Computer Graphics 3: 2D Transformations

Contents In today’s lecture we’ll cover the following: Why transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matrix multiplications Combining transformations

Why Transformations? In graphics, once we have an object described, transformations are used to move that object, scale it and rotate it Images taken from Hearn & Baker, “Computer Graphics with OpenGL” (2004)

Translation Simply moves an object from one position to another xnew = xold + dx ynew = yold + dy Note: House shifts position relative to origin y x 1 2 3 4 5 6 7 8 9 10

Translation Example y x (2, 3) (1, 1) (3, 1) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (2, 3) Translate by (5, 3) x’ = x + dx y’ = y + dy (1, 1) (3, 1)

Scaling Scalar multiplies all coordinates WATCH OUT: Objects grow and move! xnew = Sx × xold ynew = Sy × yold Note: House shifts position relative to origin y x 1 2 3 4 5 6 7 8 9 10

Scaling Example y (2, 3) (1, 1) (3, 1) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (2, 3) Scale by (2, 2) x’ = sx*x y’ = sy*y (1, 1) (3, 1)

Rotation Rotates all coordinates by a specified angle xnew = xold × cosθ – yold × sinθ ynew = xold × sinθ + yold × cosθ Points are always rotated about the origin y 6 5 4 3 2 1 x 1 2 3 4 5 6 7 8 9 10

Rotation Example y (4, 3) (3, 1) (5, 1) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (4, 3) xnew = xold × cosθ – yold × sinθ ynew = xold × sinθ + yold × cosθ 1 radian = 180/π degrees cos(π/6)=0.866 sin(π/6)=0.5 x’(4, 3) = 4*0.866 – 3*0.5 = 3.46 – 1.5 = 1.96 y’(4, 3) = 4*0.5 + 3*0.866 = 2.0 + 2.598 = 4.598 x’(3, 1) = 3*0.866 – 1*0.5 = 2.598 – 0.5 = 2.098 y’(3, 1) = 3*0.5 + 1*0.866 = 1.5 + 0.866 = 2.366 x’(5, 1) = 5*0.866 – 1*0.5 = 4.33 – 0.5 = 3.83 y’(5, 1) = 5*0.5 + 1*0.866 = 2.5 + 0.866 = 3.366 (3, 1) (5, 1)

Homogeneous Coordinates A point (x, y) can be re-written in homogeneous coordinates as (xh, yh, h) The homogeneous parameter h is a non- zero value such that: We can then write any point (x, y) as (hx, hy, h) We can conveniently choose h = 1 so that (x, y) becomes (x, y, 1)

Why Homogeneous Coordinates? Mathematicians commonly use homogeneous coordinates as they allow scaling factors to be removed from equations We will see in a moment that all of the transformations we discussed previously can be represented as 3*3 matrices Using homogeneous coordinates allows us use matrix multiplication to calculate transformations – extremely efficient!

Homogeneous Translation The translation of a point by (dx, dy) can be written in matrix form as: Representing the point as a homogeneous column vector we perform the calculation as:

Remember Matrix Multiplication Recall how matrix multiplication takes place:

Homogenous Coordinates To make operations easier, 2-D points are written as homogenous coordinate column vectors Translation: Scaling:

Homogenous Coordinates (cont…) Rotation:

Inverse Transformations Transformations can easily be reversed using inverse transformations

Combining Transformations A number of transformations can be combined into one matrix to make things easy Allowed by the fact that we use homogenous coordinates Imagine rotating a polygon around a point other than the origin Transform to centre point to origin Rotate around origin Transform back to centre point

Combining Transformations (cont…) 1 2 3 4

Combining Transformations (cont…) The three transformation matrices are combined as follows REMEMBER: Matrix multiplication is not commutative so order matters

Summary In this lecture we have taken a look at: 2D Transformations Translation Scaling Rotation Homogeneous coordinates Matrix multiplications Combining transformations Next time we’ll start to look at how we take these abstract shapes etc and get them on-screen

Exercises 1 Translate the shape below by (7, 2) y (2, 3) (1, 2) (3, 2) 1 2 3 4 5 6 7 8 9 10 (2, 3) (1, 2) (3, 2) (2, 1) x

Exercises 2 Scale the shape below by 3 in x and 2 in y y (2, 3) (1, 2) 1 2 3 4 5 6 7 8 9 10 (2, 3) (1, 2) (3, 2) (2, 1) x

Exercises 3 Rotate the shape below by 30° about the origin y (7, 3) 1 2 3 4 5 6 7 8 9 10 (7, 3) (6, 2) (8, 2) (7, 1) x

Exercise 4 Write out the homogeneous matrices for the previous three transformations Translation Scaling Rotation

Exercises 5 Using matrix multiplication calculate the rotation of the shape below by 45° about its centre (5, 3) x y 1 2 3 4 5 6 7 8 9 10 (5, 4) (6, 3) (4, 3) (5, 2)

Scratch y 1 2 3 4 5 6 7 8 9 10 x

Equations xnew = xold + dx ynew = yold + dy Translation: Scaling: xnew = Sx × xold ynew = Sy × yold Rotation xnew = xold × cosθ – yold × sinθ ynew = xold × sinθ + yold × cosθ