Different types of transformations

modeling transformation in computer graphics and explain 2d geometric transformations in detail
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Dr.ShaneMatts,United States,Teacher
Published Date:23-07-2017
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Coordinates and Transformations MIT ECCS 6.837 Wojciech Matusik many slides follow Steven Gortler’s book 1 Hierarchical modeling • Many coordinate systems: • Camera • Static scene • car • driver • arm • hand Image courtesy of Gunnar A. Sjögren on Wikimedia Commons. License: CC-BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. • ... • Makes it important to understand coordinate systems 2 Coordinates • We are used to represent points with tuples of coordinates such as • But the tuples are meaningless without a clear coordinate system could be this point could be this point in the red in the blue coordinate system coordinate system 3 Different objects • Points • represent locations • Vectors • represent movement, force, displacement from A to B • Normals • represent orientation, unit length • Coordinates • numerical representation of the above objects in a given coordinate system 4 Points & vectors are different • The 0 vector has a fundamental meaning: no movement, no force • Why would there be a special 0 point? • It’s meaningful to add vectors, not points • Boston location + NYC location =? + =? 5 Points & vectors are different • Moving car • points describe location of car elements • vectors describe velocity, distance between pairs of points • If I translate the moving car to a different road • The points (location) change • The vectors (speed, distance between points) don’t Image courtesy of Gunnar A. Sjögren on Wikimedia Commons. License: CC-BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 6 Matrices have two purposes • (At least for geometry) • Transform things • e.g. rotate the car from facing North to facing East • Express coordinate system changes • e.g. given the driver's location in the coordinate system of the car, express it in the coordinate system of the world 7 Goals for today • Make it very explicit what coordinate system is used • Understand how to change coordinate systems • Understand how to transform objects • Understand difference between points, vectors, normals and their coordinates 8 Questions? 9 Reference • This lecture follows the new book by Steven (Shlomo) Gortler from Harvard: Foundations of 3D Computer Graphics 10 Plan • Vectors • Points • Homogeneous coordinates • Normals (in the next lecture) 11 Vectors (linear space) • Formally, a set of elements equipped with addition and scalar multiplication • plus other nice properties • There is a special element, the zero vector • no displacement, no force 12 Vectors (linear space) • We can use a basis to produce all the vectors in the space: • Given n basis vectors any vector can be written as here: 13 Linear algebra notation • can be written as • Nice because it makes the basis (coordinate system) explicit • Shorthand: • where bold means triplet, t is transpose 14 Questions? 15 Linear transformation Courtesy of Prof. Fredo Durand. Used with permission. • Transformation of the vector space 16 Linear transformation Courtesy of Prof. Fredo Durand. Used with permission. • Transformation of the vector space so that • Note that it implies • Notation for transformations 17 Matrix notation • Linearity implies ? 18 Matrix notation • Linearity implies • i.e. we only need to know the basis transformation • or in algebra notation 19 Algebra notation • The are also vectors of the space • They can be expressed in the basis for example: ... • which gives us 20