Basics of Descriptive Geometry
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Mathematics Is a Gentleman's Art: Analysis and Synthesis in American College Geometry Teaching, 1790-1840 Amy K
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2000 Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 Amy K. Ackerberg-Hastings Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Higher Education and Teaching Commons, History of Science, Technology, and Medicine Commons, and the Science and Mathematics Education Commons Recommended Citation Ackerberg-Hastings, Amy K., "Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 " (2000). Retrospective Theses and Dissertations. 12669. https://lib.dr.iastate.edu/rtd/12669 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margwis, and improper alignment can adversely affect reproduction. in the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. -
Proofs with Perpendicular Lines
3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units. -
Elements of Descriptive Geometry
Livre de Lyon Academic Works of Livre de Lyon Science and Mathematical Science 2020 Elements of Descriptive Geometry Francis Henney Smith Follow this and additional works at: https://academicworks.livredelyon.com/sci_math Part of the Geometry and Topology Commons Recommended Citation Smith, Francis Henney, "Elements of Descriptive Geometry" (2020). Science and Mathematical Science. 13. https://academicworks.livredelyon.com/sci_math/13 This Book is brought to you for free and open access by Livre de Lyon, an international publisher specializing in academic books and journals. Browse more titles on Academic Works of Livre de Lyon, hosted on Digital Commons, an Elsevier platform. For more information, please contact [email protected]. ELEMENTS OF Descriptive Geometry By Francis Henney Smith Geometry livredelyon.com ISBN: 978-2-38236-008-8 livredelyon livredelyon livredelyon 09_Elements of Descriptive Geometry.indd 1 09-08-2020 15:55:23 TO COLONEL JOHN T. L. PRESTON, Professor of Latin Language and English Literature, Vir- ginia Military Institute. I am sure my associate Professors will vindicate the grounds upon which you arc singled out, as one to whom I may appro- priately dedicate this work. As the originator of the scheme, by which the public guard of a State Arsenal was converted into a Military School, you have the proud distinction of being the “ Father of the Virginia Military Institute ” You were a member of the first Board of Visitors, which gave form to the organization of the Institution; you were my only colleague during the two first and trying years of its being; and you have, for a period of twenty-eight years, given your labors and your influence, in no stinted mea- sure, not only in directing the special department of instruc- tion assigned to you, but in promoting those general plans of development, which have given marked character and wide- spread reputation to the school. -
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 . -
Holley GM LS Race Single-Plane Intake Manifold Kits
Holley GM LS Race Single-Plane Intake Manifold Kits 300-255 / 300-255BK LS1/2/6 Port-EFI - w/ Fuel Rails 4150 Flange 300-256 / 300-256BK LS1/2/6 Carbureted/TB EFI 4150 Flange 300-290 / 300-290BK LS3/L92 Port-EFI - w/ Fuel Rails 4150 Flange 300-291 / 300-291BK LS3/L92 Carbureted/TB EFI 4150 Flange 300-294 / 300-294BK LS1/2/6 Port-EFI - w/ Fuel Rails 4500 Flange 300-295 / 300-295BK LS1/2/6 Carbureted/TB EFI 4500 Flange IMPORTANT: Before installation, please read these instructions completely. APPLICATIONS: The Holley LS Race single-plane intake manifolds are designed for GM LS Gen III and IV engines, utilized in numerous performance applications, and are intended for carbureted, throttle body EFI, or direct-port EFI setups. The LS Race single-plane intake manifolds are designed for hi-performance/racing engine applications, 5.3 to 6.2+ liter displacement, and maximum engine speeds of 6000-7000 rpm, depending on the engine combination. This single-plane design provides optimal performance across the RPM spectrum while providing maximum performance up to 7000 rpm. These intake manifolds are for use on non-emissions controlled applications only, and will not accept stock components and hardware. Port EFI versions may not be compatible with all throttle body linkages. When installing the throttle body, make certain there is a minimum of ¼” clearance between all linkage and the fuel rail. SPLIT DESIGN: The Holley LS Race manifold incorporates a split feature, which allows disassembly of the intake for direct access to internal plenum and port surfaces, making custom porting and matching a snap. -
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 « » « -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
Descriptive Geometry Section 10.1 Basic Descriptive Geometry and Board Drafting Section 10.2 Solving Descriptive Geometry Problems with CAD
10 Descriptive Geometry Section 10.1 Basic Descriptive Geometry and Board Drafting Section 10.2 Solving Descriptive Geometry Problems with CAD Chapter Objectives • Locate points in three-dimensional (3D) space. • Identify and describe the three basic types of lines. • Identify and describe the three basic types of planes. • Solve descriptive geometry problems using board-drafting techniques. • Create points, lines, planes, and solids in 3D space using CAD. • Solve descriptive geometry problems using CAD. Plane Spoken Rutan’s unconventional 202 Boomerang aircraft has an asymmetrical design, with one engine on the fuselage and another mounted on a pod. What special allowances would need to be made for such a design? 328 Drafting Career Burt Rutan, Aeronautical Engineer Effi cient travel through space has become an ambi- tion of aeronautical engineer, Burt Rutan. “I want to go high,” he says, “because that’s where the view is.” His unconventional designs have included every- thing from crafts that can enter space twice within a two week period, to planes than can circle the Earth without stopping to refuel. Designed by Rutan and built at his company, Scaled Composites LLC, the 202 Boomerang aircraft is named for its forward-swept asymmetrical wing. The design allows the Boomerang to fl y faster and farther than conventional twin-engine aircraft, hav- ing corrected aerodynamic mistakes made previously in twin-engine design. It is hailed as one of the most beautiful aircraft ever built. Academic Skills and Abilities • Algebra, geometry, calculus • Biology, chemistry, physics • English • Social studies • Humanities • Computer use Career Pathways Engineers should be creative, inquisitive, ana- lytical, detail oriented, and able to work as part of a team and to communicate well. -
1-1 Understanding Points, Lines, and Planes Lines, and Planes
Understanding Points, 1-11-1 Understanding Points, Lines, and Planes Lines, and Planes Holt Geometry 1-1 Understanding Points, Lines, and Planes Objectives Identify, name, and draw points, lines, segments, rays, and planes. Apply basic facts about points, lines, and planes. Holt Geometry 1-1 Understanding Points, Lines, and Planes Vocabulary undefined term point line plane collinear coplanar segment endpoint ray opposite rays postulate Holt Geometry 1-1 Understanding Points, Lines, and Planes The most basic figures in geometry are undefined terms, which cannot be defined by using other figures. The undefined terms point, line, and plane are the building blocks of geometry. Holt Geometry 1-1 Understanding Points, Lines, and Planes Holt Geometry 1-1 Understanding Points, Lines, and Planes Points that lie on the same line are collinear. K, L, and M are collinear. K, L, and N are noncollinear. Points that lie on the same plane are coplanar. Otherwise they are noncoplanar. K L M N Holt Geometry 1-1 Understanding Points, Lines, and Planes Example 1: Naming Points, Lines, and Planes A. Name four coplanar points. A, B, C, D B. Name three lines. Possible answer: AE, BE, CE Holt Geometry 1-1 Understanding Points, Lines, and Planes Holt Geometry 1-1 Understanding Points, Lines, and Planes Example 2: Drawing Segments and Rays Draw and label each of the following. A. a segment with endpoints M and N. N M B. opposite rays with a common endpoint T. T Holt Geometry 1-1 Understanding Points, Lines, and Planes Check It Out! Example 2 Draw and label a ray with endpoint M that contains N. -
Chapter 1 – Symmetry of Molecules – P. 1
Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules. -
David Eugene Smith Collection
Ms Coll\Smith,D.E.\Historical Smith, David Eugene, 1860-1944, collector. Historical papers, [ca. 1400-1899] 17.5 linear ft.(ca. 10,100 items in 31 boxes) Biography: Mathematician. Professor of mathematics at the State Normal School, Cortland, N.Y., 1884-1891; at Michigan State Normal College, 1891-1898; at New York State Normal School, Brockport, N.Y., 1898-1901; and at Teachers College, Columbia University, 1901-1944. He was the editor of the Bulletin of the American Mathematical Society, the American Mathematical Monthly, and Scripta Mathematica, a member of the International Commission on the Teaching of Mathematics, 1908-1944; and librarian of Teachers College, 1902-1920. He was the author of Rara Mathemativca (1907), The History of Mathematics (1924), and many other works on the history of mathematics as well as over forty mathematical textbooks and numerous journal articles. He also collected manuscript materials relating to the history of mathematics. Summary: Correspondence, manuscripts, and documents of mathematicians and other scientists, often dealing with politics and fields other than mathematics. Many of these concern the French Revolution. Organization: Cataloged. * Alembert letters are available on microfilm (MN#2000-1207). Lalande letters are available on microfilm (MN#3605-4). Guillaume Libri letters are available in photocopy form. William John Clarke Miller are available on microfilm (MN#95-7023 to 95-2027). Gaspard Monge letters are available on microfilm (MN#80-1582). Selected Isaac Newton letters are available on microfilm (MN#3617-5). Selected Quetelet letters are available on microfilm (MN#3623-7). Finding aids: Contents list, 19p. Donor: Gift of David Eugene Smith, 1931. -
Machine Drawing
2.4 LINES Lines of different types and thicknesses are used for graphical representation of objects. The types of lines and their applications are shown in Table 2.4. Typical applications of different types of lines are shown in Figs. 2.5 and 2.6. Table 2.4 Types of lines and their applications Line Description General Applications A Continuous thick A1 Visible outlines B Continuous thin B1 Imaginary lines of intersection (straight or curved) B2 Dimension lines B3 Projection lines B4 Leader lines B5 Hatching lines B6 Outlines of revolved sections in place B7 Short centre lines C Continuous thin, free-hand C1 Limits of partial or interrupted views and sections, if the limit is not a chain thin D Continuous thin (straight) D1 Line (see Fig. 2.5) with zigzags E Dashed thick E1 Hidden outlines G Chain thin G1 Centre lines G2 Lines of symmetry G3 Trajectories H Chain thin, thick at ends H1 Cutting planes and changes of direction J Chain thick J1 Indication of lines or surfaces to which a special requirement applies K Chain thin, double-dashed K1 Outlines of adjacent parts K2 Alternative and extreme positions of movable parts K3 Centroidal lines 2.4.2 Order of Priority of Coinciding Lines When two or more lines of different types coincide, the following order of priority should be observed: (i) Visible outlines and edges (Continuous thick lines, type A), (ii) Hidden outlines and edges (Dashed line, type E or F), (iii) Cutting planes (Chain thin, thick at ends and changes of cutting planes, type H), (iv) Centre lines and lines of symmetry (Chain thin line, type G), (v) Centroidal lines (Chain thin double dashed line, type K), (vi) Projection lines (Continuous thin line, type B).