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Fundamental Concepts of 2-D Transformations: Coordinate Systems and Object Modeling

This document explores 2-D transformations, focusing on local and world coordinate systems. It illustrates how to convert points between different coordinate systems, addressing translation and rotation. Key concepts include defining vectors in relation to coordinate origins and using scalar components for mathematical operations. The significance of combining rotation and translation as well as the formation of transformation matrices is emphasized. The principles are presented through definitions, equations, and practical examples that enable effective modeling and visualization of objects in 2-D space.

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Fundamental Concepts of 2-D Transformations: Coordinate Systems and Object Modeling

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  1. 2-D Transformations Local/Modelling Coordinates y Object descriptions • Often defined in model coordinates • Must be mapped to world coordinates • Groups of objects are combined; complete image is formed by combining primitives x World Coordinates CS-321Dr. Mark L. Hornick

  2. 2-D Transformations Local/Modelling Coordinates y Problem statement: • Convert points from coordinates in one system to a second coordinate system x World Coordinates CS-321Dr. Mark L. Hornick

  3. 2-D Transformations • First, consider the case where two coordinate systems are offset by a pure translation • Definitions • v2 – vector from origin of coordinate system xy2 to a point on the object • has scalar components (v2x,v2y) • v1 – vector from origin of coordinate system xy1 to a point on the object • has scalar components (v1x,v1y) • p – vector from origin of coordinate system xy1 to the origin of xy2 • has scalar components (px,py) y1 y2 v2 v1 x2 p x1 CS-321Dr. Mark L. Hornick

  4. 2-D Transformations:Translation • Using vector math: • v1 = p + v2 • In terms of scalar components: • v1x = px + v2x • v1y = py + v2y • And conversely: • v2 = v1 – p • v2x = v1x - px • v2y = v1y - py y1 y2 v2 v1 x2 p x1 CS-321Dr. Mark L. Hornick

  5. y2 v2 x2 2-D Transformations:Rotation • Now consider the case of a pure rotation of one coordinate system with respect to the other • A simple case is rotation by 180 degrees • In this case, vectors v1 and v2 are coincident (they lie on top of each other), but they have opposite sense, that is: • v1 = -v2 • In terms of scalar components: • v1x = - v2x • v1y = - v2y v1 x1 y1 y2 v2 v1 x2 x1 y1 CS-321Dr. Mark L. Hornick

  6. y2 y2 x2 x2 2-D Transformations:Rotation x1 • Next, consider the general case of a pure rotation of one coordinate system with respect to the other, but through an arbitrary angle  v1 v2  x1 v2 v1  y1 CS-321Dr. Mark L. Hornick

  7. y2 y2 x2 x2 2-D Transformations:Rotation x1 • Using trigonometry, it can be shown that: • In terms of scalar components: • v1x = v2xcos - v2y sin • v1y = v2ycos + v2xsin • In vector terms: • v1 = v2cos + (-v2y,v2y)sin • But what vector is (-v2y,v2y)? • It can be shown that v1 v2  x1 v2 v1  y1 CS-321Dr. Mark L. Hornick

  8. y2 y2 x2 x2 2-D Transformations:Rotation x1 • Instead of using vector representation, these equations can also be cast in matrix form: v1 v2  x1 v2 v1  y1 CS-321Dr. Mark L. Hornick

  9. y2 y2 x2 x2 2-D Transformations:Rotation x1 • Now that we have v1 in terms of v2, how do we express v2 in terms of v1 ? • Where R()-1 is the inverse of R() v1 v2  x1 v2 v1  y1 CS-321Dr. Mark L. Hornick

  10. y2 y2 x2 x2 2-D Transformations:Rotation x1 • To express v2 in terms of v1, we can think of xy1 rotated through a negative angle  v1 v2  x1 v2 v1  y1 CS-321Dr. Mark L. Hornick

  11. y2 y2 x2 x2 2-D Transformations:Rotation x1 • So the inverseR()-1 is R(-) • Note also that the inverse R()-1is simply the transpose of R(-) since the off-diagonal elements are swapped • This is a property of special matrices called orthonormal matrices; Rotation matrices are orthonormal • This makes it easy to find the inverse of a rotation matrix v1 v2  x1 v2 v1  y1 CS-321Dr. Mark L. Hornick

  12. y2 x2 Combining Rotation and Translation • The case of combined rotation and translation can be deconstructed into a pure rotation followed by a pure translation. • The pure rotation transforms vector v2 from coordinate frame xy1 to vector v*2 in frame xy*2 • The following pure translation transforms v*2 from frame xy*2 into vector v1 in frame xy1 y*2 x1 v1 v2  x*2 p y1 CS-321Dr. Mark L. Hornick

  13. y2 x2 Combining Rotation and Translation • The equations that describe this transformation are just the combination of those we have already seen: y*2 x1 v1 v2  x*2 p y1 CS-321Dr. Mark L. Hornick

  14. y2 x2 Combining Rotation and Translation • This equation can be recast using a mathematical trick: • This is called a 3x3 homogeneous transformation matrix T(,p) y*2 x1 v1 v2  x*2 p y1 CS-321Dr. Mark L. Hornick

  15. y2 x2 Combining Rotation and Translation • T(,p) can be expressed in terms of submatrices as • The inverse of T(,p) is given by y*2 x1 v1 v2  x*2 p y1 CS-321Dr. Mark L. Hornick

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