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();
 }	 
 
 ...

};

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What is a geometry / shape?

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What types of (2D / 3D) transformations can we apply to geometries / shapes?

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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]] $

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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]] $

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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?