41. Solid Geometry - Definition Of Solid Geometry In Encyclopedia In mathematics, solid geometry was the traditional name for the geometry ofthreedimensional Euclidean space — for practical purposes the kind of space we http://encyclopedia.laborlawtalk.com/Solid_geometry

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law In mathematics solid geometry was the traditional name for the geometry of three-dimensional Euclidean space space we live in. It was studied as a sequel to plane geometry Stereometry deals with the measurements of volumes of various solid figures cylinder , circular cone , truncated cone, sphere prisms blades wine casks The Pythagoreans had dealt with the sphere and regular solids , but the pyramid , prism, cone and cylinder were but little known until the Platonists took them in hand. Eudoxus established their mensuration, proving the pyramid and cone to have one-third the content of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volumes of spheres are as the cubes of their radii See also: Archimedes Demiurge Johannes Kepler planimetry ... Plato's Timaeus ...partly from the 1911 Encyclopaedia Britannica Basic topics of solid geometry Basic topics are: incidence of planes and lines dihedral angle and solid angle the cube cuboid parallelopiped the tetrahedron and other pyramids prisms octahedron dodecahedron ... cones and cylinders the sphere other quadrics spheroid ellipsoid paraboloid and hyperboloids Other topics More advanced are the study of projective geometry of three dimensions leading to proof of Desargues' theorem by using an extra dimension further polyhedra descriptive geometry Analytic geometry and vector techniques have a major impact by allowing the systematic use of

42. Solid Geometry -- Facts, Info, And Encyclopedia Article solid geometry. Categories Euclidean geometry In (A science (or group ofrelated sciences) dealing Basic topics of solid geometry. Basic topics are http://www.absoluteastronomy.com/encyclopedia/s/so/solid_geometry.htm

Solid geometry [Categories: Euclidean geometry] In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics solid geometry was the traditional name for the (The pure mathematics of points and lines and curves and surfaces) geometry of three-dimensional (A space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional) Euclidean space (An area reserved for some particular purpose) space we live in. It was studied as a sequel to (The geometry of 2-dimensional figures) plane geometry Stereometry deals with the (The act or process of measuring) measurement s of (The magnitude of sound (usually in a specified direction)) volume s of various (A three-dimensional shape) solid figure s: (A surface generated by rotating a parallel line around a fixed line) cylinder , circular (Cone-shaped mass of ovule- or spore-bearing scales or bracts) cone , truncated cone, (A particular environment or walk of life) sphere (Optical device having a triangular shape and made of glass or quartz; used to deviate a beam or invert an image) prism s

43. 3.4 Constructive Solid Geometry With The Stencil Buffer 3.4 Constructive solid geometry with the Stencil Buffer. http://www.opengl.org/resources/tutorials/advanced/advanced97/notes/node11.html

Next: 4 Geometry and Transformations Up: 3 Modelling Previous: 3.3 Capping Clipped Solids 3.4 Constructive Solid Geometry with the Stencil Buffer Before continuing, the it may help for the reader to be familiar with the concepts presented in Section Constructive solid geometry (CSG) models are constructed through the intersection ( ), union ( ), and subtraction ( ) of solid objects, some of which may be CSG objects themselves[ ]. The tree formed by the binary CSG operators and their operands is known as the CSG tree. Figure shows an example of a CSG tree and the resulting model. The representation used in CSG for solid objects varies, but we will consider a solid to be a collection of polygons forming a closed volume. ``Solid'', ``primitive'', and ``object'' are used here to mean the same thing. CSG objects have traditionally been rendered through the use of ray-casting, which is slow, or through the construction of a boundary representation (B-rep). B-reps vary in construction, but are generally defined as a set of polygons that form the surface of the result of the CSG tree. One method of generating a B-rep is to take the polygons forming the surface of each primitive and trimming away the polygons (or portions thereof) that don't satisfy the CSG operations. A B-rep models are typically generated once and then manipulated as a static model because they are slow to generate. Drawing a CSG model using stencil usually means drawing more polygons than a B-rep would contain for the same model. Enabling stencil also may reduce performance. Nonetheless, some portions of a CSG tree may be interactively manipulated using stencil if the remainder of the tree is cached as a B-rep.

44. 8.4 Constructive Solid Geometry With The Stencil Buffer 8.4 Constructive solid geometry with the Stencil Buffer. http://www.opengl.org/resources/tutorials/advanced/advanced96/node33.html

Next: 8.4.1 Union Up: 8 Using the Stencil Previous: 8.3 Finding Depth Complexity 8.4 Constructive Solid Geometry with the Stencil Buffer Constructive Solid Geometry (CSG) is the process of building complex solids from logical combinations of simpler ones [ ]. Basic logical operations, such as union, intersection, and inverse are used as relationship operators between pairs of three dimensional solids. In these algorithms, the solids are assumed to be defined by closed surfaces composed of polygons. The polygons composing the primitives are defined in a consistent way so that the front face of each polygon faces out from the surface. CSG is used as a method to create complex solids as part of a computer aided mechanical design process. Constructive solid geometry can be created using the stencil buffer functionality. It can be thought of as a ray-casting technique, counting the intersections the pixels of an solid's surface makes with other surfaces as it's projected onto the screen. The techniques shown here are image space techniques and are limited to visualizing CSG solids. They don't produce a list of vertices that could be used for further processing. They also tend to leave the depth buffer with values that don't match the visible surfaces. As a result, CSG solids often can't be used as components of a complex depth-buffered scene. Creating CSG solids is a multipass technique. Two objects are chosen, and the desired boolean relationship between the two is discovered. The basic CSG stencil operation is the ability to find the part of one solid's surface that exists in the volume defined by another solid. Pieces of each solids surface is progressively rendered, until the entire solid is completed. The solids created with CSG are rendered as pieces of the surfaces of component solids; no new surfaces are created.

45. Computer Graphics : Surface Modeling : 15 / 30 : Constructive Solid-Geometry Met The method of Constructive solid geometry arose from the observation that manyindustrial components derive from combinations of various simple geometric http://escience.anu.edu.au/lecture/cg/surfaceModeling/CSG.en.html

Contact Links Previous : 14 / 30 : Patch Representation vs. Polygon Mesh Up ... Next : 16 / 30 : Patch Representation vs. Polygon Mesh Constructive Solid-Geometry Methods (CSG) Constructive models represent a solid as a combination of primitive solids. (CSG) This modeling technique combine the volumes occupied by overlapping 3D shapes using set boolean operations. This creates a new volume by applying the union, intersection, or difference operation to two volumes. Each primitive is defined as a combination of half-spaces. Typical standard primitives are: cone, cylinder, sphere, torus, block, closed spline surface, right angular wedge. swept solids (a revolution or linear sweep of a planar face which may contain holes). The method of Constructive Solid Geometry arose from the observation that many industrial components derive from combinations of various simple geometric shapes such as spheres, cones, cylinders and rectangular solids. In fact the whole design process often started with a simple block which might have simple shapes cut out of it, perhaps other shapes added on etc. in producing the final design. For example consider the simple solid below: This simple component could be produced by gluing two rectangular blocks together and then drilling the hole. Or in CSG terms the the

46. Journal Of Online Mathematics And Its Applications | Search Results 5.3 solid geometry, 5.4 Analytic Geometry, 5.5 Projective Geometry All documents on 5.3 solid geometry. Page 1 of 1. Featured Results http://joma.org/mathDL/4/?pa=content&sa=browseNode&subjectId=55

47. Science Search > Solid Geometry Current location Math Geometry solid geometry. 1. The Geometry of the SphereThis material was the text for part of the Advanced Mathematics course in http://www.science-search.org/index/Math/Geometry/Solid_Geometry/

Search for: Current Category Everything What's new Top Searches Statistics Science News ... Home Current location: Math Geometry > Solid Geometry The Geometry of the Sphere This material was the text for part of the Advanced Mathematics course in the High School Teachers Program at the IAS/Park City Mathematics Institute at the Institute for http://math.rice.edu/~pcmi/sphere/ detailed information Rating: [6.00] Votes: [1903] Geometry Machine 3D virtual reality geometrical images of vectors, lines and planes. Needs a VRML viewer or plugin such as live3D. http://www.scit.wlv.ac.uk/~cm1993/geometry/INTRO.HTM detailed information Rating: [5.00] Votes: [2239] About Us Contact Us Site Guide Add Your Site ... Add Science Search to Your Web Site The Science Search Directory is based on DMOZ. Science Search has modified and enhanced this data. Submitting to DMOZ: Help build the largest human-edited directory on the web. Submit a Site Open Directory Project Become an Editor Science Search's Sponsors: Goom NL

48. Constructive Solid Geometry Constructive solid geometry. Constructive solid geometry. In CSG, solids aredescribed as combinations of simple primitives or other solids in a series http://cadd.web.cern.ch/cadd/cad_geant_int/thesis/node21.html

[an error occurred while processing this directive] Next: Boundary models Up: Constructive models Previous: Half-space models Constructive solid geometry In CSG, solids are described as combinations of simple primitives or other solids in a series of Boolean operations, i.e. a CSG model is constructed using a so called building block approach. A user operates only on parameterised instances of solid primitives and Boolean operations on them. Each primitive is defined as a combination of half-spaces. The construction of CSG models is exactly the same as in half-space models, except that only finite sides of the half-spaces are included. Using Boolean operations on finite solid primitives, the resulting solid is always guaranteed to be finite as well. Any primitive for which all the operation procedures can be written can be included into a CSG modeller. A normal addition to a CSG modeller is sweeping primitives. For example, a translational sweeping primitive is defined by a bounded face and the sweeping vector. Typical standard primitives are cone, cylinder, sphere, torus, block and right angular wedge. Operations are union, intersection and difference. Other CSG primitives are swept solids, i.e. a revolution or linear sweep of a planar face which may contain holes. Composite CSG solids may also be subjected to Boolean operations. Using a Boolean operation, a new solid is constructed from two intersecting solids. This technique is common in mechanical engineering since it gives a precise, analytic description of the model. The convenience of a CSG modeller depends on the set of primitives and Boolean operations available. A simple CSG model with a Boolean subtraction is shown in Figure

49. Exploring Solid Geometry Exploring solid geometry Students enjoy exploring the world of solid geometrywith these three compelling videos. Animation, graphics, and live action http://www.mathteacherstore.com/middle/midlvido/category1/titles/eesg/eesgmain.h

Home Middle School Math Videos Math Videos Exploring Solid Geometry VIDEOS Grades 8-12 30 min. each To add to a secure Shopping Cart, click on the Buy Button. Exploring Solid Geometry PART 1 on-line Exploring Solid Geometry PART 2 on-line Exploring Solid Geometry PART 3 on-line Exploring Solid Geometry SET OF 3 on-line To buy by PHONE, FAX, or with PURCHASE ORDER, print out this page and complete the Order Form. Introduce the principles of solid geometry in a way that captivates and motivates your students. The fascinating journey that started with the World of Geometry introductory video and continued with the Exploring Plane Geometry video set concludes with Exploring Solid Geometry This 3-video set provides a comprehensive exploration of solid geometry. Topics include basic properties and figures, volume, surface area, polyhedral angles, and non-Euclidean geometry. Exploring Solid Geometry illustrates numerous real-world examples of geometric principles in action through computer animation, graphics, and live action sequences. Explanations of concepts are clear and thorough. These videos make a wonderful classroom supplement or an excellent resource for use at home.

50. MathFiction: Solid Geometry (Ian McEwan) I wanted to tell you about the short story solid geometry by Ian McEwan solid geometry is about a man discovering mathematical diagrams in a diary http://math.cofc.edu/faculty/kasman/MATHFICT/mfview.php?callnumber=mf391

51. The Cornell Library Historical Mathematics Monographs Document name solid geometry. Go to. page NA Production Note, page NA Titlepage, page iii Prefatory matter, page v Table of contents http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=03980001&se

52. Constructive Solid Geometry Constructive solid geometry (CSG) essentially consists of Boolean set operationson closed primitives in 3D space. The three CSG operations are union, http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/DODGSON/p4c.html

Constructive Solid Geometry Dr Neil Dodgson University of Cambridge Computer Laboratory Constructive solid geometry (CSG) essentially consists of Boolean set operations on closed primitives in 3D space. The three CSG operations are union intersection and difference CSG is covered in FvDFH sections 12.7 and 15.10.3. The following example illustrates the three CSG operations in use on simple three dimensional primitives. The two primitives: a sphere and a box. The union of the two primitives. The intersection of the two primitives. The difference of the two primitives: box minus sphere. The following example, based on FvDFH Plate III.2, shows an object for which CSG is (probably) the only sensible modelling technique. The object rendered in the right-hand image is constructed from the primitives shown in the left-hand image. It is mostly made out of cylinders. At right, I tried the " Godel, Escher, Bach " treatment on my initials. Unfortunately the letters N, A, and D are not as amenable to this as the letters G, E, and B: notice that the shadow of the N has a slight curve at its top right, owing to the N's intersection with the curve on the D and the slope on the A. Various other arrangements of the three letters were tried, all of which gave more noticable artefacts than this. Each of the letters is a CSG object (the D, for example, is constructed from cylinders and boxes). The final effect is produced simply by intersecting the three letters. Exercises Work out how to represent a Lego technic brick as a CSG object. You may assume that you have box and cylinder primitives.

53. CVonline: Constructive Solid Geometry/Set-Theoretic Modeling Settheoretic or Constructive-solid-geometry Modellers (Adrian Bowyer) Constructivesolid geometry (PDF, Section 9.4.3, page 9) (Computer Vision, http://homepages.inf.ed.ac.uk/cgi/rbf/CVONLINE/entries.pl?TAG738

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55. Solid Geometry (from Euclidean Geometry) --  Encyclopædia Britannica solid geometry (from Euclidean geometry) The most important difference betweenplane and solid Euclidean geometry is that human beings can look at the plane http://www.britannica.com/eb/article-235570

Home Browse Newsletters Store ... Subscribe Already a member? Log in Content Related to this Topic This Article's Table of Contents Introduction Fundamentals Plane geometry Congruence of triangles Similarity of triangles Areas Pythagorean theorem ... Conic sections and geometric art Solid geometry Volume Regular solids Additional Reading Print this Table of Contents Shopping Price: USD $1495 Revised, updated, and still unrivaled. The Official Scrabble Players Dictionary (Hardcover) Price: USD $15.95 The Scrabble player's bible on sale! Save 30%. Merriam-Webster's Collegiate Dictionary Price: USD $19.95 Save big on America's best-selling dictionary. Discounted 38%! More Britannica products Euclidean geometry Page 10 of 13 Solid geometry Euclidean geometry... (75 of 2526 words) var mm = [["Jan.","January"],["Feb.","February"],["Mar.","March"],["Apr.","April"],["May","May"],["June","June"],["July","July"],["Aug.","August"],["Sept.","September"],["Oct.","October"],["Nov.","November"],["Dec.","December"]]; To cite this page: MLA style: "Euclidean geometry."

56. Constructive Solid Geometry (CSG) Constructive solid geometry (CSG) Constructive solid geometry (CSG). It isthe concept of CSG to provide solid 3D primitives which are described a set http://www.ipf.tuwien.ac.at/fr/buildings/diss/node38.html

Next: Solid modelling and building Up: Modelling of man-made solid Previous: Primitive instancing Constructive Solid Geometry (CSG) It is the concept of CSG to provide solid 3D primitives which are described a set of parameters reflecting the object dimensions (Figure ). The CSG primitives are simple objects such as cubes, boxes, tetrahedrons or quadratic pyramids. They are considered to be bounded point sets in 3D space, and they can easily be combined using Boolean set operations ( union intersection and difference ) in order to represent more complex objects consisting of more than one primitive (cf. section ). In theory, the (bounded) primitives themselves can be considered to be the intersections of half spaces containing all points P for which the inequality is fulfilled, where f P ) is a characteristic function of the point P and f P ) = describes the bounding surface of the point set, e.g. a plane in 3D space ( halfspace models The most natural way to represent a CSG model is the CSG tree which can be defined as follows: where is an instance of one of the primitives of the primitive data base

57. Solid Geometry - YourDictionary.com - American Heritage Dictionary Search Mamma.com for solid geometry . TYPE IN YOUR WORD CLICK GO! solid geometry n. The branch of mathematics that deals with threedimensional http://www.yourdictionary.com/ahd/s/s0548800.html

Search Mamma.com for "solid geometry" Search: Normal Definitions Short defs (Pronunciation Key) solid geometry n. The branch of mathematics that deals with three-dimensional figures and surfaces. Back to Search Back The American Heritage Dictionary of the English Language, Fourth Edition

58. Royal Horticultural Society - RHS Flower Show At Tatton Park 2004: Andrew Loudon solid geometry is a garden inspired by the work of sculptor and designer IsamuNoguchi and his use of natural materials and geometric forms. http://www.rhs.org.uk/tatton/2004/exhibitors/back2back_gardens/andrewloudon.asp

RHS Home RHS Events Visitor Information Tickets ... SHOW GARDENS Back-to-Back Gardens Solid Geometry To view a hi-res version of the image click on the image Designers: Sponsor: Coniston Copper Mine Cottages Contractor: Solid Geometry is a garden inspired by the work of sculptor and designer Isamu Noguchi and his use of natural materials and geometric forms. In 1968 Noguchi wrote: "An empty space has no visual dimension or significance. Scale and meaning enter when some thoughtful object or line is introduced. This is why sculptures, or rather sculptural objects, create space. Their function is illusionist". Solid Geometry has been designed with this in mind, using geometric patterns and forms in stone and wood, combined with lush herbaceous planting, to demonstrate how natural materials can be used to create a small garden with a contemporary look. Accessibility Help Contact Us SEARCH: Registered charity Number 222879 back to top

59. Visual Solid Geometry Software Visual solid geometry helps students develop an intuitive understanding ofgeometric properties, relationships, and formulas, paving the way to success and http://www.mathematicssoftware.com/mathematics-software-html/solid-geometry-soft

60. POV-Ray: Documentation: 2.4.6 Constructive Solid Geometry 2.4.6 Constructive solid geometry. In addition to all of the primitive shapesPOVRay supports, you can also combine multiple simple shapes into complex http://www.povray.org/documentation/view/3.6.1/302/

Download Documentation Resources Community ... Documentation : 2.4.6 Constructive Solid Geometry POV-Ray 3.6 Documentation Online View 2.4.5 Parametric Object 2.4.6.1 Inside and Outside Constructive Solid Geometry In addition to all of the primitive shapes POV-Ray supports, you can also combine multiple simple shapes into complex shapes using Constructive Solid Geometry (CSG). There are four basic types of CSG operations: union, intersection, difference, and merge. CSG objects can be composed of primitives or other CSG objects to create more, and more complex shapes. 2.4.5 Parametric Object 2.4.6.1 Inside and Outside

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