Chapter 08 Multiview Drawings Study Guide
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An Analytical Introduction to Descriptive Geometry
An analytical introduction to Descriptive Geometry Adrian B. Biran, Technion { Faculty of Mechanical Engineering Ruben Lopez-Pulido, CEHINAV, Polytechnic University of Madrid, Model Basin, and Spanish Association of Naval Architects Avraham Banai Technion { Faculty of Mathematics Prepared for Elsevier (Butterworth-Heinemann), Oxford, UK Samples - August 2005 Contents Preface x 1 Geometric constructions 1 1.1 Introduction . 2 1.2 Drawing instruments . 2 1.3 A few geometric constructions . 2 1.3.1 Drawing parallels . 2 1.3.2 Dividing a segment into two . 2 1.3.3 Bisecting an angle . 2 1.3.4 Raising a perpendicular on a given segment . 2 1.3.5 Drawing a triangle given its three sides . 2 1.4 The intersection of two lines . 2 1.4.1 Introduction . 2 1.4.2 Examples from practice . 2 1.4.3 Situations to avoid . 2 1.5 Manual drawing and computer-aided drawing . 2 i ii CONTENTS 1.6 Exercises . 2 Notations 1 2 Introduction 3 2.1 How we see an object . 3 2.2 Central projection . 4 2.2.1 De¯nition . 4 2.2.2 Properties . 5 2.2.3 Vanishing points . 17 2.2.4 Conclusions . 20 2.3 Parallel projection . 23 2.3.1 De¯nition . 23 2.3.2 A few properties . 24 2.3.3 The concept of scale . 25 2.4 Orthographic projection . 27 2.4.1 De¯nition . 27 2.4.2 The projection of a right angle . 28 2.5 The two-sheet method of Monge . 36 2.6 Summary . 39 2.7 Examples . 43 2.8 Exercises . -
Axonometric (Isometric Projection)
AXONOMETRIC (ISOMETRIC PROJECTION) Representation Systems Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Elevation Profile Plan Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Isometric Into the ISOMETRIC PERSPECTIVE, the isometric axes form a 120º angle between one another Front View Technical Drawing Name and Surname: Axonometric System Draw the AXONOMETRIC PERSPECTIVE of these pieces. -
Viewing in 3D
Viewing in 3D Viewing in 3D Foley & Van Dam, Chapter 6 • Transformation Pipeline • Viewing Plane • Viewing Coordinate System • Projections • Orthographic • Perspective OpenGL Transformation Pipeline Viewing Coordinate System Homogeneous coordinates in World System zw world yw ModelViewModelView Matrix Matrix xw Tractor Viewing System Viewer Coordinates System ProjectionProjection Matrix Matrix Clip y Coordinates v Front- xv ClippingClipping Wheel System P0 zv ViewportViewport Transformation Transformation ne pla ing Window Coordinates View Specifying the Viewing Coordinates Specifying the Viewing Coordinates • Viewing Coordinates system, [xv, yv, zv], describes 3D objects with respect to a viewer zw y v P v xv •A viewing plane (projection plane) is set up N P0 zv perpendicular to zv and aligned with (xv,yv) yw xw ne pla ing • In order to specify a viewing plane we have View to specify: •P0=(x0,y0,z0) is the point where a camera is located •a vector N normal to the plane • P is a point to look-at •N=(P-P)/|P -P| is the view-plane normal vector •a viewing-up vector V 0 0 •V=zw is the view up vector, whose projection onto • a point on the viewing plane the view-plane is directed up Viewing Coordinate System Projections V u N z N ; x ; y z u x • Viewing 3D objects on a 2D display requires a v v V u N v v v mapping from 3D to 2D • The transformation M, from world-coordinate into viewing-coordinates is: • A projection is formed by the intersection of certain lines (projectors) with the view plane 1 2 3 ª x v x v x v 0 º ª 1 0 0 x 0 º « » « -
'The "Perfyt Scyens" of the Map; a Study of the Meaning and Interpretation of Local Maps in Early Tudor England 1509-1547'
'The "Perfyt Scyens" of the Map; a Study of the Meaning and Interpretation of Local Maps in Early Tudor England 1509-1547' Lewis John Kaye Roberts Queen Mary, University of London 70,056 words (excluding bibliography) This work was supported by the AHRC (BGP award reference: 0673) A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy, University of London 1 Statement of Originality. I, Lewis Roberts, confirm that the research included within this thesis is my own work or that where it has been carried out in collaboration with, or supported by others, that this is duly acknowledged below and my contribution indicated. Previously published material is also acknowledged below. I attest that I have exercised reasonable care to ensure that the work is original, and does not to the best of my knowledge break any UK law, infringe any third party’s copyright or other Intellectual Property Right, or contain any confidential material. I accept that the College has the right to use plagiarism detection software to check the electronic version of the thesis. I confirm that this thesis has not been previously submitted for the award of a degree by this or any other university. The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author. Signature: Date: 16th January 2014 2 Abstract. This thesis begins by examining an unexplored contextual background for sixteenth century local maps. It argues that the architectural drawing techniques developed by master masons in the late twelfth century continued to be taught to the King’s masons well into the sixteenth, and that these drawing techniques lie behind the innovations in sixteenth century topographical mapping. -
Understanding Projection Systems
Understanding Projection Systems Understanding Projection Systems A Point: A point has no dimensions, a theoretical location that has neither length, width nor height. A point shows an exact location in space. It is important to understand that a point is not an object, but a position. We represent a point by placing a dot with a pencil. A Line: A line is a geometric object that has length and direction but no thickness. A line may be straight or curved. A line may be infinitely long. If a line has a definite length it is called a line segment or curve segment. A straight line is the shortest distance between two points which is known as the true length of the line. A line is named using letters to indicate its endpoints. B B A A AB - Straight Line Segment AB – Curved Line Segment A line may be seen as the locus of a point as it travels between two points. A B A line can graphically represent the intersection of two surfaces, the edge view of a surface, or the limiting element of a surface. B A Plane: A plane is a flat surface which is infinitely large with zero thickness. Just as a point generates a line, a line can generate a plane. A A portion of a plane is referred to as a lamina. A Plane may be defined in a number of different ways. - 1 - Understanding Projection Systems A plane may be defined by; (i) 3 non-linear points (ii) A line and a point (iii) Two intersecting lines (iv) Two Parallel Lines (The point can not lie on the line) Descriptive Geometry: refers to the representation of 3D objects in a 2D format using points, lines and planes. -
Squaring the Circle in Panoramas
Squaring the Circle in Panoramas Lihi Zelnik-Manor1 Gabriele Peters2 Pietro Perona1 1. Department of Electrical Engineering 2. Informatik VII (Graphische Systeme) California Institute of Technology Universitat Dortmund Pasadena, CA 91125, USA Dortmund, Germany http://www.vision.caltech.edu/lihi/SquarePanorama.html Abstract and conveying the vivid visual impression of large panora- mas. Such mosaics are superior to panoramic pictures taken Pictures taken by a rotating camera cover the viewing with conventional fish-eye lenses in many respects: they sphere surrounding the center of rotation. Having a set of may span wider fields of view, they have unlimited reso- images registered and blended on the sphere what is left to lution, they make use of cheaper optics and they are not be done, in order to obtain a flat panorama, is projecting restricted to the projection geometry imposed by the lens. the spherical image onto a picture plane. This step is unfor- The geometry of single view point panoramas has long tunately not obvious – the surface of the sphere may not be been well understood [12, 21]. This has been used for mo- flattened onto a page without some form of distortion. The saicing of video sequences (e.g., [13, 20]) as well as for ob- objective of this paper is discussing the difficulties and op- taining super-resolution images (e.g., [6, 23]). By contrast portunities that are connected to the projection from view- when the point of view changes the mosaic is ‘impossible’ ing sphere to image plane. We first explore a number of al- unless the structure of the scene is very special. -
Basic Engineering Drawings
Today’s Thoughts The thing that’s so wonderful about using beautiful, appropriate [CAD] tools is that they become an extension of you, your body, you fingertips, and your mind. They get out of the way and let you directly interact with the problem you are solving. Everyone’s tried to remove a screw without a screwdriver; a task quickly becomes impossible that otherwise would be trivial. — Luke Crawford Here is one of the few effective keys to the design problem — the ability of the designer to recognize as many of the constraints as possible — his willingness and enthusiasm for working within these constraints. Constraints of price, of size, of strength, of balance, of surface, of time and so forth. — Charles Eames Section Views ME 172 Outline • Full Section • Half Section • Offset Section • Broken-out Section • Revolved Section • Removed Section • Sectioning Problems • Sectioning Quiz A Section View Technical Drawing– by Giesecke Visualizing a Section View Technical Drawing– by Giesecke Visualizing a Section View Technical Drawing– by Giesecke A Cut Plane Line Technical Drawing– by Giesecke Full Section View Blueprint Reading Basics – by Warren Hammer Full Section View Technical Drawing– by Giesecke Correct Full Section View Technical Drawing– by Giesecke Correct Full Section View Technical Drawing– by Giesecke Correct Full Section View Technical Drawing– by Giesecke Sectioning Symbols Technical Drawing– by Giesecke Correct Sectioning Lines Technical Drawing– by Giesecke ProblemAligned Full with Section a Full Section Blueprint Reading -
3D Viewing Week 8, Lecture 15
CS 536 Computer Graphics 3D Viewing Week 8, Lecture 15 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 Overview • 3D Viewing • 3D Projective Geometry • Mapping 3D worlds to 2D screens • Introduction and discussion of homework #4 Lecture Credits: Most pictures are from Foley/VanDam; Additional and extensive thanks also goes to those credited on individual slides 2 Pics/Math courtesy of Dave Mount @ UMD-CP 1994 Foley/VanDam/Finer/Huges/Phillips ICG Recall the 2D Problem • Objects exist in a 2D WCS • Objects clipped/transformed to viewport • Viewport transformed and drawn on 2D screen 3 Pics/Math courtesy of Dave Mount @ UMD-CP From 3D Virtual World to 2D Screen • Not unlike The Allegory of the Cave (Plato’s “Republic", Book VII) • Viewers see a 2D shadow of 3D world • How do we create this shadow? • How do we make it as realistic as possible? 4 Pics/Math courtesy of Dave Mount @ UMD-CP History of Linear Perspective • Renaissance artists – Alberti (1435) – Della Francesca (1470) – Da Vinci (1490) – Pélerin (1505) – Dürer (1525) Dürer: Measurement Instruction with Compass and Straight Edge http://www.handprint.com/HP/WCL/tech10.html 5 The 3D Problem: Using a Synthetic Camera • Think of 3D viewing as taking a photo: – Select Projection – Specify viewing parameters – Clip objects in 3D – Project the results onto the display and draw 6 1994 Foley/VanDam/Finer/Huges/Phillips ICG The 3D Problem: (Slightly) Alternate Approach • Think of 3D viewing as taking a photo: – Select Projection – Specify -
Drawing the “Boundaries”, the Start of an Urban Planning Project
Geographia Technica, Vol. 12, Issue 2, 2017, pp 73 to 81 DRAWING THE “BOUNDARIES”, THE START OF AN URBAN PLANNING PROJECT Jordi GOMIS1, Carlos TURÓN DOI: 10.21163/GT_2017.122.07 ABSTRACT: The graphic representation of the boundaries of urban planning areas is a sufficiently important element of the representation of urban planning as not to allow the slightest hint of doubt or imprecision to arise in regard to the dividing lines it indicates. It can include everything from the growth limit of the urban land as a whole to small inner areas of the city subject to improvement, renovation or change of planning conditions. It is this first concept that the town planner must include/draw on his working plan to begin his task. Throughout the history of urban planning, this first “boundary” has been represented in various ways using different graphic and/or visual mechanisms. This article presents a journey through the history of the resources used by technicians for their representation and their evolution. Key-words: Graphic representation, Technical drawing, Urban Planning, City drawing, Urbanism. 1. INTRODUCTION The demarcation of geographical, territorial, municipal, regional, etc. areas has never been a problem dissociated from society. Since ancient times the specification of geographic boundaries and limits of influence has occupied and preoccupied people. “Boundaries” and “frontiers” have always given rise to disputes that people have attempted to resolve and settle, sometimes graphically, many others, unfortunately, through the use of force. Various inventories of land, people, property, crops, etc. were made ever since ancient times to inform kings and noblemen of the quantity and quality of their belongings and properties. -
CS 4204 Computer Graphics 3D Views and Projection
CS 4204 Computer Graphics 3D views and projection Adapted from notes by Yong Cao 1 Overview of 3D rendering Modeling: * Topic we’ve already discussed • *Define object in local coordinates • *Place object in world coordinates (modeling transformation) Viewing: • Define camera parameters • Find object location in camera coordinates (viewing transformation) Projection: project object to the viewplane Clipping: clip object to the view volume *Viewport transformation *Rasterization: rasterize object Simple teapot demo 3D rendering pipeline Vertices as input Series of operations/transformations to obtain 2D vertices in screen coordinates These can then be rasterized 3D rendering pipeline We’ve already discussed: • Viewport transformation • 3D modeling transformations We’ll talk about remaining topics in reverse order: • 3D clipping (simple extension of 2D clipping) • 3D projection • 3D viewing Clipping: 3D Cohen-Sutherland Use 6-bit outcodes When needed, clip line segment against planes Viewing and Projection Camera Analogy: 1. Set up your tripod and point the camera at the scene (viewing transformation). 2. Arrange the scene to be photographed into the desired composition (modeling transformation). 3. Choose a camera lens or adjust the zoom (projection transformation). 4. Determine how large you want the final photograph to be - for example, you might want it enlarged (viewport transformation). Projection transformations Introduction to Projection Transformations Mapping: f : Rn Rm Projection: n > m Planar Projection: Projection on a plane. -
Perspective Projection
Transform 3D objects on to a 2D plane using projections 2 types of projections Perspective Parallel In parallel projection, coordinate positions are transformed to the view plane along parallel lines. In perspective projection, object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection) 2 Perspective Projection 3 Parallel Projection 4 PROJECTIONS PARALLEL PERSPECTIVE (parallel projectors) (converging projectors) One point Oblique Orthographic (one principal (projectors perpendicular (projectors not perpendicular to vanishing point) to view plane) view plane) Two point General (Two principal Multiview Axonometric vanishing point) (view plane parallel (view plane not parallel to Cavalier principal planes) to principal planes) Three point (Three principal Cabinet vanishing point) Isometric Dimetric Trimetric 5 Perspective v Parallel • Perspective: – visual effect is similar to human visual system... – has 'perspective foreshortening' • size of object varies inversely with distance from the center of projection. Projection of a distant object are smaller than the projection of objects of the same size that are closer to the projection plane. • Parallel: It preserves relative proportion of object. – less realistic view because of no foreshortening – however, parallel lines remain parallel. 6 Perspective Projections • Characteristics: • Center of Projection (CP) is a finite distance from object • Projectors are rays (i.e., non-parallel) • Vanishing points • Objects appear smaller as distance from CP (eye of observer) increases • Difficult to determine exact size and shape of object • Most realistic, difficult to execute 7 • When a 3D object is projected onto view plane using perspective transformation equations, any set of parallel lines in the object that are not parallel to the projection plane, converge at a vanishing point. -
U. S. Department of Agriculture Technical Release No
U. S. DEPARTMENT OF AGRICULTURE TECHNICAL RELEASE NO. 41 SO1 L CONSERVATION SERVICE GEOLOGY &INEERING DIVISION MARCH 1969 U. S. Department of Agriculture Technical Release No. 41 Soil Conservation Service Geology Engineering Division March 1969 GRAPHICAL SOLUTIONS OF GEOLOGIC PROBLEMS D. H. Hixson Geologist GRAPHICAL SOLUTIONS OF GEOLOGIC PROBLEMS Contents Page Introduction Scope Orthographic Projections Depth to a Dipping Bed Determine True Dip from One Apparent Dip and the Strike Determine True Dip from Two Apparent Dip Measurements at Same Point Three Point Problem Problems Involving Points, Lines, and Planes Problems Involving Points and Lines Shortest Distance between Two Non-Parallel, Non-Intersecting Lines Distance from a Point to a Plane Determine the Line of Intersection of Two Oblique Planes Displacement of a Vertical Fault Displacement of an Inclined Fault Stereographic Projection True Dip from Two Apparent Dips Apparent Dip from True Dip Line of Intersection of Two Oblique Planes Rotation of a Bed Rotation of a Fault Poles Rotation of a Bed Rotation of a Fault Vertical Drill Holes Inclined Drill Holes Combination Orthographic and Stereographic Technique References Figures Fig. 1 Orthographic Projection Fig. 2 Orthographic Projection Fig. 3 True Dip from Apparent Dip and Strike Fig. 4 True Dip from Two Apparent Dips Fig. 5 True Dip from Two Apparent Dips Fig. 6 True Dip from Two Apparent Dips Fig. 7 Three Point Problem Fig. 8 Three Point Problem Page Fig. Distance from a Point to a Line 17 Fig. Shortest Distance between Two Lines 19 Fig. Distance from a Point to a Plane 21 Fig. Nomenclature of Fault Displacement 23 Fig.