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Ray Tracing
Published in Daniel Malacara-Hernández, Brian J. Thompson, Fundamentals and Basic Optical Instruments, 2017
Ricardo Flores-Hernández, Armando Gómez-Vieyra
In his paper, E. Delano defined a pure scalar determinant operation, given above, at the introduction of this section (Equation 4.123). Positive physical distances τ correspond to vectors τ pointing in the positive z-axis direction and negative distances between surfaces to vectors pointing in the negative z-direction, as evident from Equation 4.126. Remember also that the magnitude of a vector product of two vectors is equal to the area of the associated rhomboid parallelogram spanned by them. To simplify the interpretation of a Delano diagram, the magnitude of the distance between two consecutive surfaces, represented by the line segment connecting their diagram points, is interpreted as one half the rhomboid area, that is, as the area of the triangle formed by both point vectors and the line segment defined by them.
Induction Mass Heating
Published in Valery Rudnev, Don Loveless, Raymond L. Cook, Handbook of Induction Heating, 2017
Valery Rudnev, Don Loveless, Raymond L. Cook
Such an approach can particularly be beneficial when designing systems to heat irregular-shaped billets or components with various masses and complex cross sections including triangular, trapezoidal, rhomboid, hexagonal, parallelogram, polygonal, and others. The ability of IH to provide highly efficient and rapid (if required) bulk heating (as an induction booster) may be highly attractive since it allows a marked increase in production rate and dramatically reduces the scale formation owing to a significant reduction of the overall process time and most importantly the time when the steel surface is exposed to high temperatures.
Prism Design and Applications
Published in Paul Yoder, Daniel Vukobratovich, Opto-Mechanical Systems Design, 2017
The rhomboid prism shown in Figure 7.14 is equivalent to a combination of two right-angle prisms of aperture A with their reflecting surfaces of face length D parallel. Integral with and between these prisms may be a plane-parallel plate of dimensions A × A× C. The prism is used to displace the axis transversely by a distance B without changing the axis direction. An image seen through such a device is erect in both directions. The following Design Example 7.6 and equations given therein pertain.
Technical Note: The Anti-plane Scattering of SH Waves by the Non-circular Cavity in an Infinite Strip
Published in Journal of Earthquake Engineering, 2022
It can be seen that when the mapping function method is used to solve the stress concentration around the elliptic cavity or square cavity under the action of SH wave incident, the results will have a large error due to the approximate solution. With the addition of various boundaries, the solution process becomes more and more complicated. And it is impossible to solve the SH wave scattering problem of rectangle cavity and rhomboid cavity. The finite element method is used to solve such problems in the strip domain, which only needs to simulate the SH wave in the strip domain and then directly change the cavity shape in the finite element software. Compared with the mapping function method, the finite element method is more concise, faster and more accurate in solving such problems.
Mathematical understanding in problem solving with GeoGebra: a case study in initial teacher education
Published in International Journal of Mathematical Education in Science and Technology, 2020
Alexánder Hernández, Josefa Perdomo-Díaz, Matías Camacho-Machín
When looking for an extension of the problem, we must reflect on which mathematical properties or objects of the original problem we can change, and what would happen with the problem solution. Would the same solution steps be valid? Would the solution change as well? One option could be to change the figure that circumscribes the circumference to another quadrilateral (a rhombus or rhomboid), in which a square with side 2R would again be the solution. Thinking of a non-mathematical context for a given problem requires considering which mathematical properties or objects could still be directly related, and which could stem from deduction within the context or be formulated in a non-mathematical language.
Preparation of a dicationic, p-cymene ruthenium(II) dimer, [(p-cymene)Ru(µ-Cl)(P{OCH2}3CEt)]2 2+: structure, characterization and reactivity
Published in Journal of Coordination Chemistry, 2021
Joseph G. Bazemore, Clifford W. Padgett, Brandon Quillian
Suitable crystals of 2 were grown by allowing chloroform solutions of 1 to gradually precipitate 2 as orange crystals. The single crystal X-ray structure of 2, crystallizing in the triclinic point group, is shown in Figure 3. The structure reveals a dimeric Ru(II) structure with bridging chloride ligands supporting a nearly planar Ru2Cl2 rhomboid core, which is slightly displaced from square geometry. Two sterically imposing BArF′ anions reside in the outer coordination sphere as counter ions but are not shown for clarity. The [(p-cymene)Ru(µ-Cl)(P{OCH2}3CEt)]22+ central core possesses Ci symmetry. The nearly planar Ru2Cl2 rhomboid core was also observed in [{(p-cymene)Ru(μ-Cl)}2(μ-1,8-dpmn-P,P′)]2+ and {[(p-cymene)Ru(μ-Cl)]2(μ-1,2-di(2′,2′-diethyl-1′,3′-propanedioxyphosphinoethane)}2+ [39, 40]. The disparity from square geometry of the Ru2Cl2 core is a consequence of the Cl(1)-Ru(1)-Cl(1) bond angle being slightly acute (82.29(7)°), while the Ru(1)-Cl(1)-Ru(1) bond angle is obtuse (97.71(7)°). These values are strikingly similar to those observed in previously reported examples of compounds with a Ru2Cl2 rhomboid core [39, 40]. The Cl(1)-Ru(1)-Cl(1) bond angle in 2 is, however, slightly more acute than that found in monomer (p-cymene)Ru(P{OCH2}3CEt)Cl2 (88.59(1)°) [17]. The Ru(1)-Cl(1) bond distances are equivalent (2.431(2) Å) and compare well with those in [{(p-cymene)Ru(μ-Cl)}2(μ-1,8-dpmn-P,P′)]2+ (2.439(3) Å) [33], but are slightly longer than those in {[(p-cymene)Ru(μ-Cl)]2(μ-1,2-di(2′,2′-diethyl-1′,3′-propanedioxyphosphinoethane)}2+ (2.4207(16) Å) [34]. The two ruthenium ions in 2 are displaced at a distance of 3.668 Å, which is significantly longer than the Ru–Ru bond distance (2.601 Å) found in the dimeric Ru(I) complex, [(p-cymene)Ru(μ-Cl)]2 [41, 42]. We do not suppose the diamagnetic 2 possesses metal-metal boding due to the presence of the two counter anions. The two phosphorus ligands in 2 reside on opposite sides, nearly perpendicular to the Ru2Cl2 plane (P(1)-Ru(1)-Cl(1) bond angle = 89.16(8)°) with a Ru(1)-P(1) bond distance of 2.275(2) Å, which is slightly longer than that previously reported for dichloride monomer, (p-cymene)Ru(Cl)2(P{OCH2}3CEt) (Ru − P bond distance = 2.2529(4) Å) [18].