Apparent Shape of the Mach Book

Perception ◽  
1996 ◽  
Vol 25 (3) ◽  
pp. 313-319 ◽  
Author(s):  
Richard A Clement
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Two Dimensional ◽  
Piecewise Linear Functions ◽  
Common Edge ◽  
Edge Orientation ◽  
Line Drawings ◽  
Alternative Hypotheses

The Mach book is a two-dimensional figure which looks three-dimensional. Despite the impression of depth in the figure, the apparent shape has not been determined. It has been suggested that the book appears as part of a ‘cubic corner’, ‘as flat as possible’, or with each half rotated about its long diagonal. Alternative hypotheses as to the three-dimensional orientation of the book were tested by means of a probe-line technique. It was found that, although no hypothesis matched the results of all of the subjects, the probe-line settings of individual subjects were approximately linear or piecewise linear functions of the angles in the picture. The technique was also applied to asymmetric versions of the figure and it was found that the subjects modified their settings in accord with the constraint that the two halves of the figure must join in depth along their common edge. The findings are in agreement with models of the interpretation of line drawings in which local estimates of edge orientation in depth are formed, and subsequently checked for consistency.

Modeling of three-dimensional surfaces using high-level canonical piecewise-linear functions in cylindrical coordinates

2018 ◽  
Vol 37 (4) ◽  
pp. 5500-5513
Author(s):  
Victor M. Tlapa-Carrera ◽  
Victor M. Jimenez-Fernandez ◽  
Hector Vazquez-Leal ◽  
Uriel A. Filobello-Nino ◽  
Jesus Garcia-Guzman ◽  
...  
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Cylindrical Coordinates ◽  
High Level

Long-Time Solutions to Heat-Conduction Transients with Time-Dependent Inputs

1980 ◽  
Vol 102 (1) ◽  
pp. 115-120 ◽  
Author(s):  
H. T. Ceylan ◽  
G. E. Myers
Keyword(s):  
Heat Conduction ◽  
Lower Cost ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Time Dependent ◽  
Time Interval ◽  
Piecewise Linear Functions ◽  
One Dimensional ◽  
Long Time

An economical method for obtaining long-time solutions to one, two, or three-dimensional heat-conduction transients with time-dependent forcing functions is presented. The conduction problem is spatially discretized by finite differences or by finite elements to obtain a system of first-order ordinary differential equations. The time-dependent input functions are each approximated by continuous, piecewise-linear functions each having the same uniform time interval. A set of response coefficients is generated by which a long-time solution can be carried out with a considerably lower cost than for conventional methods. A one-dimensional illustrative example is included.

scholarly journals Exploring the use of two-dimensional piecewise-linear functions as an alternative model for representing and processing grayscale-images

2016 ◽  
Vol 14 (5) ◽  
pp. 311-318 ◽  
Author(s):  
Victor Manuel Jimenez-Fernandez ◽  
Héctor Vazquez-Leal ◽  
Uriel Antonio Filobello-Nino ◽  
Maribel Jimenez-Fernandez ◽  
Luis Javier Morales-Mendoza ◽  
...  
Keyword(s):  
Alternative Model ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Two Dimensional ◽  
Piecewise Linear Functions ◽  
Grayscale Images

scholarly journals A posteriori error estimates for semilinear optimal control problems

2021 ◽  
Author(s):  
Alejandro Allendes ◽  
Francisco Fuica ◽  
Enrique Otarola ◽  
Daniel Quero
Keyword(s):  
Optimal Control ◽  
Piecewise Linear ◽  
Control Variable ◽  
Three Dimensional ◽  
Posteriori Error ◽  
Linear Functions ◽  
Adjoint Equations ◽  
Error Estimator ◽  
Piecewise Linear Functions ◽  
A Posteriori Error

In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.

Modelling the single-electron transistor with piecewise linear functions

2009 ◽  
Author(s):  
Arturo Sarmiento-Reyes ◽  
Luis Hernandez-Martinez ◽  
Miguel Angel Gutierrez de Anda ◽  
Francisco Javier Castro Gonzalez
Keyword(s):  
Piecewise Linear ◽  
Linear Functions ◽  
Single Electron ◽  
Single Electron Transistor ◽  
Piecewise Linear Functions

Mesh duality and Legendre duality

1990 ◽  
Vol 428 (1875) ◽  
pp. 351-377 ◽  
Keyword(s):  
Piecewise Linear ◽  
Voronoi Tessellation ◽  
Tangent Plane ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Dual Transformation ◽  
Delaunay Mesh ◽  
Definition Of

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

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