Geometric Transformations Basic 2D geometric transformation Homogeneous coordinates 2D transformations Translation Scaling Rotation Composite transformations Other transformation: Shearing, Reflection 3D transformations Other 3D transformation: Shearing, Reflection OpenGL Matrix stack Lecture 7 Transformation

1. Basic 2D Transformation 2D Translation A translation moves a vertex to a new position by adding a displacement vector to it. Eg, The following translates the vertex to by adding (3,13) (7,7) Lecture 7 Transformation

2D Rotation r   (x’,y’) (x,y) x=r cos y=r sin x’ = r cos(+) = r cos cos  r sin sin = x cos - y sin y’ = r sin(+) = r sin cos + r cos sin = x sin + y cos y=r sin Lecture 7 Transformation

2D Scaling x’ = x sx y’ = y sy (x, y) (x’, y’) Lecture 7 Transformation

2. Linear Transformations on vertices A linear transformation moves a vertex to a new position by multiplying it with a non-singular matrix. Eg, The following transforms the vertex (2, 3) to (3, 13) (3,13) (2,3) Lecture 7 Transformation

Affine Transformations on vertices An affine transformation is a linear transformation followed by a translation. Its 2D general form is Affine transformations preserve lines. Many geometric movements of objects, eg, translations, rotations, and scalings are affine transformations. Lecture 7 Transformation

3. Homogeneous coordinates A point in homogeneous coordinates (x, y, w), w ≠ 0, corresponds to the 2-D vertex (x/w, y/w) in Cartesian coordinates Conceive that the Cartesian coordinates axes lies on the plane of w = 1. The intersection of the plane and the line connecting the origin and (x, y, w) gives the corresponding Cartesian coordinates w (x, y, w) w = 1 (x/w, y/w, 1) x y y x w = 0 Lecture 7 Transformation

E.g. both the points (6, 9, 3) and (4, 6, 2) in the homogeneous coordinates corresponds to (2, 3) in the Cartesian coordinates. Conversely, the point (2, 1) of the Cartesian corresponds to (2, 1, 1), (4, 2, 2) or (6, 3, 3) of the homogeneous w 6, 3, 3) (4, 2, 2) w = 1 (2, 1, 1) x y y x w = 0 Lecture 7 Transformation

Homogeneous coordinates of vectors A 2-D vector (x, y) of Cartesian corresponds to (x, y, 0) in homogeneous coordinates and vice versa. Note that vectors and vertices have different representations in homogeneous coordinates. On the other hand, the transformation formulas for both vectors and vertices in homogeneous coordinates are identical. Such generalization enhances simplicity, and thus the reliability, of a graphics system Lecture 7 Transformation

4. 2D Transformation in Homogeneous form 2D Translation in Homogeneous form (1,1) (5,1) (1,3) (x,y) (3,2) (7,2) (3,4) (x’, y’) Lecture 7 Transformation

2D Rotation in Homogeneous form Lecture 7 Transformation

Eg, to rotate 45o (/4 radian) sin /4 = cos /4 = 0.7071 (1,1) (5,1) (1,3) (x,y) (0,1.4) (2.8, 4.2) (-1.4, 2.8) (x’, y’) Lecture 7 Transformation

2D Translation in Homogeneous (1,1) (5,1) (1,3) (x,y) (2,1/2) (10,1/2) (2,3/2) (x’, y’) Lecture 7 Transformation

5. Composite a sequence of transformations Translate the right-angle vertex to the origin (Tx = -1, Ty = -1) Rotate 45o (/4 radian) sin /4 = cos /4 = 0.7071 (1,1) (5,1) (1,3) (0,0) (4,0) (0,2) (2.8,2.8) (-1.4,1.4) Lecture 7 Transformation

Lecture 7 Transformation

6. Composite Transformation Implementation All practical transformations on vertices and vectors in homogeneous coordinates can be expressed as pure matrix multiplications. Such uniformity greatly enhance the run-time efficiency of a graphics system. Suppose that a set of 1 million vertices needs to go through three transformations representing by the matrices M1, M2, and M3. A straightforward formula of the overall transformations for a vertex v is v´ = M3*M2*M1*v. As matrix multiplication is associative, we can compute M = M3*M2*M1 in advance, then transform all 1 million vertices using the formula v´ = M*v. Lecture 7 Transformation

7. Other 2D Transformation Shearing Uneven scaling and shearing change the shape of an object An example of 2D shearing in x direction (To alter x-coordinates by an amount proportional to the y value.) y x (0,0) (0,1) (1,0) (1,1) (2,1) (3,1) x’ = x + 2y y’ = y Lecture 7 Transformation

x’ = x y’ = 2x + y y x y x (0,0) (0,1) (1,2) (1,3) (0,0) (0,1) (1,0) (1,1) x’ = x y’ = 2x + y Lecture 7 Transformation

(x,y) (x,-y) Reflection about the x axis Reflection about the y axis ? Reflection w.p.t the origin ? Reflection w.p.t the x = y Lecture 7 Transformation

y =mx+b Reflection along a line Translate (0, -b) so that the line passes through the origin Rotate the line onto the x axis by -o Reflect about the x axis Backward rotate backward translate (Be reminded that these operations must be specified in reverse order.) b  Lecture 7 Transformation