2D Transformations By: KanwarjeetSingh 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations Matrix math Is there a difference between possible representations? 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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? 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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  4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations More rotation Another convention, we’ll take  to be counterclockwise, as in our example. R, the rotation matrix, looks like: 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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. 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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? 4 June 2018 Week 5-2D Transformations

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. 4 June 2018 Week 5-2D Transformations

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) 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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: 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations Now what? 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations And? Rotations: Scales: 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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? 4 June 2018 Week 5-2D Transformations

Is matrix multiplication associative? 4 June 2018 Week 5-2D Transformations

Is matrix multiplication commutative? 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations Order of operations So, it does matter. Let’s look at an example: 4 June 2018 Week 5-2D Transformations

Week 5-2D Transformations 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 4 June 2018 Week 5-2D Transformations

Other transformations Reflection: x-axis y-axis 4 June 2018 Week 5-2D Transformations

Other transformations Reflection: origin line x=y 4 June 2018 Week 5-2D Transformations

Other transformations Shear: x-direction y-direction 4 June 2018 Week 5-2D Transformations

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 4 June 2018 Week 5-2D Transformations

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? 4 June 2018 Week 5-2D Transformations

Matrix multiplication Since we don’t use the bottom row of the 2D transformation matrix, we could have a special transform composition operation: 4 June 2018 Week 5-2D Transformations