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| <h1>Transformations in KLayout</h1> |
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| <p> |
| KLayout supports a subset of affine transformations with the following contributions: |
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| <ul> |
| <li><b>Rotation and/or mirroring:</b> rotation by a given angle or mirroring at a given axis.</li> |
| <li><b>Scaling:</b> magnification by the given factor.</li> |
| <li><b>Translation:</b> a displacement by the given vector.</li> |
| </ul> |
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| <p> |
| The execution order is "displacement after rotation, mirroring and scaling". |
| Transformations are used for example to describe the instantiation of a cell. The content of |
| a cell appears in the parent cell after the given transformation has been applied to the |
| content of the cell. |
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| <p> |
| The transformations supported by KLayout cover the transformations employed within GDS2, OASIS and |
| other layout formats. KLayout does not support shearing currently. |
| </p> |
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| <p> |
| The following figure illustrates the effect of the transformation "r90 *2 7,9". This notation |
| specifies a transformation composed of a rotation by 90 degree, a scaling with factor 2 and |
| a displacement by 7 units in x- and 9 units in y-direction. In that example, the "F" shape is |
| first rotated by 90 degree around the origin. Because the "F" is already displaced from the origin, |
| this will also move the "F" shape. The shape then is scaled. Again it will move because |
| every point of the polygon moves away from the origin. Finally it is displaced by the |
| given displacement vector. |
| </p> |
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| <img src="images/transformation_overview.svg"/> |
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| <p> |
| The notation shown here is used in many places within KLayout. It is basically composed of the following parts |
| which are combined putting one or more blanks in between. The order the parts are specified |
| is arbitrary: the displacement is always applied after the rotation. |
| </p> |
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| <ul> |
| <li><x><b>,</b><y>: A displacement (applied after rotation and scaling) in micron units. If no displacement is specified, "0,0" is assumed.</li> |
| <li><b>r</b><a> or <b>m</b><a>: A rotation by angle "a" (in degrees) or mirroring at the "a" axis (the x axis rotated by "a" degree). If no rotation or mirroring is specified, no rotation is assumed.</li> |
| <li><b>*</b><s>: A scaling by the factor "s". If no scaling is specified, no scaling is assumed.</li> |
| </ul> |
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| <p> |
| Here are some examples: |
| </p> |
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| <ul> |
| <li><b>0,100</b>: shift 100 units up.</li> |
| <li><b>r90</b>: rotation by 90 degree counterclockwise (positive in the mathematical sense).</li> |
| <li><b>m0</b>: mirroring at the x-axis.</li> |
| <li><b>m45 100,-200</b>: swap x and y (mirror at 45 degree axis), shift 100 units to the right and 200 units down.</li> |
| <li><b>r22.5 *1.25</b>: rotate by 22.5 degree and scale by factor 1.25.</li> |
| </ul> |
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| <p> |
| The distance units are usually micron. In some cases (i.e. transformations inside a database), |
| the unit is database units and dx and dy are integer values. |
| </p> |
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| <p> |
| Mirroring and rotation are exclusive and mirroring includes a rotation. In fact, a mirror operation at a certain axis is |
| identical to a mirror operation at the x-axis, followed by a rotation by twice the angle "a". |
| The following figure illustrates rotation and mirroring with the eight basic transformations involving rotations by |
| multiples of 90 degree: |
| </p> |
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| <img src="images/transformation_basic.svg"/> |
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| <p> |
| KLayout is not restricted to these basic operations. Arbitrary angles are supported (i.e. "r45" or "m22.5"). Usually however, |
| this implies grid snapping and other issues. This also is true for arbitrary scaling values. KLayout is also more effective when using |
| simple transformations involving only rotations by multiples of 90 degree and do not use scaling. |
| </p> |
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