affine

affine.md

@@ -5,39 +5,100 @@ title: Environment
 
 ## Affine Transforms & Vector Math
 
+	class Transformable {
+	  
+	  // attaributes
+	  PVector _position;
+	  float   _rotation;
+	  float   _scale;
+
+	  ...
+
+	public void display() {
+	    pushMatrix();
+	    translate(_position.x, _position.y);
+	    rotate(_rotation);        
+	    scale(_scale);
+	    draw_shape();
+	    popMatrix();
+	 }	 
+	 
+	 ...
+
+	};
+
+---
+
+What is a geometry / shape?
+
+![](http://onemathematicalcat.org/Math/Geometry_obj/graphics/vertex_vertices.png)
+
+---
+
+What types of (2D / 3D) transformations can we apply to geometries / shapes?
+
+![](http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/coverage_tools/transform_arc3.gif)
+
+---
+
 <!-- ASCIImath -->
 
-Matrix multiplication  
+What is a matrix Matrix?
+
+$
+[[a, b, c], [d, e, f], [g, h, i]]
+$
+
+
+Matrix [3 x 3] multiplication with a Vector [1 x 3]
+
 $
 [[a, b, c], [d, e, f], [g, h, i]] [[x], [y], [z]] = [[ax + by + cz], [dx + ey + fz], [gx + hy + iz]]
 $
 
+---
+
+What types of geometrical transformations (vertices transformation) can we perform using a Matrix?
+
 
 Translation  
+
 $
 [[1, 0, tx], [0, 1, ty], [0, 0, 1]]  [[x], [y], [1]] = [[dot x], [dot y], [1]]
 $
 
 Scaling  
+
 $
 [[sx, 0, 0], [0, sy, 0], [0, 0, 1]]  [[x], [y], [1]] = [[dot x], [dot y], [1]]
 $
 
 Rotation $theta$  
+
 $
 [[cos theta, -sin theta, 0], [sin theta, cos theta, 0], [0, 0, 1]]  [[x], [y], [1]] = [[dot x], [dot y], [1]]
 $
 
 Shearing  
+
 $
 [[1, shx, 0], [shy, 1, 0], [0, 0, 1]]  [[x], [y], [1]] = [[dot x], [dot y], [1]]
 $
 
-Affine Transform (Translation + Scaling + Rotation + Shearing)  
+---
+
+What is an Affine Transform?
+
+Affine Transform Matrix comprises a Translation, Scaling, Rotation and Shearing
+
 $
 [[sx * cos theta, -sin theta * shx, tx], [sin theta * shy, sy * cos theta, ty], [0, 0, 1]]  [[x], [y], [1]] = [[dot x], [dot y], [1]]
 $
 
+Yes, but what other types of transformations exists?
+
+![](https://geodata.ethz.ch/resources/tutorials/L3GeodataAndSpatialRelation/en/images/transformation.png)
+
 
 
 

index.md

@@ -28,7 +28,7 @@ This course not only focuses on tools, but on how these tools can become a part
 
 [Objects](objects.html)
 
-[Environment](/)
+Environment - [Affine Transformatino](affine.html) and [Oscillatiors](/)
 
 [Simulation](/)