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2D Transformations By: KanwarjeetSingh

2D Transformations By: KanwarjeetSingh. Matrix math. Is there a difference between possible representations?. Pick a convention. We’ll use the column-vector representation for a point.

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2D Transformations By: KanwarjeetSingh

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  1. 2D TransformationsBy: KanwarjeetSingh Week 5-2D Transformations

  2. Matrix math Is there a difference between possible representations? Week 5-2D Transformations

  3. Pick a convention We’ll use the column-vector representation for a point. Which implies that we use pre-multiplication of the transformation – it appears before the point to be transformed in the equation. What if we needed to switch to the other convention (to use some library, for instance)? How could we do that? Week 5-2D Transformations

  4. Translation A translation moves all points in an object along the same straight-line path to new positions. The path is represented by a vector, called the translation or shift vector. We can write the components: p'x= px + tx p'y= py + ty or in matrix form: P' = P+T Week 5-2D Transformations

  5. Rotation A rotation repositions all points in an object along a circular path in the plane centered at the pivot point. First, we’ll assume the pivot is at the origin. We can write the components: p'x = px cos  – py sin  p'y = px sin  + py cos  or in matrix form: P' = R •P Week 5-2D Transformations

  6. More rotation Another convention, we’ll take  to be counterclockwise, as in our example. R, the rotation matrix, looks like: Week 5-2D Transformations

  7. Scaling Scaling alters the size of an object. Scales are about the the origin. Scale factors between 0 and 1 shrink objects. Scale factors greater than 1 enlarge objects. We can write the components: p'x = sx•px p'y = sy•py or in matrix form: P' = S •P The scale factors need not be the same in each direction. Week 5-2D Transformations

  8. More scaling We write a scale matrix as: Scaling also translates objects; away from the origin if the scale factor is greater than 1, or towards the origin if the scale factor is less than 1. What does scaling by 1 do? What is that matrix called? What does scaling by a negative value do? Week 5-2D Transformations

  9. Combining transformations We have a general transformation of a point: P' = M•P + A When we scale or rotate, we set M, and A is the additive identity. When we translate, we set A, and M is the multiplicative identity. To combine multiple transformations, we must explicitly compute each transformed point. It’d be nicer if we could use the same matrix operation all the time. But we’d have to combine multiplication and addition into a single operation. Week 5-2D Transformations

  10. A less than obvious solution Let’s move our problem into 3D. Let point (x, y) in 2D be represented by point (x, y, 1) in the new space. Scaling our new point by any value a puts us somewhere along a particular line: (ax, ay, a). We can always map back to the original 2D point by dividing by the last coordinate. The fact that all the points along each line can be mapped back to the same point in 2D gives this coordinate system its name – homogeneous coordinates. w (x,y,1) w = 1 x,y (0,0,0) Week 5-2D Transformations

  11. So what? Well, now we can wedge some addition into our multiplicative matrix. Our point now has three coordinates. So our matrix is needs to be 3x3. We want a matrix which gives us: Week 5-2D Transformations

  12. Now what? Week 5-2D Transformations

  13. And? Rotations: Scales: Week 5-2D Transformations

  14. What of it? We can represent any of our three transformations as a single matrix. No special cases when transforming a point – matrix • vector. Composite transformations – matrix • matrix. Composite transformations: Rotate about an arbitrary point – translate, rotate, translate Scale about an arbitrary point – translate, scale, translate Change coordinate systems – translate, rotate, scale Does the order of operations matter? Week 5-2D Transformations

  15. Is matrix multiplication associative? Week 5-2D Transformations

  16. Is matrix multiplication commutative? Week 5-2D Transformations

  17. Order of operations So, it does matter. Let’s look at an example: Week 5-2D Transformations

  18. Useful compositions Rotate about a pivot point: T(pivot) • R() • T(–pivot) • P Scale about a fixed point: T(fixed) • S(scale) • T(–fixed) • P General scaling directions: R(–) • S(scale) • R() • P Week 5-2D Transformations

  19. Other transformations Reflection: x-axis y-axis Week 5-2D Transformations

  20. Other transformations Reflection: origin line x=y Week 5-2D Transformations

  21. Other transformations Shear: x-direction y-direction Week 5-2D Transformations

  22. Coordinate system transformations We often need to transform points from one coordinate system to another: • We might model an object in non-Cartesian space (polar) • Objects may be described in their own local system • Other reasons: textures, display, etc Week 5-2D Transformations

  23. Matrix multiplication So we can do a lot with one basic operation. We’d better make this operation as fast as possible. Let’s start with the form of the matrix: Why haven’t we used the bottom row of the matrix? Week 5-2D Transformations

  24. Matrix multiplication Since we don’t use the bottom row of the 2D transformation matrix, we could have a special transform composition operation: Week 5-2D Transformations

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