Difference between revisions of "Translation"

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{{Navigation|before=[[Transformations]]|overview=[[Transformations]]|next=[[Rotation]]}}
 
{{Navigation|before=[[Transformations]]|overview=[[Transformations]]|next=[[Rotation]]}}
  
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[File:translation1.png|right|300px]
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[[File:translation1.png|right|250px]]
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Translation is the easiest kind of transformation. Translating a point <math>p</math> means that it is shifted by a translation vector. So the translation vector <math>\vec{\mathbf{t}}</math> is added to the position vector <math>\vec{\mathbf{p}}</math> of <math>p</math>. The position vector <math>\vec{\mathbf{p}_t}</math> of the resulting transformed point <math>p_t</math> is calculated as follows:<br/>
+
Translation is the easiest kind of transformation. Translating a point <math>q_0</math> means that it is shifted by a translation vector. So the translation vector <math>\vec{\mathbf{p}}</math> is added to the position vector <math>\vec{\mathbf{q}}_0</math> of <math>q_0</math>. The position vector <math>\vec{\mathbf{q}}_1</math> of the resulting transformed point <math>q_1</math> is calculated as follows:<br/>
 
:<math>
 
:<math>
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\vec{\mathbf{p}_t}=\vec{\mathbf{p}}+\vec{\mathbf{t}}
+
\vec{\mathbf{q}}_1=\vec{\mathbf{q}}_0+\vec{\mathbf{p}}
 +
</math>
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The figure on the right shows an example in two-dimensional space. In robotics usually three dimensions are regarded. Considering the particular components of the vectors, a translation looks as follows:<br/>
 +
:<math>
 +
\left[\begin{array}{c}
 +
x_1\\
 +
y_1\\
 +
z_1
 +
\end{array}\right]=
 +
\left[\begin{array}{c}
 +
x_0\\
 +
y_0\\
 +
z_0
 +
\end{array}\right]+
 +
\left[\begin{array}{c}
 +
p_x\\
 +
p_y\\
 +
p_z
 +
\end{array}\right]=
 +
\left[\begin{array}{c}
 +
x_0+p_x\\
 +
y_0+p_y\\
 +
z_0+p_z
 +
\end{array}\right]
 
</math>
 
</math>
 
For further information about vector addition and examples, please have a look at the article about [[Simple arithmetic operations|simple arithmetic vector operations]].
 
For further information about vector addition and examples, please have a look at the article about [[Simple arithmetic operations|simple arithmetic vector operations]].

Latest revision as of 17:31, 24 November 2017

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Translation1.png

Translation is the easiest kind of transformation. Translating a point q_0 means that it is shifted by a translation vector. So the translation vector \vec{\mathbf{p}} is added to the position vector \vec{\mathbf{q}}_0 of q_0. The position vector \vec{\mathbf{q}}_1 of the resulting transformed point q_1 is calculated as follows:

\vec{\mathbf{q}}_1=\vec{\mathbf{q}}_0+\vec{\mathbf{p}}

The figure on the right shows an example in two-dimensional space. In robotics usually three dimensions are regarded. Considering the particular components of the vectors, a translation looks as follows:

\left[\begin{array}{c} x_1\\ y_1\\ z_1 \end{array}\right]= \left[\begin{array}{c} x_0\\ y_0\\ z_0 \end{array}\right]+ \left[\begin{array}{c} p_x\\ p_y\\ p_z \end{array}\right]= \left[\begin{array}{c} x_0+p_x\\ y_0+p_y\\ z_0+p_z \end{array}\right]

For further information about vector addition and examples, please have a look at the article about simple arithmetic vector operations.

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