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The Polycons: the Sphericon (Or Tetracon) Has Found Its Family
The polycons: the sphericon (or tetracon) has found its family David Hirscha and Katherine A. Seatonb a Nachalat Binyamin Arts and Crafts Fair, Tel Aviv, Israel; b Department of Mathematics and Statistics, La Trobe University VIC 3086, Australia ARTICLE HISTORY Compiled December 23, 2019 ABSTRACT This paper introduces a new family of solids, which we call polycons, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that of other developable rollers such as the oloid and particular polysphericons. The paper concludes with a discussion of the polycons as stationary and kinetic works of art. KEYWORDS sphericon; polycons; tetracon; ruled surface; developable roller 1. Introduction In 1980 inventor David Hirsch, one of the authors of this paper, patented `a device for generating a meander motion' [9], describing the object that is now known as the sphericon. This discovery was independent of that of woodturner Colin Roberts [22], which came to public attention through the writings of Stewart [28], P¨oppe [21] and Phillips [19] almost twenty years later. The object was named for how it rolls | overall in a line (like a sphere), but with turns about its vertices and developing its whole surface (like a cone). It was realised both by members of the woodturning [17, 26] and mathematical [16, 20] communities that the sphericon could be generalised to a series of objects, called sometimes polysphericons or, when precision is required and as will be elucidated in Section 4, the (N; k)-icons. These objects are for the most part constructed from frusta of a number of cones of differing apex angle and height. -
United States Patent (11) 3,572,646 72 Inventor Heinz Kocher 390,533 8/1965 Switzerland
United States Patent (11) 3,572,646 72 Inventor Heinz Kocher 390,533 8/1965 Switzerland.................. 18/12(SF) Centralstrasse 41, Neuhausen, Switzerland 21 Appl. No. 721,816 Primary Examiner-William I. Price 22 Filed Apr. 16, 1968 Attorneys-Ralph E. Bucknam, Jesse D. Reingold, Robert R. 45 Patented Mar. 30, 1971 Strack and Henry A. Marzullo, Jr. (32) Priority Apr. 17, 1968 (33) Austria 31 A-3592/67 54 APPARATUS FORTRANSFORMING AND MXNG ABSTRACT: Apparatus for transforming and mixing DEFORMABLE MEDIA deformable media, such as plastics, comprises a cylindrical 11 Claims, 11 Drawing Figs. Casing having a rotor mounted for coaxial driven rotation in 52) U.S. Cl..................................................... 259/25 the cylinder which is provided with an inlet for the medium. 5 i Int. Cl................ ... B01f 7/02 The rotor is provided with a conveyor screw for the medium, and following the screw in the direction of flow the rotor com 50) Field of Search............................... ... 25917, 9, prises diverging diffusor portions alternating with cylindrical 10, 25, 26; 18/12 (SF), 12 (SE) rotor portions. The end of the rotor adjacent the outlet of the 56 References Cited cylinder is formed as a converging diffusor. The different rotor UNITED STATES PATENTS portions coact with surrounding wall portions of the cylinder having portions with a surface finish differing from that of the 2,810,159 10/1957 Teichmann................... 18/12(SE) rotor portions to subject the medium flowing in the space 3,239,882 3/1966 Yokana........................ 18/12(SE) between the rotor and the surrounding casing wall to differing FOREIGN PATENTS frictional resistance along the external and internal layers 1,007, 140 10/1965. -
||||||||||||III USOO540.8598A United States Patent (19) 11 Patent Number: 5,408,598 Pryor, Jr
||||||||||||III USOO540.8598A United States Patent (19) 11 Patent Number: 5,408,598 Pryor, Jr. 45 Date of Patent: Apr. 18, 1995 54 METHOD FOR FAST GENERATION OF 4,943,935 7/1990 Sate ..................................... 364/518 PARAMETRC CURVES EMPLOYNGA 4,949,281 8/1990 Hillenbrand et al. ... 364/518 CULA NUMB 4,999,789 3/1991 Fiasconaro .......................... 395/127 SEE SEREEE 5,133,052 7/1992 Bier et al. .......... ... 395/133 X 5,214,754 5/1993 Okamoto et al. ................... 395/142 DETERMINED ERROR THRESHOLD saw is is 75 Inventor: Duaine W. Pryor, Jr., Berkeley, OTHER PUBLICATIONS Calif. Computer Science Press, Inc., “Algorithms for Graph s ics and Image Processing', Theo Pavlidis Bell Labora 73 Assignee: tail Rte Mahines tories, pp. 221-231., 1982. is all y o a I. D. Faux et al., "Computational Geometry for Design 21 Appl. No.: 202,678 and Manufacture', Dept. of Mathematics 1979, pp. 127-145. Publisher: E. Horwood Ltd. 22 Fed:e Feb. 25, 1994 Wolfgang Dahmen, “Subdivision Algorithms Con verge Quadratically”, 1988, pp. 145-158, Journal of Related U.S. Application Data Computational and Applied Mathimatics. 63 Continuation of Ser. No. 705,041, May 23, 1991, aban- s doned. Primary Examiner-Almis R. Jankus 51 Int. Cl6 11/00 Attorney, Agent, or Firm-Perman & Green 52 U.S.C.. 395/2,39571.43 (57) ABSTRACT 58 Field of Search ................................ 395/141-143, A method enables the prediction of the number of sub 395/119, 123, 127, 133, -140, 155, 150, 151 divisions of a curve that will be required by control (56) References Cited polygons to assure that a resulting straight line repre sentation of the curve will not exceed a preset error U.S. -
Specialty Vision Products Collection
Specialty Vision Products Collection The largest portfolio and educational resource for specialty lenses in the industry. Scleral Spherical Ortho K Hybrids Irregular Cornea Custom Soft Bifocal/Multifocal Elevate your confidence and fitting success. Wet Labs Specialty Business Lunch & Learns Reviews Scleral Lens Highly Skilled Bootcamps Consultation Team Staff Trainings Custom Soft www.abboptical.com 1.800.772.3911 SPECIALTY VISI N PRODUCT CONSULTATION TEAM Our acclaimed team of Specialty Vision Consultants are your go-to when looking for strategies, educational opportunities and resources to grow the specialty lens side of your practice. Let our highly skilled team help you with tools like Lunch and Learns, Wet Labs, Scleral Lens Boot Camps and fitting assistance. With more than 59 years of experience in manufacturing specialty gas permeable and custom soft lenses for eye care practitioners, you can count on our consultants’ extraordinary expertise to provide you with the best in service and product choices to fit even your most difficult GP and custom soft lens patients. Arch Holcomb M.Sc. Ann Shackelford Dede Reyes Scleral Lens Expert Hybrid Lens Expert Experience: Experience: Experience: Over 35 years in the Over 40 years in the 20 years Clinical Contact Lens Industry Contact Lens Industry Contact Lens Fitter Taught at Ohio State NCLC Certified 1997 12 years Contact Lens University, College NCLC AC Certified 2007 Industry Consultant of Optometry NCLE-AC certified 1990 Fellowship CLSA 1995 Ext. 5375 • [email protected] Ext. 5823 • [email protected] Ext. 5860 • [email protected] Alika Mackley Grant Parson Crystal Bailey Scleral Lens Expert Experience: Experience: Experience: Over 35 years in the Over 35 years in the Over 20 years in the Contact Lens Industry Contact Lens Industry Contact Lens Industry NCLC certified 2003 NCLC Certified 2013 NCLC Certified 2002 NCLC-AC certified 2007 Ext. -
Adjoints and Canonical Forms of Polypols
Adjoints and Canonical Forms of Polypols Kathl´enKohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Rainer Sinn, Miruna-Stefana Sorea, Simon Telen August 27, 2021 Abstract Polypols are natural generalizations of polytopes, with boundaries given by non- linear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric per- spective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numer- ator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics. 1 Introduction Polytopes are very familiar geometric objects, with boundaries given by linear equations. Their beautiful and important properties have been extensively studied from different perspectives and have numerous applications. The present paper studies several more general classes of real domains/shapes with non-linear algebraic boundaries, known as polypols, polycons, polypoldra, positive geometries, etc., in the existing literature. They share some of their properties with polytopes, but, in general, are quite different from the latter. They find applications for example in finite element methods, quantum physics, and algebraic statistics. -
Gas Perm and Specialty Soft Lens Designs
Gas Perm and Specialty Soft Lens Designs We Fit Your PracticeTM Our commitment to you Dear Valued Customer, This resource guide is intended to provide you with a comprehensive look at the many proprietary lens designs X-Cel manufactures. Each page is a synopsis of key information and individual lens application. X-Cel has invested millions of dollars in state-of-the-art technology to provide the best possible quality. No one in the industry gives you such an extensive selection of custom manufactured or proprietary lens designs. We have two full time R&D engineers on staff committed to continuous improvement. We employ highly sophisticated distribution and shipping systems at all twelve X-Cel locations. Each of our seven locations is staffed with NCLE certified consultants who have the knowledge and experience to assist with difficult fits. As a gas perm and custom soft lens design manufacturer, we understand that working together to combine your years of education and clinical experience with our lens-specific consultation is the best way to ensure each fit is a good one. At X-Cel, we believe the patient in your chair is our patient too. We take the same care in consultation and manufacturing that you do in patient evaluation and fitting. Ultimately, everything we do reflects on you in the eyes of your patients. Their good vision and satisfaction is your goal. Your complete satisfaction is our goal. Combine all that with our competitive pricing and the best warranties in the industry and you have an organization committed to the very best. -
Soil Survey Field and Laboratory Methods Manual
SOIL SURVEY FIELD AND LABORATORY METHODS MANUAL Soil Survey Investigations Report No. 51 Version 1.0 Compiled and Edited by Rebecca Burt National Soil Survey Center Natural Resources Conservation Service U.S. Department of Agriculture Lincoln, Nebraska SOIL SURVEY FIELD AND LABORATORY METHODS MANUAL Soil Survey Investigations Report No. 51 Version 1.0 Compiled and Edited by Rebecca Burt National Soil Survey Center Natural Resources Conservation Service U.S. Department of Agriculture Lincoln, Nebraska Issued 2009 Trade names are used in this manual solely for the purpose of providing specific information. Mention of a trade name does not constitute a guarantee of the product by USDA nor does it imply an endorsement by USDA. USDA Nondiscrimination Statement The U.S. Department of Agriculture (USDA) prohibits discrimination in all its programs and activities on the basis of race, color, national origin, age, disability, and where applicable, sex, marital status, familial status, parental status, religion, sexual orientation, genetic information, political beliefs, reprisal, or because all or a part of an individual’s income is derived from any public assistance program. (Not all prohibited bases apply to all programs.) Persons with disabilities who require alternative means for communication of program information (Braille, large print, audiotape, etc.) should contact USDA’s TARGET Center at (202) 720-2600 (voice and TDD). To file a complaint of discrimination write to USDA, Director, Office of Civil Rights, 1400 Independence Avenue, S.W., Washington, D.C. 20250-9410 or call (800) 795-3272 (voice) or (202) 720-6382 (TDD). USDA is an equal opportunity provider and employer. Citation: Soil Survey Staff. -
12 Mar 2019 More Natural and More Elegant, to Which the Sphericon Gives Birth
The polycons: the sphericon (or tetracon) has found its family David Hirscha and Katherine A. Seatonb a Nachalat Binyamin Arts and Crafts Fair, Tel Aviv, Israel; b Department of Mathematics and Statistics, La Trobe University VIC 3086, Australia ARTICLE HISTORY Compiled March 14, 2019 ABSTRACT This paper introduces a new family of solids, which we call polycons, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that of other developable rollers such as the oloid and particular polysphericons. The paper concludes with a discussion of the polycons as stationary and kinetic works of art. KEYWORDS sphericon; polycons; tetracon; ruled surface; developable roller 1. Introduction In 1980 inventor David Hirsch, one of the authors of this paper, patented ‘a device for generating a meander motion’ [9], describing the object that is now known as the sphericon. This discovery was independent of that of woodturner Colin Roberts [22], which came to public attention through the writings of Stewart [29], P¨oppe [23] and Phillips [19] almost twenty years later. The object was named for how it rolls — overall in a line (like a sphere), but with turns about its vertices (like a cone). It was realised both by members of the woodturning [26, 27] and mathematical [17, 20] communities that the sphericon could be generalised to a series of objects, called sometimes poly- sphericons or, when precision is required, the (N,k)-icons. These objects are for the most part constructed from frusta of a number of cones of di↵ering apex angle and height. -
Geometric Valuations
GEOMETRIC VALUATIONS CORDELIA E. CSAR, RYAN K. JOHNSON, AND RONALD Z. LAMBERTY Work begun on June 12 2006. Ended on July 28 2006. 1 2 CORDELIAE.CSAR,RYANK.JOHNSON,ANDRONALDZ.LAMBERTY Introduction This paper is the result of a National Science Foundation-funded Research Experience for Undergraduates (REU) at the University of Notre Dame during the summer of 2006. The REU was directed by Professor Frank Connolly, and the research project was supervised by Professor Liviu Nicolaescu. The topic of our independent research project for this REU was Geometric Probability. Consequently, the first half of this paper is a study of the book Introduction to Geometric Probability by Daniel Klain and Gian-Carlo Rota. While closely following the text, we have attempted to clarify and streamline the presentation and ideas contained therein. In particular, we highlight and emphasize the key role played by the Radon transform in the classification of valuations. In the second part we take a closer look at the special case of valuations on polyhedra. Our primary focus in this project is a type of function called a “valuation”. A valuation as- sociates a number to each ”reasonable” subset of Rn so that the inclusion-exclusion principle is satisfied. Examples of such functions include the Euclidean volume and the Euler characteristic. Since the objects we are most interested lie in an Euclidean space and moreover we are interested in properties which are independent of the location of the objects in space, we are motivated to study “invariant” valuations on certain subsets of Rn. The goal of the first half of this paper, therefore, is to characterize all such valuations for certain natural collections of subsets in Rn.