"}] | — |
modifiedAffine transformations of the plane5dba9bcbe4d7
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| anchors | [{"section":"Affine transformation","snippet":"Affine transformations of the Euclidean plane are transformations that map lines to lines"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{pmatrix}A_{1,1}&A_{2,1}&b_{1}\\\\A_{1,2}&A_{2,2}&b_{2}\\\\0&0&1\\end{pmatrix}}{\\begin{pmatrix}x\\\\y\\\\1\\end{pmatrix}}={\\begin{pmatrix}x'\\\\y'\\\\1\\end{pmatrix}}.}"}] | — |
addedSwitching one axis reverses orientation82966534143e
modifiedStandard basisa3e1b6b703ae
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|---|
| anchors | [{"section":"Representing a vector in the standard basis","snippet":"are unit vectors in the direction of the x -axis and y -axis respectively, generally referred to as the standard basis"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {r} =x\\mathbf {i} +y\\mathbf {j} +z\\mathbf {k} ,}"}] | — |