In computer graphics and robotics, we can define (2D) scaling, rotations, and translations in Cartesian coordinates using the following operations: x 0 X x x' cos-sinð x x' = = sinᎾ cos Ꮎ + [M]-6 ©D] M-0-0 S That is, scaling and rotations are done using matrix multiplication and translation by adding a vector. We can combine these three operations (in that order, scaling, roation, and then translation) into a single matrix multiplication using homogeneous coordinates. In homogeneous coordinates, we add a third coordinate to a point (vector). Instead of being represented by a pair of numbers (x, y), each point (vector) is represented as a triple (x, y, w) where w 0. For convenience, w is usually taken to be 1. Thus, [Sx cos - sysin 0 Sx sin e 0 Sycose 0 tx] [x₁ 96-8 ty = transforms the vector (x, y)' into a new vector that has been scaled in the x direction by S and in the y direction by sy, rotated counter-clockwise by 0, and translated by (tx, ty)'. Note that in robotics, the these operations are done on rigid bodies so the scaling is always unity.

In computer graphics and robotics, we can define (2D) scaling, rotations, and translations in Cartesian coordinates using the following operations: x 0 X x x' cos-sinð x x' = = sinᎾ cos Ꮎ + [M]-6 ©D] M-0-0 S That is, scaling and rotations are done using matrix multiplication and translation by adding a vector. We can combine these three operations (in that order, scaling, roation, and then translation) into a single matrix multiplication using homogeneous coordinates. In homogeneous coordinates, we add a third coordinate to a point (vector). Instead of being represented by a pair of numbers (x, y), each point (vector) is represented as a triple (x, y, w) where w 0. For convenience, w is usually taken to be 1. Thus, [Sx cos - sysin 0 Sx sin e 0 Sycose 0 tx] [x₁ 96-8 ty = transforms the vector (x, y)' into a new vector that has been scaled in the x direction by S and in the y direction by sy, rotated counter-clockwise by 0, and translated by (tx, ty)'. Note that in robotics, the these operations are done on rigid bodies so the scaling is always unity.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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