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Three-Dimensional Shapes
Published in F. Brent Neal, John C. Russ, Measuring Shape, 2017
The basic two-dimensional transformations illustrated in Chapter 1, Figure 1.22 are translation, scaling, rotation, and shear. It is most convenient to describe and to implement these using a matrix notation, called homogeneous coordinate form, that specifies the transformed coordinates (x’, y’) in terms of the original coordinates (x, y) and the translation (Tx, Ty), scaling (Sx, Sy), rotation (H), and shear (Hx, Hy). For translation, the relationship is [x′y′1]=[xy1]⋅[100010TxTy1]
System Modelling
Published in Richard Leach, Stuart T. Smith, Basics of Precision Engineering, 2017
The homogeneous transformation matrix (HTM) is a mathematical representation that maps a position vector in homogeneous coordinates between relative coordinate systems (Maxwell 1961, Stadler 1995). By utilising homogeneous coordinates, rotation, translation, scaling and perspective information can be included in a single matrix operation. Homogeneous transformation coordinates are often used in robotics to analyse the kinematics and dynamics of complicated serial and/or parallel mechanisms. HTM modelling techniques are used in mechanism design because of their ability to combine the motions of complex systems in a structured manner. The role of HTMs in precision machine/instrument design is primarily focused on representing and quantifying the influence of error motions of a system and using this mathematical representation to determine critical aspects of a design.
Geometric Transformations
Published in Aditi Majumder, M. Gopi, Introduction to Visual Computing, 2018
There are several other practical advantages of this representation. In this chapter we will consider the 3D world with 4D homogeneous points for all our discussions. First, let us consider how you would represent points at infinity using 3D coordinates? The only option we have is (∞,∞,∞) $ (\infty , \infty , \infty ) $ . Now, this representation is pretty useless since it is the same for all points in infinity, even if they are in different directions from the origin. However, using a 4D homogeneous coordinate, points at infinity can be represented as (x,y,z,0) $ (x, y, z, 0) $ , where (x,y,z) $ (x, y, z) $ is the direction of the point from the origin. When we normalize this to get the 3D point back, we get (∞,∞,∞) $ (\infty , \infty , \infty ) $ as expected. As a consequence, homogeneous coordinates provides a way to represent directions (vectors) and distinguish them from the representation of points. w ≠ 0 signifies a point and w = 0 signifies a direction.
Non-destructive inspection system for MAG welding processes by combining multimodal data
Published in Quantitative InfraRed Thermography Journal, 2021
Katharina Simmen, Benjamin Buch, Andreas Breitbarth, Gunther Notni
The columns represent all points in homogeneous coordinates by adding 1 as the last entry. The representation in homogeneous coordinates enables a uniform treatment of the transformations. The calculation of the map of the reference image is done in the same way as . The transformation matrix is calculated using Equation (3).
Deep homography estimation in dynamic surgical scenes for laparoscopic camera motion extraction
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2022
Martin Huber, Sébastien Ourselin, Christos Bergeles, Tom Vercauteren
Two images are related by a homography if both images view the same plane from different angles and distances. Points on the plane, as observed by the camera from different angles in homogeneous coordinates are related by a projective homography (Malis and Vargas 2007)