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Transforms

Transforms. Types of Transforms. Translate Rotate Scale Shear. Affine Transform Math. Transforms are cumulative Transforms change the whole world view Order matters!. Rotate 90. Reflect X. Rotate 90. Reflect X. Making Shapes work together.

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Transforms

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  1. Transforms

  2. Types of Transforms Translate Rotate Scale Shear

  3. Affine Transform Math • Transforms are cumulative • Transforms change the whole world view • Order matters!

  4. Rotate 90 Reflect X Rotate 90 Reflect X

  5. Making Shapes work together • Create an object that has state variables corresponding to its parts • Draw from the center of the overall object • Perform transforms: 1. translate to center of overall object 2. rotate if wanted 3. scale/shear if wanted

  6. Matrices • Rectangular arrangement of numbers • [1 2 3] 2 -3 0 3 7 -1 • Dimensions • Diagonal • Transpose: interchange rows and cols • Identity: Square matrix, diagonal is ones, rest is zeros

  7. Matrix Operations • Adding and subtracting: matrices must have same dimensions • Add or subtract element-wise [1 2 3] + [4 5 6] = [5 7 9] • Scalar multiplication: multiply all elements by the scalar 2 * [1 2 3] = [2 4 6]

  8. Multiplying Matrices 3 [4 5 6] x 2 = 4*3 + 5*2 + 6*1 = [28] 1 (m x n) x (n x p) yields (mxp) Even if the matrices are square, multiplication is not commutative

  9. Matrices and Graphics • A point can be describedusing a vector:p = (x,y)p = xi+yj • Or a matrix: [x y] • Transposes are matrices that get post-multiplied to the point to produce a new point. (3,2)

  10. Scaling • Multiply by the matrix a 0where a is the scale 0 bfactor in the x direction,and b is the scale factor in the y direction • If a and b are the same, it’s like multiplying by a scalar. (Balanced or Uniform scaling)

  11. Scaling [3 2] 4 0 = [12 8] 0 4 (12,8) (3,2)

  12. Reflecting • Reflection in the x–axis: 1 0 0 -1 • Reflection in the y–axis: -1 0 0 1 • In general, multiplying by negative numbers combines a reflection and a scale.

  13. Reflection [ 3 2] -1 0 = [-3 -2] 0 -1 (3,2) (-3,-2)

  14. Rotations: Searching for a Pattern -1 0 rotates through 180 degrees 0 -1 (pi radians) 1 0 rotates through 360 degrees 0 1 (2 pi radians) 0 1 causes a rotation of 90 degrees -1 0 (pi/2 radians) 0 -1 causes a rotation of 270 degrees 1 0 (3pi/2 radians)

  15. In general… • The pattern emerges: To rotate through an angle Ө, multiply by cos Ө sin Ө -sin Ө cos Ө

  16. Transforming Polygons • To transform a line, transform its end points and connect the new points. • Polygons are collections of lines. • Multiplying several points is the same as making a matrix where each point is a row. • True for Affine Transformations • preserves points on line and parallelism • preserves proportions (i.e., the midpoint remains the midpoint)

  17. Transforming Polygons R*(18,8) P*(3,6) 1 3 3 0 3 63 0 0 2 = 9 06 4 18 8 R(6,4) P(1,3) Q*(9,0) Q(3,0)

  18. Multiple transformations • Order in transformations matters. • Matrix multiplication is not commutative!

  19. Translations • To translate, we add rather than multiply matrices. • To shift a point over by a in the x direction and b in the y direction, add the matrix [ a b]

  20. Transformations [ 3 2 ] + [ 2 4] = [5 6] (5,6) (2,4) (3,2)

  21. Homogeneous Coordinates • You can’t use a 2x2 multiplication matrix to model translation. Bummer! • But, we can add a third dimension of homogeneous coordinates, and now all 2D transformations can be expressed in a 3x3 matrix.

  22. Homogeneous Translation • Add a 1 to the point vector [3 2 1] = (3,2) • To translate over 2 and up 4, multiply by 1 0 0 0 1 02 4 1 [3 2 1] 1 0 0 0 1 0 = [5 6 1] 2 4 1

  23. Homogeneous Scale • Scale by a in x direction and b in ya 0 00 b 00 0 1 • Reflect in x-axis:1 0 00 -1 00 0 1

  24. Homogeneous Rotation • Rotate about the origin: cos Ө sin Ө 0 -sin Ө cos Ө 0 0 0 1

  25. Rotation about a point • How would we rotate about a point other than the origin? I want to rotate this shape about its center, which is (5,2).

  26. Rotation about a point • Step 1 – translate coordinates so that your point of rotation becomes the origin. • 1. Multiply by: • 0 0 • 0 1 0 • -5 -2 1

  27. Rotation about a point • Step 2 – Do your rotation 2. (For 90 degrees) Multiply by: 0 1 0 -1 0 0 0 0 1

  28. Rotation about a point • Step 3 – go back to original position • 3. Undoing step 1: • 0 0 • 0 1 0 • 5 2 1

  29. 3D • How do you think it is done in 3D?

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