Computer Graphics: 2D Transformations. Course Website:

Transcription

1 Computer Graphics: D Transformations Course Website:

2 5 Contents Wh transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications Combining transformations

3 5 Wh Transformations? In graphics, once we have an object described, transformations are used to move that object, scale it and rotate it

4 4 5 Translation Simpl moves an object from one position to another new = old + d new = old + d Note: House shifts position relative to origin

5 5 5 Translation Eample (, ) (, ) (, )

6 6 5 Scaling Scalar multiplies all coordinates WATCH OUT: Objects grow and move! new = S old new = S old Note: House shifts position relative to origin

7 7 5 Scaling Eample (, ) (, ) (, )

8 8 5 Rotation Rotates all coordinates b a specified angle new = old cosθ old sinθ new = old sinθ + old cosθ Points are alwas rotated about the origin

9 9 5 Rotation Eample (4, ) (, ) (5, )

10 5 Homogeneous Coordinates A point (, ) can be re-written in homogeneous coordinates as ( h, h, h) The homogeneous parameter h is a nonzero value such that: h We can then write an point (, ) as (h, h, h) We can convenientl choose h = so that (, ) becomes (,, ) h h h

11 5 Wh Homogeneous Coordinates? Mathematicians commonl use homogeneous coordinates as the allow scaling factors to be removed from equations We will see in a moment that all the transformations we discussed previousl can be represented as * matrices Using homogeneous coordinates allows us use matri multiplication to calculate transformations etremel efficient!

12 5 Homogeneous Translation The translation a point b (d, d) can be written in matri form as: d d Representing the point as a homogeneous column vector we perform the calculation as: d d * * d* * * d * * * * d d

13 5 Remember Matri Multiplication Recall how matri multiplication takes place: z i h g z f e d z c b a z i h g f e d c b a * * * * * * * * *

14 4 5 Homogenous Coordinates To make operations easier, -D points are written as homogenous coordinate column vectors v d d T v d d d d ), ( ' : v s s S v s s s s ), ( ' : Translation: Scaling:

15 5 5 Homogenous Coordinates (cont ) v R v ) ( ' : cos sin sin cos cos sin sin cos Rotation:

16 6 5 Inverse Transformations Transformations can easil be reversed using inverse transformations d d T s s S cos sin sin cos R

17 7 5 Combining Transformations A number transformations can be combined into one matri to make things eas Allowed b the fact that we use homogenous coordinates Imagine rotating a polgon around a point other than the origin Transform to centre point to origin Rotate around origin Transform back to centre point

18 8 5 Combining Transformations (cont ) House (H ) T ( d, d) H R ( ) T ( d, d) H T ( d, d) R( ) T ( d, d) H 4

19 9 5 Combining Transformations (cont ) The three transformation matrices are combined as follows cos sin sin cos d d d d REMEMBER: Matri multiplication is not commutative so order matters v d d T R d d T v ), ( ) ( ), ( '

20 5 Summar In this lecture we have taken a look at: D Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications Combining transformations Net time we ll start to look at how we take these abstract shapes etc and get them onscreen

21 5 Eercises Translate the shape below b (7, ) (, ) (, ) (, ) (, )

22 5 Eercises Scale the shape below b in and in (, ) (, ) (, ) (, )

23 5 Eercises Rotate the shape below b about the origin (7, ) (6, ) (8, ) (7, )

24 4 5 Eercise 4 Write out the homogeneous matrices for the previous three transformations Translation Scaling Rotation

25 5 5 Eercises 5 Using matri multiplication calculate the rotation the shape below b 45 about its centre (5, ) 5 4 (5, 4) (4, ) (6, ) (5, )

26 6 5 Scratch

27 7 5 Equations Translation: new = old + d new = old + d Scaling: new = S old new = S old Rotation new = old cosθ old sinθ new = old sinθ + old cosθ