9/24/2002, COORDINATES-DISTANCES O. Knill, Math 21aInstructor: Oliver KnillOffice: SC 434Phone: 55549Email: knill@mathUrl: www.math.harvard.edu/˜knillCA: Rene Shen, rshen@fas, 3-0094CA: Eduardo Saverin, saverin@fas, 345-9638Book: Multivariable calculus by J. StewartWeb: www.courses.fas.harvard.edu/˜math21aHomework: Due ThursdaysDue 9/26: 9.1: 5, 8, 10, 14, 28QC: Su-Th, 8-10 Loker CommonsCARTESIAN COORDINATE SYSTEMS. Points on the line are labeled by 1 coordinate x, points in the planeare labeled by 2 coordinates (x, y). Points in space are determined by three coordinates (x, y, z).1D space = line = 2 half lines 2D space = plane = 4 quadrants3D space = space =8 octantsxxy(x,y)xyz(x,y,z)CHOICEOF COOR-DINATESYSTEMThe directions of the x, y or z-coordinate axes determine one of many possible coordinate systems. On earthfor example, the z-axes usually points up. But this direction of course depends on the place.QUESTION: in 3D graphics (computer games, virtual reality, ray tracing) it is custom to have the y axis is up,the x axis to the right and the z axis in front. This is the ”photographers” point of view”. The photograph isthe x-y plane, the debth is the z axes. Is this a left or righthanded coordinate system?(Remark: The area in graphics memory reserved for storing the z-axis is called the ”z-buffer”. This is ueful for”hidden line removal” in 2D renderinging of a 3D scene. The z-axis is perpendicular to the screen with valuesincreasing towards the viewer. Any point whose z coordinate is less than the corresponding z-buffer value willbe hidden behind some feature which has already been plotted.)DISTANCE. The distance between two points P = (x, y, z) and Q = (a, b, c) isd(P, Q) =p(x − a)2+ (y − b)2+ (z − c)2.This can be derived from Pythagoras.PARITY. We usually work with a right handed coordinate system.The the ”right hand rule”: thumb=x-direction index finger=y-direction and middle finger=z-direction to check that the coordi-nate system is ”right handed”.Partity plays a role in Biology (orientation of DNA or Proteins)or particle physics, (”parity violation”: physical laws are not thesame when we look at them in the mirror).GEOMETRICAL OBJECTS. curves, surfaces and bodies are examples of geometrical objects which can bedescribed using functions of several variables. We look at some of them here to get some feel about space.The objects will be treated later in more detail.SPHERE. The sphere is the collection of points which have a fixed distance r from a given point (a, b, c). Theequation is (x − a)2+ (y − b)2+ (z − c)2= r2. ICE Completion of square.PROBLEM. Draw all points (x, y) which satisfy x ≥3 and y ≤ 2}.SOLUTION. It is a quadrant with corner at (3, 2).PROBLEM.The coordinate axes x = 0, y = 0 divide the plane into 4 regions called quadrants. Similarly, the coordinateplanes x = 0, y = 0 and z = 0 devide the space into 8 regions called octants. How many ”hyperregions” arethere in four dimensional ”hyperspace” which is labeled by points with 4 coordinates (t, x, y, z)? Give a pointfor each of these hyperregions.SOLUTION. There are 16 regions. Points are (±1, ±1, ±1 ± 1)IN CLASS EXERCICE (ICE). Tracking by PDA’s. (Triangularisation).PROBLEM.We want to draw the set of all points (x, y, z) which satisfy x + 2y − 3z = 2.a) One way to do this is to figure out, where the set intersects the x-coordinate axes, the y-coordiante axes andthe z-coordinate axes and put a plane through these three points. Find these intersection points.b) An other way to visualize the set is to find the traces, the intersections with the coordinate planes x = 0,y = 0 or z = 0. Find these traces.c) Make a drawing of the plane which shows the intersepts found in a) and the traces found in b).PROBLEM. We have been drawing the coordinate axes in a particular way. Other coordinate axes can beobtained by rotation. Draw a coordinate system which can not be turned into the usual coordinate system.SOLUTION. Use the left instead of the right hand in the ”right hand rule”.HISTORICAL. Ren´e Descartes (1596-1650) is credited for intro-ducting the Cartesian coordinate system.Annectote: ”In 1649 Queen Christina of Sweden persuadedDescartes to go to Stockholm. However the Queen wanted to drawtangents at 5 a.m. and Descartes broke the habit of his lifetime ofgetting up at 11 o’clock. After only a few months in the cold northernclimate, walking to the palace for 5 o’clock every morning, he died
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