4. Two Dimensional Transformations

Transcription

1 4. Two Dimensional Transformations CS362 Introduction to Computer Graphics Helena Wong, 2 In man applications, changes in orientations, sizes, and shapes are accomplished with geometric transformations that alter the coordinate descriptions of objects. Basic geometric transformations are: Translation Rotation Scaling Other transformations: Reflection Shear

2 4. Basic Transformations CS362 Introduction to Computer Graphics Helena Wong, 2 Translation We translate a 2D point b adding translation distances, t and t, to the original coordinate position (,): ' + t, ' + t Alternativel, translation can also be specified b the following transformation matri: t t

3 CS362 Introduction to Computer Graphics Helena Wong, 2 Then we can rewrite the formula as: ' ' t t For eample, to translate a triangle with vertices at original coordinates (,2), (,), (2,) b t 5, t, we compute as followings: Translation of verte (,2): ' ' * * 2 * * * 2 * 5* * 2 * 3 5

4 Translation of verte (,): ' 5 * + * + 5* 5 ' * + * + * 2 * + * + * CS362 Introduction to Computer Graphics Helena Wong, 2 Translation of verte (2,): ' 5 2 * 2 + * + 5* 25 ' * 2 + * + * 2 * 2 + * + * The resultant coordinates of the triangle vertices are (5,3), (5,2), and (25,2) respectivel. Eercise: translate a triangle with vertices at original coordinates (,25), (5,), (2,) b t 5, t 5. Roughl plot the original and resultant triangles.

5 Rotation About the Origin CS362 Introduction to Computer Graphics Helena Wong, 2 To rotate an object about the origin (,), we specif the rotation angle?. Positive and negative values for the rotation angle define counterclockwise and clockwise rotations respectivel. The followings is the computation of this rotation for a point: ' cos? - sin? ' sin? + cos? Alternativel, this rotation can also be specified b the following transformation matri: cos sin sin cos

6 CS362 Introduction to Computer Graphics Helena Wong, 2 Then we can rewrite the formula as: ' ' cos sin sin cos For eample, to rotate a triangle about the origin with vertices at original coordinates (,2), (,), (2,) b 3 degrees, we compute as followings: cos sin sin cos cos3 sin 3 sin 3 cos

7 Rotation of verte (,2): CS362 Introduction to Computer Graphics Helena Wong, 2 '.866 ' * + (.5) * 2 + * * * 2 + * * + * 2 + * Rotation of verte (,): '.866 ' * + (.5) * + * * * + * 3.66 * + * + *

8 Rotation of verte (2,): CS362 Introduction to Computer Graphics Helena Wong, 2 '.866 ' * 2 + (.5) * + * * * + * 8.66 * 2 + * + * The resultant coordinates of the triangle vertices are (-.34,22.32), (3.6,3.66), and (2.32,8.66) respectivel. Eercise: Rotate a triangle with vertices at original coordinates (,2), (5,), (2,) b 45 degrees. Roughl plot the original and resultant triangles.

9 Scaling With Respect to the Origin CS362 Introduction to Computer Graphics Helena Wong, 2 We scale a 2D object with respect to the origin b setting the scaling factors s and s, which are multiplied to the original verte coordinate positions (,): ' * s, ' * s Alternativel, this scaling can also be specified b the following transformation matri: s s

10 CS362 Introduction to Computer Graphics Helena Wong, 2 Then we can rewrite the formula as: ' ' s s For eample, to scale a triangle with respect to the origin, with vertices at original coordinates (,2), (,), (2,) b s 2, s.5, we compute as followings: Scaling of verte (,2): ' ' * * 2 * *.5* 2 * * * 2 2 * 3 2

11 Scaling of verte (,): CS362 Introduction to Computer Graphics Helena Wong, 2 ' 2 '.5 2 * + * + * 2 * +.5* + * 5 * + * + * Scaling of verte (2,): ' 2 ' * 2 + * + * 4 * 2 +.5* + * 5 * 2 + * + *

12 CS362 Introduction to Computer Graphics Helena Wong, 2 The resultant coordinates of the triangle vertices are (2,3), (2,5), and (4,5) respectivel. Eercise: Scale a triangle with vertices at original coordinates (,25), (5,), (2,) b s.5, s 2, with respect to the origin. Roughl plot the original and resultant triangles. 4.2 Concatenation Properties of Composite Matri I. Matri multiplication is associative: A B C (A B) C A (B C)

13 CS362 Introduction to Computer Graphics Helena Wong, 2 Therefore, we can evaluate matri products using these associative grouping. For eample, we have a triangle, we want to rotate it with the matri B, then we translate it with matri A. Then, for a verte of that triangle represented as C, we compute its transformation as: C'A (B C) But we can also change the computation method as: C' (A B) C The advantage of computing it using C' (A B) C instead of C'A (B C) is that, for computing the 3 vertices of the triangle, C, C 2, C 3, the computation time is shortened:

14 Using C'A (B C):. compute B C and put the result into I 2. compute A I and put the result into C ' 3. compute B C 2 and put the result into I 2 4. compute A I 2 and put the result into C 2 ' 5. compute B C 3 and put the result into I 3 6. compute A I 3 and put the result into C 3 ' CS362 Introduction to Computer Graphics Helena Wong, 2 Using C' (A B) C:. compute A B and put the result into M 2. compute M C and put the result into C ' 3. compute M C 2 and put the result into C 2 ' 4. compute M C 3 and put the result into C 3 '

15 CS362 Introduction to Computer Graphics Helena Wong, 2 Eample: Rotate a triangle with vertices (,2), (,), (2,) about the origin b 3 degrees and then translate it b t 5, t, We compute the rotation matri: B cos3 sin 3 sin 3 cos And we compute the translation matri: A 5 Then, we compute MA B

16 CS362 Introduction to Computer Graphics Helena Wong, 2 M M * *.5 + 5* * *.5 + * * *.5 + * *.5 + * * *.5 + * * *.5 + * * * + * + 5* * + * + * * + * + *.866 M Then, we compute the transformations of the 3 vertices:

17 Transformation of verte (,2): CS362 Introduction to Computer Graphics Helena Wong, 2 '.866 ' * + (.5) * * * * 2 + * * + * 2 + * Transformation of verte (,): '.866 ' * + (.5) * + 5 * * * + * * + * + *

18 Transformation of verte (2,): CS362 Introduction to Computer Graphics Helena Wong, 2 '.866 ' * 2 + (.5) * + 5 * * * + * * 2 + * + * The resultant coordinates of the triangle vertices are (3.66,32.32), (8.66,23.66), and (7.32,28.66) respectivel.

19 II. Matri multiplication ma not be commutative: CS362 Introduction to Computer Graphics Helena Wong, 2 A B ma not equal to B A This means that if we want to translate and rotate an object, we must be careful about the order in which the composite matri is evaluated. Using the previous eample, if ou compute C' (A B) C, ou are rotating the triangle with B first, then translate it with A, but if ou compute C' (B A) C, ou are translating it with A first, then rotate it with B. The result is different. Eercise: Translate a triangle with vertices (,2), (,), (2,) b t 5, t and then rotate it about the origin b 3 degrees. Compare the result with the one obtained previousl: (3.66,32.32), (8.66,23.66), and (7.32,28.66) b plotting the original triangle together with these 2 results.

20 CS362 Introduction to Computer Graphics Helena Wong, Composite Transformation Matri Translations B common sense, if we translate a shape with 2 successive translation vectors: (t, t ) and (t 2, t 2 ), it is equal to a single translation of (t + t 2, t + t 2 ). This additive propert can be demonstrated b composite transformation matri:

21 CS362 Introduction to Computer Graphics Helena Wong, 2 t t 2 2 t t * + * + t 2 * *+ * + t * 2 *+ * + * * + *+ t * + *+ t 2 2 * * * + *+ * * t * t * t + * t + * t + * t + t 2 * + t 2 * + * Y t t + + t t 2 2 This demonstrates that 2 successive translations are additive.

22 Rotations CS362 Introduction to Computer Graphics Helena Wong, 2 B common sense, if we rotate a shape with 2 successive rotation angles: a and?, about the origin, it is equal to rotating the shape once b an angle a +? about the origin. Similarl, this additive propert can be demonstrated b composite transformation matri: cos sin sin cos cosα sin α sin α cosα coscosα + ( sin )*sin α + * sin cosα + cos*sin α + * *cosα + *sin α + * cos*( sin α) + ( sin )*cosα + * sin *( sin α) + cos*cosα + * *( sin α) + *cosα + * cos* + ( sin )* + * sin * + cos* + * * + * + *

23 CS362 Introduction to Computer Graphics Helena Wong, 2 coscosα sin sin α sin cosα + cossin α (cossin α + sin cosα) sin sin α + coscosα cos( α + ) sin( α + ) sin( α + ) cos( α + ) This demonstrates that 2 successive rotations about the origin are additive. Scaling With Respect to the Origin B common sense, if we scale a shape with 2 successive scaling factor: (s, s ) and (s 2, s 2 ), with respect to the origin, it is equal to a single scaling of (s * s 2, s * s 2 ) with respect to the origin.

24 This multiplicative propert can be demonstrated b composite transformation matri: CS362 Introduction to Computer Graphics Helena Wong, 2 s 2 s 2 s s s2 *s + * + * *s + s2 * + * *s + * + * s 2 * * + *s + s 2 *s * + *s + * + * + * s2 * + * + * * + s2 * + * * + * + * s *s 2 s *s 2

25 CS362 Introduction to Computer Graphics Helena Wong, 2 This demonstrates that 2 successive scalings with respect to the origin are multiplicative. General Pivot-Point Rotation Rotation about an arbitrar pivot point is not as simple as rotation about the origin. The procedure of rotation about an arbitrar pivot point is:. Translate the object so that the pivot-point position is moved to the origin. 2. Rotate the object about the origin. 3. Translate the object so that the pivot point is returned to its original position.

26 CS362 Introduction to Computer Graphics Helena Wong, 2 The corresponding composite transformation matri is: r r cos sin sin cos r r

27 CS362 Introduction to Computer Graphics Helena Wong, 2 cos sin sin cos r r r r cos sin sin cos r r cos + sin r r sin + cos + r r General Fied-Point Scaling Scaling with respect to an arbitrar fied point is not as simple as scaling with respect to the origin. The procedure of scaling with respect to an arbitrar fied point is:

28 CS362 Introduction to Computer Graphics Helena Wong, 2. Translate the object so that the fied point coincides with the origin. 2. Scale the object with respect to the origin. 3. Use the inverse translation of step to return the object to its original position.

29 CS362 Introduction to Computer Graphics Helena Wong, 2 The corresponding composite transformation matri is: f f s s f f ) s ( s ) s ( s f f General Scaling Direction Scaling along an arbitrar direction is not as simple as scaling along the - ais. The procedure of scaling along and normal to an arbitrar direction (s and s 2 ), with respect to the origin, is:. Rotate the object so that the directions for s and s 2 coincide with the and aes respectivel. 2. Scale the object with respect to the origin using (s, s 2 ). 3. Use an opposite rotation to return points to their original orientation.

30 CS362 Introduction to Computer Graphics Helena Wong, 2 The corresponding composite transformation matri is: ) cos( ) sin( ) sin( ) cos( s s 2 cos sin sin cos

31 CS362 Introduction to Computer Graphics Helena Wong, Other Transformations Reflection Reflection about the ais: ' ' ie. '; '-

32 CS362 Introduction to Computer Graphics Helena Wong, 2 Reflection about the ais: ' ' ie. '-; ' Flipping both and coordinates of a point relative to the origin: ' ' ie. '-; '-

33 CS362 Introduction to Computer Graphics Helena Wong, 2 Reflection about the diagonal line : ' ' ie. '; ' Reflection about the diagonal line -: ' ' ie. '-; '-

34 CS362 Introduction to Computer Graphics Helena Wong, 2 Shear X-direction shear, with a shearing parameter sh, relative to the -ais: ' sh ' ie. '+*sh ; '- Eercise: Think of a -direction shear, with a shearing parameter sh, relative to the -ais.

35 4.5 Transformation Between 2 Cartesian Sstems CS362 Introduction to Computer Graphics Helena Wong, 2 For modeling and design applications, individual objects ma be defined in their own local Cartesian references (modeling coordinates). The local coordinates must then be transformed to position the objects within the overall scene coordinate sstem (world coordinates). Suppose we want to transform object descriptions from the sstem to the '' sstem: The composite transformation is: cos( ) sin( ) sin( ) cos( ) r r