Changes between Version 2 and Version 3 of DevWikiAffineParameters

Timestamp:

Sep 13, 2011, 9:07:32 AM (13 years ago)

Author:

bnordgren

Comment:

—

DevWikiAffineParameters

@@ -19 +19 @@
  * x' = xcosθ − ysinθ
  * y' = xsinθ + ycosθ
+
+To rotate in the clockwise direction: 
+
+[[Image(CW_rotation.png)]]
+
+ * x' = xcosθ + ysinθ
+ * y' = − xsinθ + ycosθ
+
+=== Scaling ===
+
+Scaling is used to set the size of the raster's grid cells in the x and y direction. The transformation is as follows: 
+
+[[Image(scaling.png)]]
+
+ * x' = x s,,x,,
+ * y' = y s,,y,,
+
+=== Shearing ===
+ 
+Shearing is visually equivalent to a "slanting" which is parallel to either the x or the y axis. This is a less common operation than rotation and scaling. These are presented as individual operations: one for each axis.
+
+Shearing parallel to the x axis takes the following form.
+
+[[Image(shear_x.png)]]
+
+ * x' = x + k,,x,,y
+ * y' = y
+
+While shearing parallel to the y axis has this form: 
+
+[[Image(shear_y.png)]]
+
+ * x' = x
+ * y' = k,,y,,x + y
+
+== Combining individual operations == 
+
+Whenever more than one of the above operations is required, they may be combined using [http://en.wikipedia.org/wiki/Matrix_multiplication matrix multiplication]. As an example, all of the above matrices will be combined into one. The result of such a combination is still not an affine transform, however. It is just a 2x2 matrix has all the individual functions aggregated into it.