Geometric Computing Based on Computerized Descriptive Geometric

Computer-aided Design （CAD）, video games and other computer graphic related technology evolves substantial processing to geometric elements. A novel geometric computing method is proposed with the integration of descriptive geometry, math and computer algorithm. Firstly, geometric elements in general position are transformed to a special position in new coordinate system. Then a 3D problem is projected to new coordinate planes. Finally, according to 2D/3D correspondence principle in descriptive geometry, the solution is constructed computerized drawing process with ruler and compasses. In order to make this method a regular operation, a two-level pattern is established. Basic Layer is a set algebraic packaged function including about ten Primary Geometric Functions （PGF） and one projection transformation. In Application Layer, a proper coordinate is established and a sequence of PGFs is sought for to get the final results. Examples illustrate the advantages of our method on dimension reduction, regulatory and visual computing and robustness.

New three-dimensional guidance law for BTT missiles based on differential geometry and Lie-group

A new non-decoupling three-dimensional guidance law is proposed for bank-to-turn （BTT） missiles with the motion coupling problem. In this method, the different geometry is taken for theoretically modeling on B-IT missiles＇ motion within the threedimensional style without information loss, and meanwhile, Liegroup is utilized to describe the line-of-sight （LOS） azimuth when the terminal angular constraints are considered. Under these cir- cumstances, a guidance kinematics model is established based on differential geometry. Then, corresponding to no terminal angular constraint and terminal angular constraints, guidance laws are re- spectively designed by using proportional control and generalized proportional-derivative （PD） control in SO（3） group. Eventually, simulation results validate that this developed method can effectively avoid the complexity of pure Lie-group method and the information loss of the traditional decoupling method as well.

Significance of Traditional Descriptive Geometry in Teaching and Learning Graphic Science

Graphic science is the subject which teaches geometry and graphics, and is taught in early undergraduate curricula at many Japanese universities as a liberal arts subject or as a basic subject for design and drawing. In traditional graphic science courses, de-scriptive geometry based on hand drawings was taught. However, in recent years, there continues to be a rapid spread in the use of 3D-CAD in the field of engineering design and drawing, and there is also increasing use of CG in many fields such as for visualiza-tion of computer simulation results in science, and for image display in the movie and game entertainment fields. So there is a need for graphic presentation education including the competence in the use of 3D-CAD/CG, or ＂graphics literacy （or visual literacy） education＂ for a wide range of students. In order to realize graphics literacy education, from 2007, a new graphic science curriculum has been started at the College of Arts and Sciences of the University of Tokyo. The main part of the curriculum consists of Graphic Science I and Graphic Science II. With Graphic Science I, as before, traditional descriptive geometry is taught with hand drawing as the base. With Graphic Science II, commercial graphic processing software can be experienced. In this course, by introducing geo- metric problems as examples and assignments, it is designed to mutually complement with descriptive geometry education （Graphic Science I）. With the spread of 3D-CAD/CG, some people say that there is no longer any need for descriptive geometry, but for the following reasons, it has been decided to teach descriptive geometry. 1） Traditional descriptive geometry is an excellent method in teaching and learning geometry of projection and of three-dimensional objects, and concepts and/or procedures in descriptive geome-try can be applied in solving geometric design problems by the use of 3D-CAD/CG. 2） Even in this age of 3D-CAD/CG, hand draw-ing is still being used （especially for free-hand sketches）. 3） Hand drawing is an effective method of developing spatial ability of students. However, with the spread of 3D-CAD/CG, the descriptive geometry techniques in analyzing shapes and forms of three-dimensional objects are now loosing their earlier practical importance. So emphasis is not being placed on the education of practical techniques, but is being placed on teaching the theory behind the techniques, i.e., geometry of projection and of three-dimensional objects. This paper reports specific examples of classes in order to describe the importance of descriptive geometry education, and the need to switch from education focused on techniques to education on the theory behind the techniques.

Recent Applications of Descriptive Geometry to Mechanical Engineering

This is a brief summary of a paper presented at the 5th Japan-China Joint Conference on Graphics Education in Osaka 2001.

Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber

Gaseous detonation propagating in a toroidal chamber was numerically studied for hydrogen/oxygen/nitrogen mixtures. The numerical method used is based on the three-dimensional Euler equations with detailed finiterate chemistry. The results show that the calculated streak picture is in qualitative agreement with the picture recorded by a high speed streak camera from published literature. The three-dimensional flow field induced by a continuously rotating detonation was visualized and distinctive features of the rotating detonations were clearly depicted. Owing to the unconfined character of detonation wavelet, a deficit of detonation parameters was observed. Due to the effects of wall geometries, the strength of the outside detonation front is stronger than that of the inside portion. The detonation thus propagates with a constant circular velocity. Numerical simulation also shows three-dimensional rotating detonation structures, which display specific feature of the detonation- shock combined wave. Discrete burning gas pockets are formed due to instability of the discontinuity. It is believed that the present study could give an insight into the interest- ing properties of the continuously rotating detonation, and is thus beneficial to the design of continuous detonation propulsion systems.

Fractal Geometry:Axioms,Fractal Derivative and Its Geometrical Meaning

Physics success is largely determined by using mathematics.Physics often themselves create the necessary mathematical apparatus.This article shows how you can construct a fractal calculus-mathematics of fractal geometry.In modem scientific literature often write from a firm that"there is no strict definition of fractals",to the more moderate that"objects in a certain sense,fractal and similar."We show that fractal geometry is a strict mathematical theory,defined by their axioms.This methodology allows the geometry of axiomatised naturally define fractal integrals and differentials.Consistent application on your input below the axiom gives the opportunity to develop effective methods of measurement of fractal dimension,geometri-cal interpretation of fractal derivative gain and open dual symmetry.

Study on CAD Data Interface to Geometry Description System

The basic way and method to apply the equipment s CAD data to a geometry description system(GDS) are presented,and its interface to GDS is set up.The basic function,flow and implementation technique of the interface are analyzed.Special computer software is programmed,and an application example is given also.The research results indicate that this interface can assist to derive desired data from the CAD data,and provide powerful technical support for the development of a practical data transferring interface.

Spectroscopic Studies of a Three-dimensional, Five-coordinated Copper(Ⅱ) Complex via Hydrogen Bonds: [Cu(PDA)(H_2O)_2](H_2PDA=Pyridine-2,6-dicarboxylic Acid)

A new copper(Ⅱ) complex [Cu(PDA)(H 2O) 2] was synthesized and its structure was determined. Cu(Ⅱ) is five coordinated in a tetragonal pyramid geometry. The two coordinating water molecules are different and the two Cu-O bond lengths differ by nearly 0.02 nm. The whole crystal is linked to form a three dimensional network by means of hydrogen bonds. The X band ESR spectrum shows three different g tensors with a well resolved hyperfine structure in the g z signal, giving the ESR parameters g x=2 05, g y =2 065 and g z =2 29. The covalency of the coordinate bonds and the deviation from tetragonal pyramid geometry for the complex are discussed based on the ESR spectra.

Method for detector description conversion from DD4hep to Filmbox

DD4hep serves as a generic detector description toolkit recommended for offline software development in next-generation high-energy physics(HEP)experiments.Conversely,Filmbox(FBX)stands out as a widely used 3D modeling file format within the 3D software industry.In this paper,we introduce a novel method that can automatically convert complex HEP detector geometries from DD4hep description into 3D models in the FBX format.The feasibility of this method was dem-onstrated by its application to the DD4hep description of the Compact Linear Collider detector and several sub-detectors of the super Tau-Charm facility and circular electron-positron collider experiments.The automatic DD4hep–FBX detector conversion interface provides convenience for further development of applications,such as detector design,simulation,visualization,data monitoring,and outreach,in HEP experiments.

A DIRECTED GRAPH ALGORITHM OF VARIATIONAL GEOMETRY BASED ON GEOMETRIC REASONING

The undirected graph to express engineering drawings is discussed .The principle to re-solve and reason the undirected graph is presented, and the algorithm finally transforms theundirected graph into the resolvable directed graph. Therefore,a rapid and simple way is suppliedfor variational design. A prototype of this algorithm has been implemented, and some examplesare given.

The Counterintuitive Increase of Information Due to Extra Spacetime Dimensions of a Black Hole and Dvoretzky’s Theorem

As per Hawking and Bekenstein’s work on black holes, information resides on the surface and there is a limit on it amounting to a bit for every Planck area. It would seem therefore that extra dimensions would logically lead to a hyper-surface for a black hole and consequently a reduction of the corresponding information density due to the dilution effect of these additional dimensions. The present paper argues that the counterintuitive opposite of the above is what should be expected. This surprising result is a consequence of a well known theorem on measure concentration due to I. Dvoretzky.

Exact Geometric Relationships, Symmetry Breaking and Structural Stability for Single-Walled Carbon Nanotubes

We pioneered a study about how the geometric relationship of single-walled carbon nanotubes(SWCNT) is influenced by curvature factor and non-planar geometry factor in cylindrical coordinate system based on the assumption of complete symmetry. The bond length and angle of every carbon-carbon bonds are determined by using the principle of the minimum energy. The results of the paper include(1) From the calculation result, the symmetry breaking appears for chiral carbon nanotubes, while the part symmetry appears for achiral carbon nanotubes with increasing curvature.(2) The synergistic effect of bond lengths and bond angles is first found.(3) We conclude that the influence of non-planar geometry factor can be completely ignored on bond lengths and bond angles when the curvature parameter has been included in the model.(4)The two fractal dimensions are given from the nanoscale to the macroscale for zigzag topology and armchair topology respectively. Fractal dimensions of SWCNT show special characteristics, varying with the length of SWCNT until the lengths approach infinity. The close and inevitable correlations among curvature, symmetry breaking and stability of SWCNTs can be summed up as: the increase of curvature causes symmetry breaking,and such symmetry breaking will further reduce the structural stability.

Studying Relationships between the Fractal Dimension of the Drainage Basins and Some of Their Geomorphological Characteristics

Complex nonlinear dynamic systems are ubiquitous in the landscapes and phenomena studied by earth sciences in general and by geomorphology in particular. Many natural landscape features have an aspect such as fractals. In the many geomorphologic phenomena such as river networks and coast lines this is visible. In recent years fractal geometry offers as an option for modeling river geometry and physical processes of rivers. The fractal dimension is a statistical quantity that indicates how a fractal scales with the shrink, the space occupied. This theory has the mathematical basis but also applied in geomorphology and also shown great success. Physical behavior of many natural processes as well as using fractal geometry is predictable relations. Behavior of complex natural phenomena as rivers has always been of interest to researchers. In this study using data basic maps, drainage networks map and Digital Elevation Model of the ground was prepared. Then applying the rules Horton-Strahler river network, fractal dimensions were calculated to examine the relationship between fractal dimension and some rivers geomorphic features were investigated. Results showed fractal dimension of watersheds have meaningful relations with factors such as shape form, area, bifurcation ratio and length ratio in the watersheds.

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

NONLINEAR STRAIN COMPONENTS OF GENERAL SHELLS WITH INITIAL GEOMETRIC IMPERFECTIONS

On the basis of nonlinear strain component formulations of three-dimensional continuum, this paper has derived the nonlinear strain component formulations of shells with initial geometric imperfections. The derivation is not confined to a special shell, therefore they possess general properties. These formulations provide the theoretical basis of the strain analysis for geometric nonlinear problems of shells with initial geometric imperfections.

Fractal Description of the Shearing-Surface of Tools

In this paper, the basic methods are introduced to calculate the fractaldimensions of the shearing surface of some tools. The fractal dimension of the shearing surface ofexperimental sampling is obtained and the fractal characteristics are also discussed. We can applythe fractal method to identify types of tools used by burglars and to do the job of individualrecognition. New theories and methods are provided to measure and process the shearing surfaceprofile of tools.

Development of PC-based Electronic Teaching and Learning Materials for De- scriptive Geometry

Descriptive geometry is very important and recognized as a basic skill and knowledge for mechanical engineering student. In this study, PC-based electronic teaching/learning materials for descriptive geometry are created using Flash, which is a typical animation creator. Furthermore, several axonometric representations, created by 3D-CAD, SolidWorks, for 3D objects are auxiliary materials to promote understanding of descriptive geometry. The axonometric representations in 3D-CAD are also dynamic, in other words, a viewpoint can be moved free. The movement of 3D model in a PC monitor can be recorded using a normal function of SolidWorks and replayed by typical animation software. The developed materials are excellent at accuracy of drawing, repeatability of self-study and visual attraction in comparison to oral presentation using still image and inaccurate drawing on a textbook or blackboard in a classroom. Actually, questionnaire survey results present favorable impressions from student-users, although they point out the further improvement in the replaying speed. The replaying speed can be controlled easily by using a normal function of Flash. In addition, usual playback software for animation has functions of pause and replay on demand and, thus, it is not contro-versial.

AN ALGORITHM FOR LINE GENERATION BASED ON SEGMENT CODE IN PAN-EUCLIDEAN GEOMETRY

In Comparison with the traditional point-by-point line generation method,the method we present is based on segment code in Pan-Euclidean geometric space and is quite different in re- spect of running speed and theoretical basis.This paper presents an approach of using segment code to draw straight lines and shows the characteristics of a digital line.It is a newly proposed al- gorithm applicable in CAD.

立体几何展开方法在毛绒玩具开版中的应用

为形成统一方便、立体效果显著的毛绒玩具开版方法,结合几何体展开方法与服装结构设计方法,实现从二维图纸向三维造型的转换。首先通过市场调研,发现年轻消费群体更加偏爱动物形象毛绒玩具,且十分重视毛绒玩具的造型;其次结合立体几何展开原理与部分服装结构设计方法,将毛绒玩具各部分概括为立体几何图形,进行局部造型展开分析和开版制作;最后以一款毛绒熊玩具为例,主材料为针织绒面布料,实践验证运用立体几何展开方法便于二维纸样到三维实物的转换。