Technical Briefs

Some Advancements to Visualizing Constrained Functions and Inequalities of Two Variables

[+] Author and Article Information
P. A. Simionescu

 Texas A&M University Corpus Christi, 6300 Ocean Drive, Unit 5797, Corpus Christi, TX 78412pa.simionescu@tamucc.edu

D_3D is available upon request from the author.

J. Comput. Inf. Sci. Eng 11(1), 014502 (Mar 31, 2011) (7 pages) doi:10.1115/1.3570770 History: Received November 14, 2008; Revised March 03, 2011; Published March 31, 2011; Online March 31, 2011

Based on the problem of visualizing the potential energy of a two-degree-of-freedom spring-mass system restrained by an elastic string, several advancements to visualizing functions with constraints and inequalities of two variables are introduced. These innovations include logarithmically spacing level curves (either mapped on the surface or projected on the bottom plane) and the possibility of truncating the portions of the function surface that exceed above and below the bounding box—both allowing better detailing of certain regions of the function surface, in particular the minimum and maximum areas. By selectively displaying the surface patches that either intersect or not the top and/or bottom planes of the bounding box in a truncated representation, sets of inequalities of two variables can be represented graphically in a suggestive manner. Also proposed are a new approach to producing the gradient of the function as an arrow field mapped on the bottom of the plot box that uses a finite-difference scheme applied to the 2D image-space nodes of the function surface rather than the original 3D data, and a new way of displaying the color scale in 3D surface plots.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

A double-spring-mass system constrained by a string (represented in dashed line) of length l and stiffness K. All point coordinates are in centimeters.

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Figure 2

Graph of Eq. 2 represented as a single four-sided patch, with the gradient shown as one arrow projected on the bottom plane. XOY is the screen reference frame.

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Figure 3

Graph of PE(x,y) function in Eq. 1 with 31 equally spaced level curves mapped on the surface and the gradient projected on the bottom plane. For clarity, the vertical axis was reversed. Also notice the outlines of the x=0, y=0, and z=0 planes represented in dashed lines.

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Figure 4

Plot of the potential energy of the constrained system with (a) K=1 N/cm, (b) K=10 N/cm, and (c) K=100 N/cm, featuring 20 equally spaced level curves

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Figure 5

250×250 data-point plot of the potential energy function for the constraining string with K=100 N/cm, featuring 20 level curves logarithmically spaced about the minimum within zmin=−41.8 and zmax=5678.7

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Figure 6

Plot of the function in Eq. 11 as 20 level curves logarithmically spaced away from zero in both directions for zmin=−207 and zmax=261

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Figure 7

Plot of the function in Eq. 10 as equally spaced around zero level curves (17)

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Figure 8

Graph of Eq. 2 represented as 6×6 four-sided patches, bounded between zmin=−2 and zmax=2. Various effects can be obtained by selectively displaying the patches located inside, outside, or intersecting the upper or lower planes of the bounding box.

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Figure 9

Intersection variants of an initial four-sided patch (white) with a horizontal plane. The light-color polygons belong to the areas of the surface inside the bounding box, and the dark-color polygons belong to the areas outside the bounding box. The last four cases correspond to a local extremum or a saddle point occurring on the given patch and require two additional polygons: one light colored and one dark colored (12).

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Figure 10

Equally spaced level curves mapped on the truncated surface of the potential energy function PE(x,y) in Eq. 1 with K=100 N/cm

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Figure 11

Plot of the potential energy function in Eq. 1 with K=1099 N/cm. The elongated patches intersected by the z=400 plane were not represented, while the patches aligned with the bottom plane are a plot of inequality 12. Note how the color scale was combined with the z axis, another innovation implemented in D_3D , compared with known plotting software where the color scale is provided separately.

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Figure 12

Graphical representation of inequality 12 for 16×16 and 91×91 data points

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Figure 13

Overlap of potential energy graph and dynamic settling simulation results of the double-spring-mass system constrained by an ideally inelastic string. The arrow markers are placed at equal time intervals of 0.01 s, thus giving an indication of the speed at which the process occurs.




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