Notes 19: Envelopes

Total Page:16

File Type:pdf, Size:1020Kb

Notes 19: Envelopes Notes 19: Envelopes For each value of t we are given the circle with center at (x = t; y = 0) and radius 1: Along the lines y = ±1 the curves become infinitely dense. Another way of stating this is to say that as we travel along the curve y = 1 or y = −1; we are tangent to each of the circles we touch. Such a curve is called an envelope. We give a second example. Consider the family of lines y − (1=a) = (−1=a2)(x − a): These are all tangent to the curve xy − 1 = 0: See the figures below. Assume that we are given a family, parametrized by t, of curves, that is, for each value of t we have a curve F (x; y; t) = 0: We show how to find the envelopes of this family of curves. We derive in two ways the equations for doing this. The first way is to consider the family as arising from the surface 3 in R given by F (x; y; t) = 0 and taking slices t = t0; t = t1; ··· at regular intervals and then projecting each slice to the x − y plane. The envelope occurs when the curves get close to each other. The projected curves are closer to each other exactly when the surface is steeper. Indeed, the curves become infinitely close exactly when the surface is vertical. The "direction" of the surface at a point is given by its tangent plane. The envelope is the projection of the points where the tangent plane is vertical. The tangent plane is vertical when the normal to the tangent has no vertical component. The vertical component of @F the normal to the plane is @t : Thus the envelope is the locus of points (x; y; t) so that @F F (x; y; t) = 0 and = 0: @t Here is a second derivation of these equations. The envelope is a curve γ(t) = (x(t); y(t)) so that 1. γ(t0) = (x0; y0) is a point of F (x; y; t0) = 0; and 2. the tangent line to γ(t) at t = t0 is the same as the tangent line to F (x; y; t0) = 0 at (x0; y0): We find conditions that insure that if (x0; y0) is on the envelope for t = t0), then the point gotten by starting at (x0; y0) and moving along the curve in the direction of the common tangent to γ and F (x; y; t0) is on 'the next' curve F (x; y; t0 + ) = 0; very small. Please see the figure below. The tangent to F (x; y; t0) at (x0; y0) is orthogonal to rF: Thus the tangent is the line @F @F through (x0; y0) in the direction ( @y ; − @x ) at the point (x0; y0): Please check that this is orthogonal to rF: Thus the next point on the envelope beyond (x0; y0) is @F @F (x ; y ) + ( ; − ): 0 0 @y @x 1 We want this to be on the next curve in our family of curves, that is, we want it to be on the curve F (x; y; t0 + ): Explicity we want @F @F F ((x ; y ) + ( ; − ); t + ) = 0: 0 0 @y @x 0 For small this is @F @F @F @F @F F (x ; y ; t ) + + (− ) + ≈ 0: 0 0 0 @x @y @y @x @t 1. The first term F (x0; y0; t0) = 0; since we have stipulated that (x0; y0) is on the level curve for t = t0: 2. The second and third summands sum to zero. @F 3. We conclude that @t = 0: An Envelope We consider the family of circles of radius 2 and centers at the points of the parabola parmetrized by (x = t; y = t2): We can write this as the family of curves F (x; y; t) = (x − t)2 + (y − t2)2 − 4 = 0: The envelope is given by 2 2 2 f1 = (x − t) + (y − t ) − 4 = 0; 2 f2 = 2(x − t)(−1) + 2(y − t )(−2t): The GroebnerBasis of the ideal I =< f1; f2 > is 2 4 6 2 4 g1 = −1156 + 688x − 191x + 16x + 544y + 30x y − 40x y+ 225y2 − 96x2y2 + 16x4y2 − 136y3 − 32x2y3 + 16y4 3 5 3 g2 = 7327t + 6929x − 2946x + 224x − 1928ty + 2922xy − 1480x y+ 128x5y − 768ty2 − 792xy2 − 224x3y2 − 896ty3 − 544xy3 + 128x3y3 + 256ty4 − 384xy4 2 4 g3 = −952 − 431tx + 159x + 16x − 320y + 12txy+ 214x2y − 32x4y + 366y2 + 48txy2 + 32x2y2 + 80y3 + 64txy3 − 32x2y3 − 32y4 2 3 5 g4 = −697t − 23x + 288tx + 174x − 32x − 108ty − 322xy+ 112x3y + 336ty2 − 32xy2 − 32x3y2 − 64ty3 + 96xy3 2 2 4 g5 = −128 + 135t + 26tx + 111x − 16x + 64y + 40txy+ 8x2y + 32y2 + 32txy2 − 16x2y2 − 16y3 We see that the envelope is contained in the curve g1 = 0: 2 Which points on the this curve actually arise from points on the envelope? From the Closure Theorem we know that all of the points except those on a subvariety arise from points on the envelope. To see which points lift to points on the envelope we examine the polynomials g2; ··· ; g5: We write these as polynomials in the variable t and extract the coefficient of the highest power of t: We see that for g5 the highest degree of t that appears is in the term 135t2: Since 135 6= 0 we can say that every point on the curve g1 lifts to a curve on the envelope. We ae concerned with real points. The extension theorem only works over the the complex numbers. Problems may arise. Given a point (x; y) on the envelope how many circles in our family pass through that point? For each value of t there is a circle in our family with center at (t; t2): Look at equation g4: It is of the form A4(x; y))t + B4(x; y) = 0: Hence if A(x; y) 6= 0; then there is a unique circle in the family passing through that point since we can solve t = −B4(x; y)=A4(x; y): This argument also works with g3 and g2: What happens at the points where A2(x; y) = A3(x; y) = A4(x; y) = 0: We find the points where A2 = A3 = A4 = 0: We do this by using Groebner bases. We find three points: (x = 0; y = 4:25); (x = ±:9368; y = 1:63988): These look as though they may be the singular points of the envelope. As an exercies draw the two circles that are 'tangent' to the envelope at these points. Are these points the singular points of the envelope? The envelope is given by the equations g1(x; y) = 0: The singular points are the points of the variety @g @g V (g ; 1 ; 1 ): 1 @x @y To verify it is the same points we check that @g @g g ; 1 ; 1 2< A ;A ;A > and 1 @x @y 1 2 3 @g @g A2;A2;A2 2< g ; 1 ; 1 > : 1 2 3 1 @x @y 3 Figure 1: First Example of an Envelope 4 2 0 Out[50]= -2 -4 -4 -2 0 2 4 6 Out[55]= 4 Figure 2: Example of Envelope 5 4 3 Out[58]= 2 1 0 0 1 2 3 4 5 5 Figure 3: Derivation of Envelope Equation 6 Figure 4: The Envelope 6 4 2 Out[3]= 0 -2 -4 -2 0 2 4 7.
Recommended publications
  • Strophoids, a Family of Cubic Curves with Remarkable Properties
    Hellmuth STACHEL STROPHOIDS, A FAMILY OF CUBIC CURVES WITH REMARKABLE PROPERTIES Abstract: Strophoids are circular cubic curves which have a node with orthogonal tangents. These rational curves are characterized by a series or properties, and they show up as locus of points at various geometric problems in the Euclidean plane: Strophoids are pedal curves of parabolas if the corresponding pole lies on the parabola’s directrix, and they are inverse to equilateral hyperbolas. Strophoids are focal curves of particular pencils of conics. Moreover, the locus of points where tangents through a given point contact the conics of a confocal family is a strophoid. In descriptive geometry, strophoids appear as perspective views of particular curves of intersection, e.g., of Viviani’s curve. Bricard’s flexible octahedra of type 3 admit two flat poses; and here, after fixing two opposite vertices, strophoids are the locus for the four remaining vertices. In plane kinematics they are the circle-point curves, i.e., the locus of points whose trajectories have instantaneously a stationary curvature. Moreover, they are projections of the spherical and hyperbolic analogues. For any given triangle ABC, the equicevian cubics are strophoids, i.e., the locus of points for which two of the three cevians have the same lengths. On each strophoid there is a symmetric relation of points, so-called ‘associated’ points, with a series of properties: The lines connecting associated points P and P’ are tangent of the negative pedal curve. Tangents at associated points intersect at a point which again lies on the cubic. For all pairs (P, P’) of associated points, the midpoints lie on a line through the node N.
    [Show full text]
  • A Special Conic Associated with the Reuleaux Negative Pedal Curve
    INTERNATIONAL JOURNAL OF GEOMETRY Vol. 10 (2021), No. 2, 33–49 A SPECIAL CONIC ASSOCIATED WITH THE REULEAUX NEGATIVE PEDAL CURVE LILIANA GABRIELA GHEORGHE and DAN REZNIK Abstract. The Negative Pedal Curve of the Reuleaux Triangle w.r. to a pedal point M located on its boundary consists of two elliptic arcs and a point P0. Intriguingly, the conic passing through the four arc endpoints and by P0 has one focus at M. We provide a synthetic proof for this fact using Poncelet’s porism, polar duality and inversive techniques. Additional interesting properties of the Reuleaux negative pedal w.r. to pedal point M are also included. 1. Introduction Figure 1. The sides of the Reuleaux Triangle R are three circular arcs of circles centered at each Reuleaux vertex Vi; i = 1; 2; 3. of an equilateral triangle. The Reuleaux triangle R is the convex curve formed by the arcs of three circles of equal radii r centered on the vertices V1;V2;V3 of an equilateral triangle and that mutually intercepts in these vertices; see Figure1. This triangle is mostly known due to its constant width property [4]. Keywords and phrases: conic, inversion, pole, polar, dual curve, negative pedal curve. (2020)Mathematics Subject Classification: 51M04,51M15, 51A05. Received: 19.08.2020. In revised form: 26.01.2021. Accepted: 08.10.2020 34 Liliana Gabriela Gheorghe and Dan Reznik Figure 2. The negative pedal curve N of the Reuleaux Triangle R w.r. to a point M on its boundary consist on an point P0 (the antipedal of M through V3) and two elliptic arcs A1A2 and B1B2 (green and blue).
    [Show full text]
  • Final Projects 1 Envelopes, Evolutes, and Euclidean Distance Degree
    Final Projects Directions: Work together in groups of 1-5 on the following four projects. We suggest, but do not insist, that you work in a group of > 1 people. For projects 1,2, and 3, your group should write a text le with (working!) code which answers the questions. Please provide documentation for your code so that we can understand what the code is doing. For project 4, your group should turn in a summary of observations and/or conjectures, but not necessarily code. Send your les via e-mail to [email protected] and [email protected]. It is preferable that the work is turned in by Thursday night. However, at the very latest, please send your work by 12:00 p.m. (noon) on Friday, August 16. 1 Envelopes, Evolutes, and Euclidean Distance Degree Read the section in Cox-Little-O'Shea's Ideals, Varieties, and Algorithms on envelopes of families of curves - this is in Chapter 3, section 4. We will expand slightly on the setup in Chapter 3 as follows. A rational family of lines is dened as a polynomial of the form Lt(x; y) 2 Q(t)[x; y] 1 L (x; y) = (A(t)x + B(t)y + C(t)); t G(t) where A(t);B(t);C(t); and G(t) are polynomials in t. The envelope of Lt(x; y) is the locus of and so that there is some satisfying and @Lt(x;y) . As usual, x y t Lt(x; y) = 0 @t = 0 we can handle this by introducing a new variable s which plays the role of 1=G(t).
    [Show full text]
  • Unified Higher Order Path and Envelope Curvature Theories in Kinematics
    UNIFIED HIGHER ORDER PATH AND ENVELOPE CURVATURE THEORIES IN KINEMATICS By MALLADI NARASH1HA SIDDHANTY I; Bachelor of Engineering Osmania University Hyderabad, India 1965 Master of Technology Indian Institute of Technology Madras, India 1969 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 1979 UNIFIED HIGHER ORDER PATH AND ENVELOPE CURVATURE THEORIES IN KINEMATICS Thesis Approved: Thesis Adviser J ~~(}W>p~ Dean of the Graduate College To Mukunda, Ananta, and the Mothers ACKNOWLEDGMENTS I take this opportunity to express my deep sense of gratitude and thanks to Professor A. H. Soni, chairman of my doctoral committee, for his keen interest in introducing me to various aspects of research in kinematics and his continued guidance with proper financial support. I thank the "Soni Family" for many warm dinners and hospitality. I am thankful to Professors L. J. Fila, P. F. Duvall, J. P. Chandler, and T. E. Blejwas for serving on my committee and for their advice and encouragement. Thanks are also due to Professor F. Freudenstein of Columbia University for serving on my committee and for many encouraging discussions on the subject. I thank Magnetic Peripherals, Inc., Oklahoma City, Oklahoma, my em­ ployer since the fall of 1977, for the computer facilities and the tui­ tion fee reimbursement. I thank all my fellow graduate students and friends at Stillwater and Oklahoma City for their warm friendship and discussions that made my studies in Oklahoma really rewarding. I remember and thank my former professors, B.
    [Show full text]
  • Generating the Envelope of a Swept Trivariate Solid
    GENERATING THE ENVELOPE OF A SWEPT TRIVARIATE SOLID Claudia Madrigal1 Center for Image Processing and Integrated Computing, and Department of Mathematics University of California, Davis Kenneth I. Joy2 Center for Image Processing and Integrated Computing Department of Computer Science University of California, Davis Abstract We present a method for calculating the envelope of the swept surface of a solid along a path in three-dimensional space. The generator of the swept surface is a trivariate tensor- product Bezier` solid and the path is a non-uniform rational B-spline curve. The boundary surface of the solid is the combination of parametric surfaces and an implicit surface where the determinant of the Jacobian of the defining function is zero. We define methods to calculate the envelope for each type of boundary surface, defining characteristic curves on the envelope and connecting them with triangle strips. The envelope of the swept solid is then calculated by taking the union of the envelopes of the family of boundary surfaces, defined by the surface of the solid in motion along the path. Keywords: trivariate splines; sweeping; envelopes; solid modeling. 1Department of Mathematics University of California, Davis, CA 95616-8562, USA; e-mail: madri- [email protected] 2Corresponding Author, Department of Computer Science, University of California, Davis, CA 95616-8562, USA; e-mail: [email protected] 1 1. Introduction Geometric modeling systems construct complex models from objects based on simple geo- metric primitives. As the needs of these systems become more complex, it is necessary to expand the inventory of geometric primitives and design operations.
    [Show full text]
  • Safe Domain and Elementary Geometry
    Safe domain and elementary geometry Jean-Marc Richard Laboratoire de Physique Subatomique et Cosmologie, Universit´eJoseph Fourier– CNRS-IN2P3, 53, avenue des Martyrs, 38026 Grenoble cedex, France November 8, 2018 Abstract A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a “safe” domain. In the simple cases of a constant force, harmonic potential, and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic section which can be derived either by simple calculus or by geometrical considerations. The case of harmonic forces reveals a subtle property of the maximal sum of distances within an ellipse. 1 Introduction A classical problem of classroom mechanics and military academies is the border of the so-called safe domain. A projectile is set off from a point O with an initial velocity v0 whose modulus is fixed by the intrinsic properties of the gun, while its direction can be varied arbitrarily. Its is well-known that, in absence of air friction, each trajectory is a parabola, and that, in any vertical plane containing O, the envelope of all trajectories is another parabola, which separates the points which can be shot from those which are out of reach. This will be shortly reviewed in Sec. 2. Amazingly, the problem of this envelope parabola can be addressed, and solved, in terms of elementary geometrical properties. arXiv:physics/0410034v1 [physics.ed-ph] 6 Oct 2004 In elementary mechanics, there are similar problems, that can be solved explicitly and lead to families of conic trajectories whose envelope is also a conic section.
    [Show full text]
  • On the Pentadeltoid'
    ON THE PENTADELTOID' BY R. P. STEPHENS Introductory. In his Orthocentric Properties of the Plane n-Line, Professor Morley| makes use of a curve of class 2n — 1, which he calls the A2"-1. Of these curves, the deltoid (n = 2) is the well-known three-cusped hypocycloid; but the curve of class five ( n. = 3 ), which I shall call Pentadeltoid, and more frequently represent by the symbol A'', is not so well known, though its shape and a few of its properties were known to Clifford $. I shall here prove some of the properties of the pentadeltoid. My purpose in presenting this curve is two- fold: 1. the discussion of a curve of many interesting properties; 2. the illus- tration of the use of vector analysis as a tool in the metric geometry of the plane. The line equation of a curve is here uniformly expressed by means of con- jugate coordinates,§ while the point equation is expressed by means of what is commonly called the map equation, that is, a point of the curve is expressed as a function of a parameter t, which is limited to the unit circle and is, therefore, always equal in absolute value to unity. Throughout I shall consider y and b as conjugates of x and a, respectively; that is, if X and Y are rectangular coordinates of a point, then x = X+iY, y = X-iY, and similarly for a and b.\\ Some of the properties of the curves of class 2« — 1 and of degree 2n, easily inferred from those of the A5, are also proved, but there has been no attempt at making a complete study of the general case.
    [Show full text]
  • Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse
    STEINER’S HAT: A CONSTANT-AREA DELTOID ASSOCIATED WITH THE ELLIPSE RONALDO GARCIA, DAN REZNIK, HELLMUTH STACHEL, AND MARK HELMAN Abstract. The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties. Keywords curve, envelope, ellipse, pedal, evolute, deltoid, Poncelet, osculat- ing, orthologic. MSC 51M04 and 51N20 and 65D18 1. Introduction Given an ellipse with non-zero semi-axes a,b centered at O, let M be a point in the plane. The NegativeE Pedal Curve (NPC) of with respect to M is the envelope of lines passing through points P (t) on the boundaryE of and perpendicular to [P (t) M] [4, pp. 349]. Well-studied cases [7, 14] includeE placing M on (i) the major− axis: the NPC is a two-cusp “fish curve” (or an asymmetric ovoid for low eccentricity of ); (ii) at O: this yielding a four-cusp NPC known as Talbot’s Curve (or a squashedE ellipse for low eccentricity), Figure 1. arXiv:2006.13166v5 [math.DS] 5 Oct 2020 Figure 1. The Negative Pedal Curve (NPC) of an ellipse with respect to a point M on the plane is the envelope of lines passing through P (t) on the boundary,E and perpendicular to P (t) M. Left: When M lies on the major axis of , the NPC is a two-cusp “fish” curve. Right: When− M is at the center of , the NPC is 4-cuspE curve with 2-self intersections known as Talbot’s Curve [12].
    [Show full text]
  • Iterating Evolutes and Involutes
    Iterating evolutes and involutes Work in progress with M. Arnold, D. Fuchs, I. Izmestiev, and E. Tsukerman Integrability in Mechanics and Geometry: Theory and Computations June 2015 1 2 Evolutes and involutes Evolute: the envelope of the normals, the locus of the centers of curvature; free from inflections and has zero algebraic length. Involute: string construction; come in 1-parameter families. 3 Hedgehogs Wave fronts without inflection and total rotation 2π, given by their support function p(α). ....................................... ....................................... ...................................... .................. .................. π ........... .......... ........ ........ .............. α + ...... .... ...... ......... ..... .... ..... ...... .... .... ........ 2 .... .... ....... .... .... .... ...... .... .. .... ....... .. .. ........... .. ..... ......... ....... p!(α).......... .... ....... .. ........... .... .. ...... ... .......... .... ... ....... .... .......... .... .... ....... .... ......... .... .......... ..... ........ .... ......... ..... O ..... .... ......... ....... γ ..... .......... ........ ..... ........... .... ........... •......p(α) ................ .... ..................... ...... ........................... .... ....................................................................................................... ....... .... ...... ....... .... ............ .... .......... .... ....... ..... ..... ....... ..... .α The evolute map: p(α) 7! p0(α − π=2), invertible on functions with zero
    [Show full text]
  • Compendium of Definitions and Acronyms for Rail Systems
    APTA STANDARDS DEVELOPMENT PROGRAM APTA STD-ADMIN-GL-001-19 GUIDELINES First Published June 20, 2019 American Public Transportation Association 1300 I Street NW, 12th Floor East, Washington, DC, 20005 Compendium of Definitions and Acronyms for Rail Systems Abstract: This compendium was developed by the Technical Services & Innovation Department and published by the American Public Transportation Association (APTA) to provide a glossary of commonly used definitions and acronyms in documents such as standards, recommended practices, and guidelines, so there is consistency within the rail transportation industry. Summary: APTA, through its subsidiary the North American Transit Services Association (NATSA), develops standards, recommended practices and guidelines for the benefit of public rail transportation. These tasks are accomplished by working groups consisting of members from rail transit agencies, manufacturers, consultants, engineers and other interested groups. Through the development of these documents, working groups have created a wide array of terms and abbreviations, many with varying definitions. This compendium has been developed to standardize the usage of such definitions and acronyms as it relates to rail operations, maintenance practices, designs and specifications. This document is dynamic in nature in that, over time, additional definitions and acronyms will be included. It is APTA’s intention that the document be sufficiently expanded at some point in the future to provide common usage of terms encompassing all rail industry requirements. Scope and purpose: This compendium applies to all rail agencies that operate commuter rail, heavy rail (subway systems), light rail, streetcars, and trolleys. The usage of these definitions and acronyms are voluntary. Nevertheless, it is the desire of APTA that the rail industry apply these terms to all documents such as standards, recommended practices, standard operating practices, standard maintenance practices, rail agency policies and procedures, and agency rule books.
    [Show full text]
  • Evolute and Evolvente.Pdf
    Evolute and involute (evolvent) Interactive solutions to problem s of differential geometry © S.N Nosulya, V.V Shelomovskii. Topical sets for geometry, 2012. © D.V. Shelomovskii. GInMA, 2012. http://www.deoma-cmd.ru/ Materials of the interactive set can be used in Undergraduate education for the study of the foundations of differential geometry, it can be used for individual study methods of differential geometry. All figures are interactive if you install GInMA program from http://www.deoma-cmd.ru/ Conception of Evolute In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve. Apollonius (c. 200 BC) discussed evolutes in Book V of his Conics. However, Huygens is sometimes credited with being the first to study them (1673). Equation of evolute Let γ be a plane curve containing points X⃗T =(x , y). The unit normal vector to the curve is ⃗n. The curvature of γ is K . The center of curvature is the center of the osculating circle. It lies on the normal 1 line through γ at a distance of from γ in the direction determined by the sign of K . In symbols, K the center of curvature lies at the point ζ⃗T =(ξ ,η). As X⃗ varies, the center of curvature traces out a ⃗n plane curve, the evolute of γ and evolute equation is: ζ⃗= X⃗ + . (1) K Case 1 Let γ is given a general parameterization, say X⃗T =(x(t) , y(t)) Then the parametric equation of x' y' '−x ' ' y ' the evolute can be expressed in terms of the curvature K = .
    [Show full text]
  • Binocular Vision and Ocular Motility SIXTH EDITION
    Binocular Vision and Ocular Motility SIXTH EDITION Binocular Vision and Ocular Motility THEORY AND MANAGEMENT OF STRABISMUS Gunter K. von Noorden, MD Emeritus Professor of Ophthalmology Cullen Eye Institute Baylor College of Medicine Houston, Texas Clinical Professor of Ophthalmology University of South Florida College of Medicine Tampa, Florida Emilio C. Campos, MD Professor of Ophthalmology University of Bologna Chief of Ophthalmology S. Orsola-Malpighi Teaching Hospital Bologna, Italy Mosby A Harcourt Health Sciences Company St. Louis London Philadelphia Sydney Toronto Mosby A Harcourt Health Sciences Company Editor-in-Chief: Richard Lampert Acquisitions Editor: Kimberley Cox Developmental Editor: Danielle Burke Project Manager: Agnes Byrne Production Manager: Peter Faber Illustration Specialist: Lisa Lambert Book Designer: Ellen Zanolle Copyright ᭧ 2002, 1996, 1990, 1985, 1980, 1974 by Mosby, Inc. All rights reserved. No part of this publication may be reproduced or transmit- ted in any form or by any means, electronic or mechanical, including photo- copy, recording, or any information storage and retrieval system, without per- mission in writing from the publisher. NOTICE Ophthalmology is an ever-changing field. Standard safety precautions must be followed, but as new research and clinical experience broaden our knowledge, changes in treatment and drug therapy may become necessary or appropriate. Readers are advised to check the most current product information provided by the manufacturer of each drug to be administered to verify the recommended dose, the method and duration of administration, and contraindications. It is the responsibility of the treating physician, relying on experience and knowledge of the patient, to determine dosages and the best treatment for each individual pa- tient.
    [Show full text]