Isometric Projections and Other Study and Display Methods Used in Preliminary Design of I-70 Through Glenwood Canyon
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The Mathematics of Fels Sculptures
THE MATHEMATICS OF FELS SCULPTURES DAVID FELS AND ANGELO B. MINGARELLI Abstract. We give a purely mathematical interpretation and construction of sculptures rendered by one of the authors, known herein as Fels sculptures. We also show that the mathematical framework underlying Ferguson's sculpture, The Ariadne Torus, may be considered a special case of the more general constructions presented here. More general discussions are also presented about the creation of such sculptures whether they be virtual or in higher dimensional space. Introduction Sculptors manifest ideas as material objects. We use a system wherein an idea is symbolized as a solid, governed by a set of rules, such that the sculpture is the expressed material result of applying rules to symbols. The underlying set of rules, being mathematical in nature, may thus lead to enormous abstraction and although sculptures are generally thought of as three dimensional objects, they can be created in four and higher dimensional (unseen) spaces with various projections leading to new and pleasant three dimensional sculptures. The interplay of mathematics and the arts has, of course, a very long and old history and we cannot begin to elaborate on this matter here. Recently however, the problem of creating mathematical programs for the construction of ribbed sculptures by Charles Perry was considered in [3]. For an insightful paper on topological tori leading to abstract art see also [5]. Spiral and twirling sculptures were analysed and constructed in [1]. This paper deals with a technique for sculpting works mostly based on wood (but not necessarily restricted to it) using abstract ideas based on twirls and tori, though again, not limited to them. -
Sculptures in S3
Sculptures in S3 Saul Schleimer and Henry Segerman∗ Abstract We describe the construction of a number of sculptures. Each is based on a geometric design native to the three- sphere: the unit sphere in four-dimensional space. Via stereographic projection, we transfer the design to three- dimensional space. All of the sculptures are then fabricated by the 3D printing service Shapeways. 1 Introduction The three-sphere, denoted S3, is a three-dimensional analog of the ordinary two-dimensional sphere, S2. In n+ general, the n–dimensional sphere is a subset of R 1 as follows: n n+1 2 2 2 S = f(x0;x1;:::;xn) 2 R j x0 + x1 + ··· + xn = 1g: Thus S2 can be seen as the usual unit sphere in R3. Visualising objects in dimensions higher than three is non-trivial. However for S3 we can use stereographic projection to reduce the dimension from four to three. n n n Let N = (0;:::;0;1) be the north pole of S . We define stereographic projection r : S − fNg ! R by x0 x1 xn−1 r(x0;x1;:::;xn) = ; ;:::; : 1 − xn 1 − xn 1 − xn See [1, page 27]. Figure 1a displays the one-dimensional case; this is also a cross-section of the n– dimensional case. For any point (x;y) 2 S1 − fNg draw the straight line L between N and (x;y). Then L meets R1 at a single point; this is r(x;y). N x 1−y (x;y) (a) Stereographic projection from S1 − (b) Two-dimensional stereographic projection applied to the Earth. -
The Mathematics of Mitering and Its Artful Application
The Mathematics of Mitering and Its Artful Application Tom Verhoeff Koos Verhoeff Faculty of Mathematics and CS Valkenswaard, Netherlands Eindhoven University of Technology Den Dolech 2 5612 AZ Eindhoven, Netherlands Email: [email protected] Abstract We give a systematic presentation of the mathematics behind the classic miter joint and variants, like the skew miter joint and the (skew) fold joint. The latter is especially useful for connecting strips at an angle. We also address the problems that arise from constructing a closed 3D path from beams by using miter joints all the way round. We illustrate the possibilities with artwork making use of various miter joints. 1 Introduction The miter joint is well-known in the Arts, if only as a way of making fine frames for pictures and paintings. In its everyday application, a common problem with miter joints occurs when cutting a baseboard for walls meeting at an angle other than exactly 90 degrees. However, there is much more to the miter joint than meets the eye. In this paper, we will explore variations and related mathematical challenges, and show some artwork that this provoked. In Section 2 we introduce the problem domain and its terminology. A systematic mathematical treatment is presented in Section 3. Section 4 shows some artwork based on various miter joints. We conclude the paper in Section 5 with some pointers to further work. 2 Problem Domain and Terminology We will now describe how we encountered new problems related to the miter joint. To avoid misunderstand- ings, we first introduce some terminology. Figure 1: Polygon knot with six edges (left) and thickened with circular cylinders (right) 225 2.1 Cylinders, single and double beveling, planar and spatial mitering Let K be a one-dimensional curve in space, having finite length. -
Cross-Section- Surface Area of a Prism- Surface Area of a Cylinder- Volume of a Prism
S8.6 Volume Things to Learn (Key words, Notation & Formulae) Complete from your notes Radius- Diameter- Surface Area- Volume- Capacity- Prism- Cross-section- Surface area of a prism- Surface area of a cylinder- Volume of a prism- Section 1. Surface area of cuboids: Q1. Work out the surface area of each cuboid shown below: Q2. What is the surface area of a cuboid with the dimensions 4cm, 5cm and 6cm? Section 2. Volume of cuboids: Q1. Calculate the volume of these cuboids: S8.6 Volume Q2. Section 3. Definition of prisms: Label all the shapes and tick the ones that are prisms. Section 4. Surface area of prisms: Q1. Find the surface area of this triangular prism. Q2. Calculate the surface area of this cylinder. S8.6 Volume Q3. Cans are in cylindrical shapes. Each can has a diameter of 5.3 cm and a height of 11.4 cm. How much paper is required to make the label for the 20 cans? Section 5. Volume of a prism: Q1. Find the volume of this L-shaped prism. Q2. Calculate the volume of this prism. Give your answer to 2sf Section 6 . Volume of a cylinder: Q1. Calculate the volume of this cylinder. S8.6 Volume Q2. The diagram shows a piece of wood. The piece of wood is a prism of length 350cm. The cross-section of the prism is a semi-circle with diameter 1.2cm. Calculate the volume of the piece of wood. Give your answer to 3sf. Section 7. Problems involving volume and capacity: Q1. Sam buys a planter shown below. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
What's Your Theory of Effective Schematic Map Design?
What’s Your Theory of Effective Schematic Map Design? Maxwell. J. Roberts Department of Psychology University of Essex Colchester, UK [email protected] Abstract—Amongst designers, researchers, and the general general public accept or reject radical new creations. public, there exists a diverse array of opinion about good practice Furthermore, the lay-theories held by designers will affect the in schematic map design, and a lack of awareness that such products that they create and the recommendations that they opinions are not necessarily universally held nor supported by make. Similarly, the lay-theories of transport managers will evidence. This paper gives an overview of the range of opinion determine design specifications and the acceptance or rejection that can be encountered, the consequences that this has for of end-products. With so many different theories in circulation, published designs, and a framework for organising the various conflicts will be inevitable, hence the polarised response to the views. controversial Madrid Metro map of 2007, which generated a huge adverse reaction amongst residents: presumably, the Keywords—schematic mapping; lay-theories, expectations and transport authority believed that this was a sound design. prejudices; effective design; cognition; intelligence. Likewise, the adverse public response to the removal of the I. LAY-THEORIES OF DESIGN River Thames from the London Underground map in 2009. The lay-theory held by officials, presumably, was that the river Whenever I create a controversial new schematic map and was irrelevant to usability. The public disagreed, and the river publish it on the internet, the diversity of opinion that it evokes returned a few months later. -
Module 4 Electronic Diagrams and Schematics
ENGINEERING SYMBOLOGY, PRINTS, AND DRAWINGS Module 4 Electronic Diagrams and Schematics Engineering Symbology, Prints, & Drawings Electronic Diagrams & Schematics TABLE OF CONTENTS Table of Co nte nts TABLE OF CONTENTS ................................................................................................... i LIST OF FIGURES ...........................................................................................................ii LIST OF TABLES ............................................................................................................ iii REFERENCES ................................................................................................................iv OBJECTIVES .................................................................................................................. 1 ELECTRONIC DIAGRAMS, PRINTS, AND SCHEMATICS ............................................ 2 Introduction .................................................................................................................. 2 Electronic Schematic Drawing Symbology .................................................................. 3 Examples of Electronic Schematic Diagrams .............................................................. 6 Reading Electronic Prints, Diagrams and Schematics ................................................. 8 Block Drawing Symbology ......................................................................................... 13 Examples of Block Diagrams .................................................................................... -
Changes in Cross-Section Geometry and Channel Volume in Two Reaches of the Kankakee River in Illinois, 1959-94
Changes in Cross-Section Geometry and Channel Volume in Two Reaches of the Kankakee River in Illinois, 1959-94 By PAUL J. TERRIO and JOHN E. NAZIMEK U.S. GEOLOGICAL SURVEY Water-Resources Investigations Report 96 4261 Prepared in cooperation with the KANKAKEE COUNTY SOIL AND WATER CONSERVATION DISTRICT Urbana, Illinois 1997 U.S. DEPARTMENT OF THE INTERIOR BRUCE BABBITT, Secretary U.S. GEOLOGICAL SURVEY Gordon P. Eaton, Director The use of firm, trade, and brand names in this report is for identification purposes only and does not constitute endorsement by the U.S. Geological Survey. For additional information write to: Copies of this report can be purchased from: District Chief U.S. Geological Survey U.S. Geological Survey Branch of Information Services 221 N. Broadway Box 25286 Urbana, Illinois 61801 Denver, CO 80225 CONTENTS Abstract......................................................................................... 1 Introduction...................................................................................... 1 Purpose and Scope............................................................................ 3 Description of the Study Area................................................................... 4 Acknowledgments............................................................................ 4 Compilation and Measurement of Channel Cross-Section Geometry Data ..................................... 6 Momence Wetlands Reach ..................................................................... 6 Six-Mile Pool Reach ......................................................................... -
Vision & Implementation Plan
Buffalo River Greenway Vision & Implementation Plan Implementation Guidelines for the Buffalo River, Cayuga Creek, Buffalo Creek and Cazenovia Creek within the City of Buffalo and Towns of Cheektowaga and West Seneca. Prepared For: Buffalo Niagara RIVERKEEPER® Prepared By: Lynn Mason, ASLA With Assistance From: Julie Barrett O’Neill Jill Spisiak Jedlicka Lynda Schneekloth May 2006 II TABLE OF CONTENTS FIGURE LIST pg. IV INVOLVED AGENCIES pg. VI EXECUTIVE SUMMARY pg. VII I. GREENWAY VISION pg. 1 1. Greenway Benefits pg. 3 2. Buffalo River Greenway Project Area pg. 8 II. GREENWAY PLANNING HISTORY pg. 15 1. Recent Projects pg. 17 2. Recent Planning Projects pg. 28 III. EXISTING GREENWAY RESOURCES pg. 29 1. Parks & Parkways pg. 31 2. Conservation Areas pg. 37 3. Bike/Hike Trails pg. 38 4. Boat Launches and Marinas pg. 41 5. Fishing Access and Hot Spots pg. 41 6. Urban Canoe Trail and Launches pg. 42 7. Heritage Interpretation Areas pg. 44 IV. PROPOSED NEW GREENWAY ELEMENTS pg. 47 1. Introduction pg. 49 2. Buffalo River Trail Segments pg. 56 3. Site Specific Opportunities pg. 58 V. IMPLEMENTATION pg. 95 1. Introduction pg. 97 2. Implementation Strategies pg. 106 3. Leveraging Resources and Identifying Funding pg. 108 4. Legislative Action pg. 110 5. Education and Encouragement pg. 110 6. Operations and Maintenance pg. 111 7. Trail Development pg. 112 8. Design Guidelines pg. 181 APPENDIX/ BACKGROUND INFORMATION pg. 189 A. Existing Greenway Resources pg. 191 B. Prohibited Uses pg. 209 C. Phytoremediation pg. 213 D. Buffalo River Paper Streets: A Status Report pg. 215 REFERENCES pg. -
How Neurons Exploit Fractal Geometry to Optimize Their Network Connectivity Julian H
www.nature.com/scientificreports OPEN How neurons exploit fractal geometry to optimize their network connectivity Julian H. Smith1,5, Conor Rowland1,5, B. Harland2, S. Moslehi1, R. D. Montgomery1, K. Schobert1, W. J. Watterson1, J. Dalrymple‑Alford3,4 & R. P. Taylor1* We investigate the degree to which neurons are fractal, the origin of this fractality, and its impact on functionality. By analyzing three‑dimensional images of rat neurons, we show the way their dendrites fork and weave through space is unexpectedly important for generating fractal‑like behavior well‑ described by an ‘efective’ fractal dimension D. This discovery motivated us to create distorted neuron models by modifying the dendritic patterns, so generating neurons across wide ranges of D extending beyond their natural values. By charting the D‑dependent variations in inter‑neuron connectivity along with the associated costs, we propose that their D values refect a network cooperation that optimizes these constraints. We discuss the implications for healthy and pathological neurons, and for connecting neurons to medical implants. Our automated approach also facilitates insights relating form and function, applicable to individual neurons and their networks, providing a crucial tool for addressing massive data collection projects (e.g. connectomes). Many of nature’s fractal objects beneft from the favorable functionality that results from their pattern repeti- tion at multiple scales 1–3. Anatomical examples include cardiovascular and respiratory systems4 such as the bronchial tree5 while examples from natural scenery include coastlines 6, lightning7, rivers8, and trees9,10. Along with trees, neurons are considered to be a prevalent form of fractal branching behavior11. -
Have You Ever Used Two Picture Planes to Draw a Single Perspective View?
G e n e r a l a r t i c l e have you ever used Two Picture Planes to Draw a Single Perspective View? T o m á S G arc í A S A l gad o Apparently, the use of two picture planes to draw a single view has be a more versatile plane that in addition to representing not been attempted before. Most perspective methods, after Alberti, the appearance of 3D objects would also serve to measure take for granted the use of a single picture plane, disregarding its likely real dimensions? Such a plane, in Modular Perspective [3], use in dual positions. What if two picture planes are necessary to ABSTRACT draw a single view—for example, given a lack of spatial references at can be called the perspective plane (PPL). On the PPL one can ground level to estimate the distance between two objects? This article measure and draw directly the three modular coordinates of demonstrates that to draw the interior of a building from which another all points of interest of any given object in space. In other building can be seen about 190 m away, where the projection of such words, the PPL is a true three-dimensional plane, despite building on the first picture plane would be imprecise, it may be wise to actually being two-dimensional. use a second picture plane. This leads to consideration of how objects In addition, the PPL can be of any size, since all points are change shape as they move away from the viewer. -
The Convergence of Graphics and Vision
The Convergence of Graphics Cover Feature and Vision Approaching similar problems from opposite directions, graphics and vision researchers are reaching a fertile middle ground. The goal is to find the best possible tools for the imagination. This overview describes cutting- edge work, some of which will debut at Siggraph 98. Jed Lengyel t Microsoft Research, the computer vision sampled representations. This use of captured Microsoft and graphics groups used to be on opposite scenes (enhanced by vision research) yields richer Research sides of the building. Now we have offices rendering and modeling methods (for graphics) along the same hallway, and we see each other than methods that synthesize everything from A every day. This reflects the larger trend in our scratch. field as graphics and vision close in on similar problems. • Exploiting temporal and spatial coherence (sim- Computer graphics and computer vision are inverse ilarities in images) via the use of layers and other problems. Traditional computer graphics starts with techniques is boosting runtime performance. input geometric models and produces image se- • The explosion in PC graphics performance is quences. Traditional computer vision starts with input making powerful computational techniques more image sequences and produces geometric models. practical. Lately, there has been a meeting in the middle, and the center—the prize—is to create stunning images in real VISION AND GRAPHICS CROSSOVER: time. IMAGE-BASED RENDERING AND MODELING Vision researchers now work from images back- What are vision and graphics learning from each ward, just as far backward as necessary to create mod- other? Both deal with the image streams that result els that capture a scene without going to full geometric when a real or virtual camera is exposed to the phys- models.