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| <!-- content from article translation (physics) has been merged with this article {{Merge from|Translation (physics)|date=December 2010}}-->
| | The ac1st16.dll error is annoying and especially prevalent with all kinds of Windows computers. Not only does it make your computer run slower, yet it may furthermore prevent we from utilizing a range of programs, including AutoCAD. To fix this issue, you need to utilize a easy method to cure all of the potential issues which cause it. Here's what we should do...<br><br>You are able to reformat the computer to make it run quicker. This will reset a computer to whenever we first used it. Always remember to back up all files plus programs before doing this since this might remove a files from a database. Remember before we do this we need all of the motorists and installation files plus this ought to be a last resort when you are shopping for slow computer tips.<br><br>The error is basically a outcome of problem with Windows Installer package. The Windows Installer is a tool chosen to install, uninstall plus repair the many programs on the computer. Let you discuss a few points that helped a great deal of persons that facing the synonymous problem.<br><br>In purchase to remove the programs on the computer, Windows Installer must be in a healthy state. If its installation is corrupted you may get error 1721 inside Windows 7, Vista and XP throughout the program removal process. Just re-registering its component files would resolve your issue.<br><br>The [http://bestregistrycleanerfix.com/system-mechanic system mechanic] should come as standard with a back up plus restore facility. This should be an convenient to apply task.That means which should you encounter a issue with your PC following using a registry cleaning we can just restore a settings.<br><br>S/w associated error handling - If the blue screen bodily memory dump occurs after the installation of s/w application or a driver it could be which there is system incompatibility. By booting into safe mode and removing the software you can swiftly fix this error. We might furthermore try out a "program restore" to revert to an earlier state.<br><br>Your disk requires area inside order to run smoothly. By freeing up several room from your disk, you are capable to accelerate your PC a bit. Delete all file in the temporary web files folder, recycle bin, clear shortcuts plus icons from your desktop which you never use plus remove programs you never use.<br><br>There is a lot a wise registry cleaner can do for the computer. It could check for and download updates for Windows, Java plus Adobe. Keeping updates present is an significant piece of wise computer health. It will also safeguard a individual and business confidentiality and also your online protection. |
| <!-- added references {{unreferenced|date=December 2007}}-->
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| [[Image:TraslazioneOK.png|right|thumb|A translation moves every point of a figure or a space by the same amount in a given direction.]]
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| [[Image:Simx2=traslOK.png|right|thumb|A [[reflection (mathematics)|reflection]] against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.]]
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| In [[Euclidean geometry]], a '''translation''' is a function that moves every point a constant distance in a specified direction.<ref>{{cite book|authors=Osgood, William F. & Graustein, William C.|title=Plane and solid analytic geometry|publisher=The Macmillan Company|year=1921|page=330|url=http://books.google.com/books?id=mxOBAAAAMAAJ&pg=PA330}}</ref> A translation can be described as a [[Euclidean group|rigid motion]]: other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant [[vector space|vector]] to every point, or as shifting the [[Origin (mathematics)|origin]] of the [[coordinate system]]. A '''translation operator''' is an [[operator (mathematics)|operator]] <math>T_\mathbf{\delta}</math> such that <math>T_\mathbf{\delta} f(\mathbf{v}) = f(\mathbf{v}+\mathbf{\delta}).</math>
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| If '''v''' is a fixed vector, then the translation ''T''<sub>'''v'''</sub> will work as ''T''<sub>'''v'''</sub>('''p''') = '''p''' + '''v'''.
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| If ''T'' is a translation, then the [[image (mathematics)|image]] of a subset ''A'' under the [[function (mathematics)|function]] ''T'' is the '''translate''' of ''A'' by ''T''. The translate of ''A'' by ''T''<sub>'''v'''</sub> is often written ''A'' + '''v'''.
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| In a [[Euclidean space]], any translation is an [[isometry]]. The set of all translations forms the translation group ''T'', which is isomorphic to the space itself, and a [[normal subgroup]] of [[Euclidean group]] ''E''(''n'' ). The [[quotient group]] of ''E''(''n'' ) by ''T'' is isomorphic to the [[orthogonal group]] ''O''(''n'' ): | |
| :''E''(''n'' ) ''/ T'' ≅ ''O''(''n'' ).
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| == Matrix representation ==<!-- This section is linked from [[Affine transformation]] -->
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| A translation is an [[affine transformation]] with ''no'' [[fixed point (mathematics)|fixed point]]s. Matrix multiplications ''always'' have the [[origin (mathematics)|origin]] as a fixed point. Nevertheless, there is a common [[workaround]] using [[homogeneous coordinates]] to represent a translation of a [[vector space]] with [[matrix multiplication]]: Write the 3-dimensional vector '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>) using 4 homogeneous coordinates as '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>, 1).<ref> Richard Paul, 1981, [http://books.google.com/books?id=UzZ3LAYqvRkC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators], MIT Press, Cambridge, MA</ref>
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| To translate an object by a [[Vector (geometry)|vector]] '''v''', each homogeneous vector '''p''' (written in homogeneous coordinates) can be multiplied by this '''translation matrix''':
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| : <math> T_{\mathbf{v}} =
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| \begin{bmatrix}
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| 1 & 0 & 0 & v_x \\
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| 0 & 1 & 0 & v_y \\
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| 0 & 0 & 1 & v_z \\
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| 0 & 0 & 0 & 1
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| \end{bmatrix}
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| </math> | |
| As shown below, the multiplication will give the expected result:
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| : <math> T_{\mathbf{v}} \mathbf{p} =
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| \begin{bmatrix}
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| 1 & 0 & 0 & v_x \\
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| 0 & 1 & 0 & v_y\\
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| 0 & 0 & 1 & v_z\\
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| 0 & 0 & 0 & 1
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| \end{bmatrix}
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| \begin{bmatrix}
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| p_x \\ p_y \\ p_z \\ 1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1
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| \end{bmatrix}
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| = \mathbf{p} + \mathbf{v} </math>
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| The inverse of a translation matrix can be obtained by reversing the direction of the vector:
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| : <math> T^{-1}_{\mathbf{v}} = T_{-\mathbf{v}} . \! </math>
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| Similarly, the product of translation matrices is given by adding the vectors:
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| : <math> T_{\mathbf{u}}T_{\mathbf{v}} = T_{\mathbf{u}+\mathbf{v}} . \! </math>
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| Because addition of vectors is [[commutative]], multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).
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| ==Translations in physics==
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| In [[physics]], '''translation''' (Translational motion) is movement that changes the [[displacement (vector)|position]] of an object, as opposed to [[rotation]]. For example, according to Whittaker:<ref name=Whittaker>{{cite book |title=A Treatise on the Analytical Dynamics of Particles and Rigid Bodies |author=Edmund Taylor Whittaker |isbn=0-521-35883-3 |publisher=Cambridge University Press |year=1988 |url=http://books.google.com/books?id=epH1hCB7N2MC&pg=PA4&dq=rigid+bodies+translation#PPA1,M1 |edition=Reprint of fourth edition of 1936 with foreword by William McCrea |page=1}}</ref>
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| {{Quotation|If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length ''ℓ'', so that the orientation of the body in space is unaltered, the displacement is called a ''translation parallel to the direction of the lines, through a distance ℓ''. |E.T. Whittaker: ''A Treatise on the Analytical Dynamics of Particles and Rigid Bodies'', p. 1}}
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| A translation is the operation changing the positions of all points ''(x, y, z)'' of an object according to the formula
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| :<math>(x,y,z) \to (x+\Delta x,y+\Delta y, z+\Delta z)</math>
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| where <math>(\Delta x,\ \Delta y,\ \Delta z)</math> is the same [[Euclidean vector|vector]] for each point of the object. The translation vector <math>(\Delta x,\ \Delta y,\ \Delta z)</math> common to all points of the object describes a particular type of [[Displacement (vector)|displacement]] of the object, usually called a ''linear'' displacement to distinguish it from displacements involving rotation, called ''angular'' displacements.
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| A translation of space (or time) should not be confused with a translation of an object. Such translations have no [[Fixed point (mathematics)|fixed points]].
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| == See also ==
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| * [[Translational symmetry]]
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| * [[Transformation matrix]]
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| * [[Rotation matrix]]
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| * [[Scaling (geometry)]]
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| * [[Advection]]
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| ==External links==
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| {{Commons category|Translation (geometry)}}
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml Translation Transform] at [[cut-the-knot]]
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| * [http://www.mathsisfun.com/geometry/translation.html Geometric Translation (Interactive Animation)] at Math Is Fun
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| * [http://demonstrations.wolfram.com/Understanding2DTranslation/ Understanding 2D Translation] and [http://demonstrations.wolfram.com/Understanding3DTranslation/ Understanding 3D Translation] by Roger Germundsson, [[The Wolfram Demonstrations Project]].
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Translation (Geometry)}}
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| [[Category:Euclidean symmetries]]
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| [[Category:Transformation (function)]]
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The ac1st16.dll error is annoying and especially prevalent with all kinds of Windows computers. Not only does it make your computer run slower, yet it may furthermore prevent we from utilizing a range of programs, including AutoCAD. To fix this issue, you need to utilize a easy method to cure all of the potential issues which cause it. Here's what we should do...
You are able to reformat the computer to make it run quicker. This will reset a computer to whenever we first used it. Always remember to back up all files plus programs before doing this since this might remove a files from a database. Remember before we do this we need all of the motorists and installation files plus this ought to be a last resort when you are shopping for slow computer tips.
The error is basically a outcome of problem with Windows Installer package. The Windows Installer is a tool chosen to install, uninstall plus repair the many programs on the computer. Let you discuss a few points that helped a great deal of persons that facing the synonymous problem.
In purchase to remove the programs on the computer, Windows Installer must be in a healthy state. If its installation is corrupted you may get error 1721 inside Windows 7, Vista and XP throughout the program removal process. Just re-registering its component files would resolve your issue.
The system mechanic should come as standard with a back up plus restore facility. This should be an convenient to apply task.That means which should you encounter a issue with your PC following using a registry cleaning we can just restore a settings.
S/w associated error handling - If the blue screen bodily memory dump occurs after the installation of s/w application or a driver it could be which there is system incompatibility. By booting into safe mode and removing the software you can swiftly fix this error. We might furthermore try out a "program restore" to revert to an earlier state.
Your disk requires area inside order to run smoothly. By freeing up several room from your disk, you are capable to accelerate your PC a bit. Delete all file in the temporary web files folder, recycle bin, clear shortcuts plus icons from your desktop which you never use plus remove programs you never use.
There is a lot a wise registry cleaner can do for the computer. It could check for and download updates for Windows, Java plus Adobe. Keeping updates present is an significant piece of wise computer health. It will also safeguard a individual and business confidentiality and also your online protection.