Gnomonic Projection of the Surface of an Ellipsoid

1997 ◽  
Vol 50 (2) ◽  
pp. 314-320 ◽  
Author(s):  
Roy Williams
Keyword(s):  
Tangent Plane ◽  
The Body ◽  
Normal Curvature ◽  
Geodesic Mapping ◽  
Straight Line ◽  
Ellipsoid Of Revolution

When a surface is mapped onto a plane so that the image of a geodesic arc is a straight line on the plane then the mapping is known as a geodesic mapping. It is only possible to perform a geodesic mapping of a surface onto a plane when the surface has constant normal curvature. The normal curvature of a sphere of radius r at all points on the surface is I/r hence it is possible to map the surface of a sphere onto a plane using a geodesic mapping. The geodesic mapping of the surface of a sphere onto a plane is achieved by a gnomonic projection which is the projection of the surface of the sphere from its centre onto a tangent plane. There is no geodesic mapping of the ellipsoid of revolution or the spheroid onto a plane because the ellipsoid of revolution or the spheroid are not surfaces whose curvature is constant at all points. We can, however, still construct a projection of the surface of the ellipsoid from the centre of the body onto a tangent plane and we call this projection a gnomonic projection also.

Potential and Viscous Parts of the Thrust Deduction and Wake Fraction for an Ellipsoid of Revolution

1960 ◽  
Vol 4 (03) ◽  
pp. 1-16
Author(s):  
Stavros Tsakonas ◽  
Winnifred R. Jacobs
Keyword(s):  
Boundary Layer ◽  
Potential Flow ◽  
Functional Dependence ◽  
Longitudinal Axis ◽  
Layer Growth ◽  
The Body ◽  
Displacement Thickness ◽  
Layer Effect ◽  
Ellipsoid Of Revolution ◽  
Equivalent Sources

Expressions are developed for wake fraction and thrust deduction due to the potential flow and to the boundary-layer effects for a fully-submerged prolate ellipsoid of revolution. The functional dependence of wake fraction and thrust deduction on axial-propeller clearance, body slenderness, after body geometry, and Reynolds number (scale effect) are exhibited for both potential and viscous-flow cases. Closed-form expressions are derived for the potential-flow case by representing the body by a line source-sink distribution and the propeller action by a sink disk. The boundary-layer effect is determined by Lighthill's method of equivalent sources distributed on the surface having strength proportional to the displacement thickness and its derivative. The wake is replaced by a cylinder of diameter equal to twice the displacement thickness at the stern. Although in practice the propeller is usually fully submerged in the wake of the hull, in this case the substitute cylinder has been shown by computation to be no wider than the hub diameter and thus the propeller is operating in a potential field. This consideration is fundamental to the construction of a possible mathematical model having the surface sources mentioned and an equivalent sink on the longitudinal axis whose position is determined on the basis of the velocity distribution in the wake. Computational work is carried out for a modification of the airship Akron. Four different methods, with various degrees of accuracy, are used for the evaluation of the boundary-layer growth in order to ascertain the degree of sensitivity of the thrust deduction and wake fraction to the boundary-layer development.

Extended Camus Theory and Higher Order Conjugated Curves

2019 ◽  
Author(s):  
Cody Leeheng Chan ◽  
Kwun-Lon Ting
Keyword(s):  
Stationary Point ◽  
Higher Order ◽  
The Body ◽  
The Other ◽  
Body System ◽  
Straight Line ◽  
Pair Generation ◽  
Curvature Theory ◽  
Three Body ◽  
Two Bodies

Abstract According to Camus’ theorem, for a single DOF 3-body system with the three instant centers staying coincident, a point embedded on a body traces a pair of conjugated curves on the other two bodies. This paper discusses a fundamental issue not addressed in Camus’ theorem in the context of higher order curvature theory. Following the Aronhold-Kennedy theorem, in a single degree-of-freedom three-body system, the three instant centers must lie on a straight line. This paper proposes that if the line of the three instant centers is stationary (i.e. slide along itself), on the line of the instant centers a point embedded on a body traces a pair of conjugated curves on the other two bodies. Another case is that if the line of the three instant centers rotate about a stationary point, the stationary point embedded on the body also traces a pair of conjugated curves on the other two bodies. The paper demonstrates the use of instantaneous invariants to synthesize such a three-body system leading to a conjugate curve-pair generation. It is a supplement or extension of the Camus’ theorem. The Camus’ theorem may be regarded as a special singular case, in which all three instant centers are coincident.

The dispersion of contaminant released from instantaneous sources in laminar flow near stagnation points

1974 ◽  
Vol 66 (4) ◽  
pp. 753-766 ◽  
Author(s):  
P. C. Chatwin
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Bluff Body ◽  
Velocity Potential ◽  
Tangent Plane ◽  
The Body ◽  
Centre Of Mass ◽  
Stagnation Points ◽  
Molecular Diffusivity ◽  
Instantaneous Source

This paper considers the dispersion of a cloud of passive contaminant released from an instantaneous source in the steady two-dimensional laminar flow near the forward stagnation point on a bluff body. The body is replaced by its tangent plane y = 0 with x measuring distance along the plane. Far away from y = 0 the flow is irrotational with velocity potential ½l(x2 – y2), where l is a positive constant. When the boundary layer is ignored the equation governing the distribution of concentration can be solved exactly. Consequences of this solution are that for large times the centre of mass moves parallel to the body at a speed proportional to exp (lt) while the cloud spreads out along the body symmetrically about the centre of mass with the magnitude of the spread also proportional to exp (lt). However, this solution is unrealistic because most of the contaminant is confined to a layer adjoining the body of thickness of order (k/l)½, where k is the molecular diffusivity, and this layer normally lies within the boundary layer, which is of thickness of order (v/l)½, where v is the kinematic viscosity. An approximate analysis, based on ideas similar to those supporting the Pohlhausen method in boundary-layer theory, suggests that when the boundary layer is taken into account the conclusions above remain true provided that exp (lt) is replaced by exp (βlt), where β is a constant depending on v/k. Calculations give values of β ranging from 0·73 when v/k = 0·5 to 0·10 when v/k = 103.

Low Reynolds numbers flow past an ellipsoid of revolution of large aspect ratio

1965 ◽  
Vol 23 (4) ◽  
pp. 657-671 ◽  
Author(s):  
Yun-Yuan Shi
Keyword(s):  
Reynolds Number ◽  
Aspect Ratio ◽  
Three Dimensional ◽  
Major Axis ◽  
Reynolds Numbers ◽  
The Body ◽  
Large Aspect Ratio ◽  
Low Reynolds Numbers ◽  
Ellipsoid Of Revolution ◽  
Limiting Case

The results of Proudman & Pearson (1957) and Kaplun & Lagerstrom (1957) for a sphere and a cylinder are generalized to study an ellipsoid of revolution of large aspect ratio with its axis of revolution perpendicular to the uniform flow at infinity. The limiting case, where the Reynolds number based on the minor axis of the ellipsoid is small while the other Reynolds number based on the major axis is fixed, is studied. The following points are deduced: (1) although the body is three-dimensional the expansion is in inverse power of the logarithm of the Reynolds number as the case of a two-dimensional circular cylinder; (2) the existence of the ends and the variation of the diameter along the axis of revolution have no effect on the drag to the first order; (3) a formula for drag is obtained to higher order.

scholarly journals Formation surfaces of Monge by the kinematic method in AutoCAD environment

2019 ◽  
Vol 15 (2) ◽  
pp. 106-116
Author(s):  
Viktoryna A Romanova
Keyword(s):  
Tangent Plane ◽  
Sine Wave ◽  
Circular Cone ◽  
Dynamic Mode ◽  
Developable Surface ◽  
Monitor Screen ◽  
Straight Line ◽  
Kinematic Method ◽  
Conical Surfaces

Aims of research. Studying the possibility of forming Monge carved surfaces, defined by the method of their formation, creating an algorithm and program in the AutoLISP language to demonstrate the formation of surfaces in the AutoCAD environment in a dynamic mode. Methods. Monge carved surfaces are formed by a flat curve, located in the tangent plane to the fixed guide of the developable surface, when the plane and the curve roll along the guide surface without sliding. The described method of formation of these surfaces allows to perform their formation by the kinematic method in the AutoCAD environment using AutoLISP software. The article describes the construction of the Monge surfaces using cylindrical and conical surfaces as guides. A straight line and a sine wave are used as the forming lines. Results. An algorithm and a program in the AutoLISP language were created to form sets of compartments of several Monge surfaces and to visualize the formation of these surfaces in a dynamic mode by sequentially displaying the compartments on the monitor screen. The mini-film about formation of Monge surface by rolling a plane with a straight line along a circular cone is created. In the mini-film the drawings received by transformation of drawings of the AutoCAD environment are used.

Counterrolling of the eyes and its dependence on the magnitude of gravitational or inertial force acting laterally on the body

1959 ◽  
Vol 14 (4) ◽  
pp. 632-634 ◽  
Author(s):  
Richard C. Woellner ◽  
Ashton Graybiel
Keyword(s):  
Incident Angle ◽  
Inertial Force ◽  
Current Theory ◽  
The Body ◽  
Otolith Organs ◽  
Straight Line ◽  
Independent Variable ◽  
The Subject ◽  
Acting Force ◽  
Healthy Persons

Counterrolling of the eyes was measured in five healthy persons when inclined on a tilt-chair and when exposed to a change in direction of force on a human centrifuge. For equivalent changes in direction of force incident to the subject, the magnitude of the force was greater on the centrifuge. When the amount of roll was plotted as a function of the incident angle of force, divergent curves were obtained for tilt-chair and centrifuge data. When the amount of roll was plotted as a function of magnitude of laterally-acting force as the independent variable, a single curve resulted indicating a straight line relation within the range of 1 g. These findings not only constitute definitive proof that the counterrolling reflex can be released by gravitational (and inertial) force but also are consistent with the current theory of the functioning of the otolith organs. Submitted on January 20, 1959

Psychophysical Properties of the Trunk Midline

1997 ◽  
Vol 78 (1) ◽  
pp. 545-549 ◽  
Author(s):  
Giuseppe Spidalieri ◽  
Roberto Sgolastra
Keyword(s):  
Mental Representation ◽  
Direct Evidence ◽  
The Body ◽  
Image Analysis System ◽  
Anatomic Landmarks ◽  
Straight Line ◽  
Anterior Trunk ◽  
Analysis System ◽  
The Right ◽  
Male Subjects

Spidalieri, Giuseppe and Roberto Sgolastra. Psychophysical properties of the trunk midline. J. Neurophysiol. 78: 545–549, 1997. This study was carry out to obtain direct evidence that the body midline actually is perceived and to assess some psychophysical properties of this line. Twelve normal, right-handed male subjects were asked to make accurate pointing movements toward the midline of the anterior trunk on the basis of their mental representation of this line. Each hand was used to point while the head was either aligned with the trunk or tilted 30° to the right or left. Analysis of end-positions of pointing on trunk images acquired by an image analysis system indicated that the trunk midline indeed is perceived as a straight line. Three putative trunk midlines were taken into consideration on the basis of anatomic landmarks, and it was found that the mental representation of the trunk midline came nearest to the line orthogonal to the intermammary line crossing its midpoint. The performing hand and the position of the head relative to the trunk both had an effect on the mental representation of the trunk midline. These findings suggest that somatosensory signals from the trunk, as well as proprioceptive input from the neck, contribute to the elaboration of the subject's mental representation of the trunk midline.

Modeling, Analysis and Design of the Formula SAE Aerodynamics System

2020 ◽  
Author(s):  
Isaac Van Baren ◽  
Andrew Van Milligan ◽  
Scott Ashcraft ◽  
Stephen Rosser ◽  
Xiuling Wang
Keyword(s):  
Simulation Software ◽  
The Body ◽  
Analysis And Design ◽  
Formula Sae ◽  
System A ◽  
Venturi Effect ◽  
Modeling Analysis ◽  
Straight Line ◽  
Design Competition ◽  
Vehicle Chassis

Abstract This project developed a study on methods to increase downforce on the university’s Formula SAE vehicle by implementing a lightweight, efficient aerodynamic design. The team planned to improve the performance and reduce lap times of the vehicle with an undertray, which grants better wheel traction and stability while handling corners. Upon completion, the aerodynamic component would have allowed the PNW Motorsports team to more effectively compete at the FSAE design competition in the spring of 2020. While reducing drag, an undertray provides the capability to direct the air beneath the vehicle chassis in a way which adds “artificial weight” to the system. A pressure gradient of high magnitude is established between the two sides of the undertray, with a low negative pressure region found beneath the body. This design is based upon the principles of fluid dynamics, in particular the venturi effect through the use of nozzles and diffusers. In this fashion, the vehicle can receive the benefits of a heavier car around corners while maintaining the higher straight-line acceleration of a lighter car. This report describes the use of simulation software in the design of an undertray, as well as the approach to manufacture it. Two-dimensional benchmark cases were performed in the replication of results obtained in a literature search. Subsequently, the undertray model was optimized with CFD and FEA/FEM techniques to obtain a component that was prepared for manufacturing. An operating procedure was established to outline the complicated steps of its assembly. Finally, it provides future aerodynamics teams with a solid foundation upon which improvements can be made.

scholarly journals Measuring Aerodynamic Interference Drag Between a Bird Body and the Mounting Strut of a drag Balance

1990 ◽  
Vol 154 (1) ◽  
pp. 439-461 ◽  
Author(s):  
VANCE A. TUCKER
Keyword(s):  
Reynolds Numbers ◽  
The Body ◽  
Peregrine Falcon ◽  
Rigid Surface ◽  
Hand Tools ◽  
Cross Sectional ◽  
Straight Line ◽  
Aerodynamic Interference ◽  
The Relationship

1. The drag of a bird body mounted on the strut of a drag balance in a wind tunnel is more than the sum of the drags of the isolated strut and the isolated body. The strut changes the air flow around the body and generates additional drag, known as interference drag. This paper describes practical methods for measuring the drag of bird bodies: a strain-gauge drag balance, dimensions for struts made with machine or hand tools, and a procedure for correcting drag measurements for interference drag. 2. Interference drag can be measured by extrapolating a relationship between the drag of isolated struts with different crosssectional sizes and shapes and the drag of a body mounted on those struts. The interference length the length of an isolated strut that produces drag equal to the interference drag is a usefulquantity for predicting interference drag. 3. The relationship mentioned above is a straight line for a model peregrine falcon (Falco peregrinus L.) body mounted on smooth struts struts with convex cross-sectional shapes ranging from streamlined to circular. This finding simplifies the determination of interference drag in three ways: (i) the line can be found from measurements with just two struts a standard strut with low drag and a calibration strut with high drag; (ii) the two struts need not have the same shape for example, the standard strut can be changed to a calibration strut by attaching a spoiler without disturbing the body mounted on the strut and (iii) a single value of interference length (33.1mm) describes smooth struts with a range of shapes and sizes. These struts had drag coefficients between 0.33 and 0.91 at Reynolds numbers between 2100 and 10800. 4. The interference length of a strut supporting the actual falcon body with a feathered surface is not significantly different from that of the strut supporting the model body with a rigid surface. 5. As a hypothesis, interference length (hI, in metres) of a smooth strut varieswith the size of the body mounted on it: hI=0.0365m0.333 where m is the body mass (in kg) of the intact bird.

Memoirs: Materials for a Monograph of the Ascons.--I. On the Origin and Growth of the Triradiate and Quadriradiate Spicules in the Family Clathrinidæ

1898 ◽  
Vol s2-40 (160) ◽  
pp. 469-587
Author(s):  
E. A. MINCHIN
Keyword(s):  
Distal Portion ◽  
The Body ◽  
The Other ◽  
Ground Substance ◽  
Connective Tissue Layer ◽  
Dermal Layer ◽  
Straight Line ◽  
The Family ◽  
A Cell ◽  
Pore Cell

1. The first appearance of a calcareous spicule or spicular element, both ancestrally and in the actual development, was probably a minute vacuole in a cell of the dermal layer, filled with an organic substance perhaps identical with the intercellular ground substance, within which the minute sclerite appeared as a crystal or concretion. 2. The ancestral sclerite, though crystalline in structure, soon assumed a non-crystalline form as a whole, as an adaptation to its secondarily acquired function of support, and as it grew in size the contents of the vacuole formed the spicule sheath. 3. The ancestral form of spicule in the Calcarea was a simple monaxon, placed tangentially and completely embedded in the body-wall, lying between two adjacent pores. 4. From this ancestral spicule the forms of spicule now occurring in the Calcarea arose as follows: (a) the primitive monaxon acquired a distal portion projecting from the surface, as in the existing primary monaxons; (b) groups consisting each of three primitive monaxons became united by their contiguous ends to form a single triradiate system; (c) to some of the triradiate systems thus formed a fourth ray was added, secreted by the pore-cell, giving rise to the quadriradiate system ; (d) some of the triradiate systems, by loss of one ray and placing of the other two in a straight line, or by loss of two rays, perhaps became modified into secondary monaxon spicules. 5. The power of secreting a monaxon sclerite was primitively possessed by every cell of the dermal layer, and this condition appears to be retained in Leucosolenia. In Clathrina, on the other hand, all the skeletogenous cells migrate inwards from the dermal epithelium, and form a connective-tissue layer distinct in function from the contractile, undifferentiated dermal epithelium. In Leucosolenia also the actinoblasts of the triradiate systems form a deeper layer, but the dermal epithelium secretes primary monaxons--at least in the young form--and is non-contractile. 6. The forms of the spicules are the result of adaptation to the requirements of the sponge as a whole, produced by the action of natural selection upon variation in every direction.

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