• Journal for History of Mathematics
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• v.37 no.4
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• pp.93-105
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• 2024

Research on the caustic, the envelope of light rays, began with the geometric optics studies of Huygens and others in the 17th century. One of the most important problems in optics in the 17th century was focusing the sun's rays. This was a problem that had to be solved in order to manufacture various practical optical instruments at the time. Beginning with research on the caustic during this period, the concept of envelope became generalized and expanded to various fields until the 19th century. This paper examines the mathematical history involved in the study of envelope curves. We compare several methods of defining the envelope and provide an example of calculating the envelope accordingly.

• Structural Engineering and Mechanics
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• v.16 no.4
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• pp.405-422
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• 2003

Scaled brick masonry panels were tested under cyclic unialxial compression loading to evaluate its deformation characteristics. An envelope stress - strain curves, a common point curves and stability point curves were obtained for various cyclic test conditions. Loops of the stress-strain hysteresis were used to determine the energy dissipation for each cycle. Empirical expressions were proposed for the relations between energy dissipation and envelope and residual strains. These relations indicated that the decay of masonry strength starts at about two-third of peak stress.

• Korean Journal of Computational Design and Engineering
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• v.6 no.2
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• pp.89-100
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• 2001

Swept volume is the sweeping region of moving objects. It is used in various applications such as interference detection in assembly design, visualization of manipulator motions in robotics, simulation of the volume removal by a cutter in NC machining. The shape of swept volume is defined by the envelope, which is determined by the boundary of moving objects and its direction of motion. In order to implement the generation of swept volume, researchers have taken much effort to develop the techniques how to generate the envelope. However, their results are confined to envelope generated only in simple shape objects, such as polyhedra or quadric surfaces. This study provided the envelope generation algorithm of NURBS objects. Characteristic points were obtained by applying the geometric conditions of envelope to NURBS equations, and then characteristic curves were created by means of interpolating those points. Silhouette edges were determined in the following procedures. First, two adjacent surfaces which have the same edge were found from B-Rep data. Then, by taking the scalar product of velocity vector of a point on that edge with each normal vector on two surfaces, silhouette edges were discriminated. Finally, envelope was generated along moving direction in the form of ruled surfaces by using both the partial information between initial and final position of objects affecting envelope along with characteristic curves and silhouette edge. Since this developed algorithm can be applied not only to NURBS objects but also to their Boolean objects, it can be used effectively in various applications.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.121-133
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• 2008

In this article, we define Minkowski Pythagorean-hodograph (MPH) curves in the Minkowski plane $\mathbb{R}^{1,1}$ and obtain $C^1$ Hermite interpolations for MPH quintics in the Minkowski plane $\mathbb{R}^{1,1}$. We also have the envelope curves of MPH curves, and make surfaces of revolution with exact rational offsets. In addition, we present an example of $C^1$ Hermite interpolations for MPH rational curves in $\mathbb{R}^{2,1}$ from those in $\mathbb{R}^{1,1}$ and a suitable MPH preserving mapping.

• Journal of the Korean Society for Precision Engineering
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• v.15 no.5
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• pp.85-92
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• 1998

This present paper describes a mathematical model of profile shifted elliptical gears, and this model is based on the concepts of envelope theory and conjugate geometry between the blank and the straight-sided rack cutter. The geometric model of the rack cutter includes working regions generating involute curves and fillets for trocoidal curves, and furthermore the addendum modified coeff. is considered for avoiding undercutting. The addendum modified coeff. is changed linearly along with pitch curves and must be the same absolute value at both major semi-axis and minor semi-axis. If undercutting is at all pronounced, the undercut tooth not only are weakened in strength, but lose a small portion of the involute adjacent to the base circle, then this loss of involute may cause a serious reduction in the length of contact. A very effective method of avoiding undercutting is to use the so-called profile shifted gearing. Non-undercutting condition is examined with the change of eccentricity and addendum modified coeff. in elliptical gears and then the minimum number of tooth is proposed not to gernerate undercutting phenomenon.

• The Bulletin of The Korean Astronomical Society
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• v.45 no.1
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• pp.70.1-70.1
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• 2020

A type IIb supernova (SN IIb) is the result of core-collapse of a massive star which lost most of its hydrogen-rich envelope during its evolution. The pre-SN progenitor properties, such as the total radius and the mass of the hydrogen-rich envelope, can widely vary due to the mass-loss history of the progenitors. Optical light curves of SNe IIb are dominated by energy released by the hydrogen recombination and the radioactive decay of 56Ni in the early and late epochs respectively. This may result in distinctive double peaked light curves like the one observed in SN 1993J. The first peak, caused by the hydrogen recombination, can be modelled with numerical simulations providing information on the pre-SN progenitor properties. We compare two radiation-hydrodynamics codes, STELLA and SNEC, that are frequently used in SNe modelling, and investigate the effect of opacity treatment on the temporal evolution of the color temperature of SNe and eventually on the optical light curves. We find that with a proper treatment of the scattering opacity, SNe IIb models exploded from the progenitor models evolved with latest stellar evolution model hardly match the observational data. We also discuss the smaller scale features found in the models during hydrogen recombination phase.

• Structural Engineering and Mechanics
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• v.9 no.4
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• pp.375-382
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• 2000

Experimental investigation into the cyclic behaviour of sand plast brick masonry was performed on forty two square panels. The panels were subjected to cyclic uniaxial compression for two cases of loading: normal to bed joint and parallel to bed joint. Experimental data were used to plot the unloading-reloading curves for the entire range of the stress-strain curve. Mathematical expressions to predict the reloading and unloading stress-strain curves at various values of residual strain are proposed. A simple parabola and an exponential type formula are found adequate to model the unloading and reloading curves respectively. The models account for the potential effects of residual strain on these curves. Comparison of test results with the proposed mathematical expression shows good correspondence.

• Structural Engineering and Mechanics
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• v.11 no.5
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• pp.565-574
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• 2001

Experimental investigation into the behaviour of half-scale brick masonry panels were conducted under cyclic loading normal to the bed joint and parallel to the bed joint. For each cycle, full reloading was performed with the cycle peaks coinciding approximately with the envelope curve. Unloading, however, was carried out fully to zero stress level and partially to two different stress levels of 25 percent and 50 percent of peak stress. Stability point limit exhibits a unique stress-strain curve for full unloading but it could not be established for partial unloading. Common point limit was established for all unloading-reloading patterns considered, but its location depends on the stress level at which unloading is carried to. Common point curves were found to follow an exponential formula, while residual strains versus envelope strains can be expressed by a polynomial function of a single term. The relation between residual strain and envelope strain can be used to determine the stress level at which deterioration due to cyclic loading began.

• Proceedings of the Korea Concrete Institute Conference
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• 2000.04a
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• pp.599-604
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• 2000

The analytical studies for the test results are presented. From experimental envelope curve, hysteretic behaviors under cyclic loading are modelled for beam-column subassemblages. Modelled curves show relatively good reproduction or experimental curves. finally, seismic performances are investigated for 7 story plain framed structure by nonlinear dynamic analysis. The resulting responses are different by each beam-column joint design.

• Computers and Concrete
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• v.18 no.6
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• pp.1195-1211
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• 2016

In this study, seismic behaviour of reinforced concrete buildings using the pushover and incremental dynamic analysis method was investigated. A numerical study was performed for a reinforced concrete frame building. Pushover analysis according to uniform and triangular load shapes and incremental dynamic analyses were performed for selected building. For the nonlinear analysis, three ground motion records were selected to ensure compatibility with the design spectrum defined in the Turkish Seismic Code. The maximum response, dynamic pushover curve, capacity curves, interstorey drifts and moment rotation curves for various element ends of the selected building were obtained. Results were compared each other and good correlation was obtained between the dynamic analyses envelope with static pushover curves for the building.