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Fundamentals
Published in Michael Hann, The Grammar of Pattern, 2019
This chapter introduced the fundamental structural elements (including points, lines and planes) associated with the visual arts and design in general. Differentiation was made briefly between ‘organisational geometry’ and ‘content geometry’, and the necessity for designers to aim for simplicity was stressed. Simplicity is enhanced through planned organisation of constituent elements, and this is most conveniently done using a regular grid. Various fundamental issues were reviewed. For example, line divisions of 2, 3 and 4 were discussed, followed by grid division and an explanation of shape division methods. Proportions, such as 1:2; 1:3; 5:4; 1:√2; 1:√5 and 1:0.618, used commonly in graphic, as well as architectural, design were introduced. A convex polygon, which plays an important role in grid structures, was seen to be a simple figure in which no line segment between two points on the boundary goes ‘outside’ the polygon. Also, in a convex polygon, all interior angles were seen to be less than 180°. Kandinsky played an important role in reviewing the nature of fundamental elements. Also, it is worth stressing that he recognised that the use of different techniques (i.e. different means of application on different surfaces) would yield different results.
Algorithms for Planning under Uncertainty in Prediction and Sensing
Published in Shuzhi Sam Ge, Frank L. Lewis, Autonomous Mobile Robots, 2018
Jason M. O’Kane, Benjamín Tovar, Peng Cheng, Steven M. LaValle
In the context of manufacturing, a part may need to have a specific orientation before being assembled with other components. In a sensorless setting, a robot, in this case a robotic arm with a gripper, needs to orient a part without any feedback [84, 85]. The part is modeled as a convex polygon. Its initial orientation is unknown; the goal is to bring the part to a known orientation, up to symmetry. The manipulation process is shown in Figure 13.9. The part moves on the conveyor toward a fence, against which it comes to rest after possibly rotating to reach a stable orientation. The robotic arm grasps the part, changes its orientation, and drops it up again in the conveyor. This process is repeated until the part achieves the desired orientation against the fence.
Mesh Parameterization
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
In this chapter introduces mesh parameterization, an application of generalized barycentric coordinates. By “unfolding” a surface onto a 2D space, mesh parameterization has many possible applications, such as texture mapping. In this chapter we present some basic notions of topology that characterize the class of surfaces that admit a parameterization. Then we focus on the specific case of a topological disk with its boundary mapped to a convex polygon. In this setting, Tutte’s barycentric mapping theorem not only gives sufficient conditions, but also a practical algorithm to compute a parameterization. We outline the main argument of the simple and elegant proof of Tutte’s theorem by Gortler, Gotsman, and Thurston [172]. It is remarkable that their proof solely uses basic topological notions together with a counting argument. Finally, we mention the importance of the weights in the quality of the result, and demonstrate how mean value coordinates can be used to reduce the distortions.
Automated classification of thermal defects in the building envelope using thermal and visible images
Published in Quantitative InfraRed Thermography Journal, 2023
Changmin Kim, Gwanyong Park, Hyangin Jang, Eui-Jong Kim
As described above, material-related thermal bridges and air leakages are classified into different clusters according to GMM-based clustering. To classify the clusters adjacent to the window into material-related thermal bridges and air leakages, this study used the area of the window and each cluster. The window area was calculated using a segmentation map, and the area of each cluster was calculated by applying a convex hull. By calculating the convex hull, it is possible to define a convex polygon that contains each cluster point [31]. Figure 7 shows an example of dividing material-related thermal bridges and air leakages via the method proposed herein. As shown in the figure, the convex hull of the cluster corresponding to the material-related thermal bridge has a larger area than the window, and the convex polygon of the cluster corresponding to air leakage is restricted to a part of the window. Therefore, in this study, clusters with an area larger than the window area were classified as material-related thermal bridges, and small ones were classified as air leakages. The pseudo-code of the proposed classification algorithm can be found in Appendix B.
Superconvergence analysis of a nonconforming MFEM for nonlinear Schrödinger equation
Published in Applicable Analysis, 2022
Let be a convex polygon with boundary parallel to the coordinate axes, be a rectangular subdivision. Given , let the four vertices and edges be and , respectively. The MFE spaces and are defined by [24] where . stands for jump of across the edge if is an internal edge, and if is a boundary edge.
Describing the geometric difference of architectural forms in three primary shapes of circle, triangle and square
Published in Journal of Asian Architecture and Building Engineering, 2022
One of the reasons that shape analysis is delicate, despite the introduction of powerful statistical modelling techniques, is that not all variations between shapes are necessarily significant. That is to say, minor perturbations caused by noise have little effect on shape discrimination. The first operation we introduced to ignore the small shape details is the convex hull (CH). The convex hull (CH) of a polygon (P) is a conventional problem in computational geometry and is defined as the smallest convex polygon enclosing the entire polygon. Many studies handle the computing matter of CH; a representative algorithm can be referred to the study by Chen and Rokne(1992). Probable losses of crucial features along the concave part covered by CH, such as dents, indentations and cavities caused by dominant protrusions of steeple spire, can be handled by multiplying convexity as the ratio of the area of P to CH.