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Computer Graphics 3: 2D Transformations

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?.

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Computer Graphics 3: 2D Transformations

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  1. Computer Graphics 3:2D Transformations

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

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

  4. Translation y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x Note: House shifts position relative to origin • Simply moves an object from one position to another • xnew = xold + dx ynew = yold + dy

  5. Translation Example y 6 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 (2, 3) (1, 1) (3, 1)

  6. Scaling y 6 5 4 3 x 2 Note: House shifts position relative to origin 1 0 1 2 3 4 5 6 7 8 9 10 • Scalar multiplies all coordinates • WATCH OUT: Objects grow and move! • xnew = Sx × xold ynew = Sy × yold

  7. Scaling Example y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 (2, 3) (1, 1) (3, 1)

  8. 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 0 1 2 3 4 5 6 7 8 9 10

  9. Rotation Example y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 (4, 3) (3, 1) (5, 1)

  10. Homogeneous Coordinates • A point (x, y) can be re-written in homogeneous coordinates as (xh, yh, h) • The homogeneous parameterh 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)

  11. 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!

  12. 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:

  13. Remember Matrix Multiplication • Recall how matrix multiplication takes place:

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

  15. Homogenous Coordinates (cont…) Rotation:

  16. Inverse Transformations • Transformations can easily be reversed using inverse transformations

  17. 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

  18. Combining Transformations (cont…) 2 1 3 4

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

  20. 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

  21. Exercises 1 y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 • Translate the shape below by (7, 2) (2, 3) (1, 2) (3, 2) (2, 1) x

  22. Exercises 2 y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 • Scale the shape below by 3 in x and 2 in y (2, 3) (1, 2) (3, 2) (2, 1) x

  23. Exercises 3 y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 • Rotate the shape below by 30° about the origin (7, 3) (6, 2) (8, 2) (7, 1) x

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

  25. Exercises 5 y 5 (5, 4) 4 (4, 3) (6, 3) 3 2 (5, 2) 1 x 0 1 2 3 4 5 6 7 8 9 10 • Using matrix multiplication calculate the rotation of the shape below by 45° about its centre (5, 3)

  26. Scratch y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x

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

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