mirror of https://github.com/KLayout/klayout.git
More elaborate doc for transformations
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Matthias Koefferlein
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<doc>
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<title>Transformations in KLayout</title>
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<keyword name="transformation"/>
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<p>
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KLayout supports a subset of affine transformations with the following contributions:
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given displacement vector.
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</p>
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<p>
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<img src="/about/transformation_overview.png"/>
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</p>
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<p>
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The notation shown here is used in many places within KLayout. It is basically composed of the following parts
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multiples of 90 degree:
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</p>
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<p>
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<img src="/about/transformation_basic.png"/>
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</p>
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<p>
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KLayout is not restricted to these basic operations. Arbitrary angles are supported (i.e. "r45" or "m22.5"). Usually however,
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simple transformations involving only rotations by multiples of 90 degree and do not use scaling.
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</p>
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<h2>Coding transformations</h2>
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<keyword name="transformation objects"/>
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<p>
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Note that mirroring at an axis with a given angle "a" is equivalent to mirroring at the x axis followed by a rotation
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by twice the angle "a". For example:
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</p>
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<pre>m45 == m0 followed by r90</pre>
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<p>
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When coding transformations, two parameters are used to represent the rotation/mirror part:
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a rotation angle and a flag indicating mirroring at the x axis. The mirroring is applied before
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the rotation. In terms of these parameters, the basic transformations are:
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</p>
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<table>
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<tr><th>Rotation angle<br/>(degree)</th><th>Mirror flag <br/>= False</th><th>Mirror flag <br/>= True</th></tr>
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<tr><td>0</td><td>r0</td><td>m0</td></tr>
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<tr><td>90</td><td>r90</td><td>m45</td></tr>
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<tr><td>180</td><td>r180</td><td>m90</td></tr>
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<tr><td>270</td><td>r270</td><td>m135</td></tr>
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</table>
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<h3>Transformation objects</h3>
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<p>
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Transformation objects are convenient objects to operate with. They represent a
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transformation (technically a matrix) consisting of an angle/mirror and a displacement
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part. They support some basic operations:
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</p>
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<ul>
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<li>Concatenation: <tt>T = T1 * T2</tt><br/><tt>T</tt> is transformation <tt>T2</tt> applied, then <tt>T1</tt> (note this order)</li>
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<li>Inversion: <tt>TI = T.inverted()</tt><br/><tt>TI</tt> is the inverse of <tt>T</tt>, i.e. <tt>TI * T = T * TI = 1</tt> where <tt>1</tt> is the neutral transformation which does not modify coordinates</li>
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<li>Application to geometrical objects: <tt>q = T * p</tt><br/>where <tt>p</tt> is a box, polygon, path, text, point, vector etc. and <tt>q</tt> is the transformed object</li>
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</ul>
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<h3>Vectors and Points</h3>
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<keyword name="vector"/>
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<keyword name="point"/>
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<p>
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In KLayout there are two two-dimensional coordinate objects: the vector and the point.
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Basically, the vector is the difference between two points:
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</p>
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<pre>v = p2 - p1</pre>
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<p>Here <tt>v</tt> is a vector object while <tt>p1</tt> and <tt>p2</tt> are points.</p>
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<p>
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Regarding transformations, vectors and points behave differently. While for a point, the
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displacement is applied, it is not for vectors. So
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</p>
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<pre>p' = T * p = M * p + d
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v' = T * v = M * v</pre>
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<p>
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Here <tt>M</tt> is the 2x2 rotation/mirror matrix part of the transformation and <tt>d</tt> is the
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displacement vector
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</p>
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<p>
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The reason why the displacement is not applied to a vector is seen here:
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</p>
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<pre>
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v' = T * v
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= T * (p2 - p1)
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= T * p2 - T * p1
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= (M * p2 + d) - (M * p1 + d)
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= M * p2 + d - M * p1 - d
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= M * p2 - M * p1
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= M * (p2 - p1)</pre>
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<p>where the latter simply is:</p>
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<pre>v' = M * v</pre>
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<h3>Simple transformations</h3>
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<keyword name="simple transformation"/>
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<keyword name="Trans"/>
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<keyword name="DTrans"/>
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<p>
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Simple transformations are represented by <class_doc href="DTrans"/> or <class_doc href="Trans"/> objects.
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The first operates with floating-point displacements in units of micrometers while the second one with integer displacements
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in database units. "DTrans" objects act on "D" type floating-point coordinate shapes (e.g. <class_doc href="DBox"/>) while "Trans" objects act
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on the integer coordinate shapes (e.g. <class_doc href="Box"/>).
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</p>
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<p>
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The basic construction parameters of "DTrans" and "Trans" are:
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</p>
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<pre>Trans(angle, mirror, displacement)
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DTrans(angle, mirror, displacement)</pre>
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<p>"displacement" is a <class_doc href="DVector"/> (for DTrans) or a <class_doc href="Vector"/> (for Trans).</p>
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<p>"angle" is the rotation angle in units of 90 degree and "mirror" is the mirror flag:</p>
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<table>
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<tr><th>angle</th><th>mirror <br/>= False</th><th>mirror <br/>= True</th></tr>
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<tr><td>0</td><td>r0</td><td>m0</td></tr>
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<tr><td>1</td><td>r90</td><td>m45</td></tr>
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<tr><td>2</td><td>r180</td><td>m90</td></tr>
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<tr><td>3</td><td>r270</td><td>m135</td></tr>
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</table>
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<h3>Complex transformations</h3>
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<keyword name="complex transformation"/>
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<keyword name="CplxTrans"/>
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<keyword name="ICplxTrans"/>
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<keyword name="DCplxTrans"/>
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<keyword name="VCplxTrans"/>
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<p>
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Complex transformations in addition to the simple transformations feature
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scaling (magnification) and arbitrary rotation angles. Other than simple transformations
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they do not necessarily preserve a grid and rounding is implied. Furthermore they imply a shift is physical scale
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which renders them difficult to use in physical frameworks (e.g. DRC). Hence their
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use is discouraged for certain applications.
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</p>
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<p>
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The basic classes are <class_doc href="DCplxTrans"/> for the micrometer-unit (floating-point) version
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and <class_doc href="ICplxTrans"/> for the database-unit (integer) version. The
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construction parameters are:
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</p>
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<pre>ICplxTrans(angle, mirror, magnification, displacement)
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DCplxTrans(angle, mirror, magnification, displacement)</pre>
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<p>
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Here, "angle" is the rotation angle in degree (note the difference to "Trans" and "DTrans" where the
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rotation angle is in units for 90 degree. "magnification" is a factor (1.0 for "no change in scale").
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</p>
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<p>
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There are two other variants useful for transforming coordinate systems: <class_doc href="CplxTrans"/> takes
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integer-unit objects and converts them to floating-point unit objects. It can be used to convert
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from database units to micrometer units when configured with a magnification equal to the database unit value:
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</p>
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<pre>T = CplxTrans(magnification: dbu)
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q = T * p</pre>
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<p>
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The other variant is <class_doc href="VCplxTrans"/> which converts floating-point unit objects
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to integer-unit ones. These objects are generated when inverting "CplxTrans" objects.
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</p>
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</doc>
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