The Celestial Sphere the Celestial Sphere
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Angle Chasing
Angle Chasing Ray Li June 12, 2017 1 Facts you should know 1. Let ABC be a triangle and extend BC past C to D: Show that \ACD = \BAC + \ABC: 2. Let ABC be a triangle with \C = 90: Show that the circumcenter is the midpoint of AB: 3. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove that H is the incenter of 4DEF . 4. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove (i) that A; E; F; H lie on a circle diameter AH and (ii) that B; E; F; C lie on a circle with diameter BC. 5. Let ABC be a triangle with circumcenter O and orthocenter H: Show that \BAH = \CAO: 6. Let ABC be a triangle with circumcenter O and orthocenter H and let AH and AO meet the circumcircle at D and E, respectively. Show (i) that H and D are symmetric with respect to BC; and (ii) that H and E are symmetric with respect to the midpoint BC: 7. Let ABC be a triangle with altitudes AD; BE; and CF: Let M be the midpoint of side BC. Show that ME and MF are tangent to the circumcircle of AEF: 8. Let ABC be a triangle with incenter I, A-excenter Ia, and D the midpoint of arc BC not containing A on the circumcircle. Show that DI = DIa = DB = DC: 9. Let ABC be a triangle with incenter I and D the midpoint of arc BC not containing A on the circumcircle. -
Examples of Manifolds
Examples of Manifolds Example 1 (Open Subset of IRn) Any open subset, O, of IRn is a manifold of dimension n. One possible atlas is A = (O, ϕid) , where ϕid is the identity map. That is, ϕid(x) = x. n Of course one possible choice of O is IR itself. Example 2 (The Circle) The circle S1 = (x,y) ∈ IR2 x2 + y2 = 1 is a manifold of dimension one. One possible atlas is A = {(U , ϕ ), (U , ϕ )} where 1 1 1 2 1 y U1 = S \{(−1, 0)} ϕ1(x,y) = arctan x with − π < ϕ1(x,y) <π ϕ1 1 y U2 = S \{(1, 0)} ϕ2(x,y) = arctan x with 0 < ϕ2(x,y) < 2π U1 n n n+1 2 2 Example 3 (S ) The n–sphere S = x =(x1, ··· ,xn+1) ∈ IR x1 +···+xn+1 =1 n A U , ϕ , V ,ψ i n is a manifold of dimension . One possible atlas is 1 = ( i i) ( i i) 1 ≤ ≤ +1 where, for each 1 ≤ i ≤ n + 1, n Ui = (x1, ··· ,xn+1) ∈ S xi > 0 ϕi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n Vi = (x1, ··· ,xn+1) ∈ S xi < 0 ψi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n So both ϕi and ψi project onto IR , viewed as the hyperplane xi = 0. Another possible atlas is n n A2 = S \{(0, ··· , 0, 1)}, ϕ , S \{(0, ··· , 0, −1)},ψ where 2x1 2xn ϕ(x , ··· ,xn ) = , ··· , 1 +1 1−xn+1 1−xn+1 2x1 2xn ψ(x , ··· ,xn ) = , ··· , 1 +1 1+xn+1 1+xn+1 are the stereographic projections from the north and south poles, respectively. -
Constructing a Galactic Coordinate System Based on Near-Infrared and Radio Catalogs
A&A 536, A102 (2011) Astronomy DOI: 10.1051/0004-6361/201116947 & c ESO 2011 Astrophysics Constructing a Galactic coordinate system based on near-infrared and radio catalogs J.-C. Liu1,2,Z.Zhu1,2, and B. Hu3,4 1 Department of astronomy, Nanjing University, Nanjing 210093, PR China e-mail: [jcliu;zhuzi]@nju.edu.cn 2 key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, PR China 3 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, PR China 4 Graduate School of Chinese Academy of Sciences, Beijing 100049, PR China e-mail: [email protected] Received 24 March 2011 / Accepted 13 October 2011 ABSTRACT Context. The definition of the Galactic coordinate system was announced by the IAU Sub-Commission 33b on behalf of the IAU in 1958. An unrigorous transformation was adopted by the Hipparcos group to transform the Galactic coordinate system from the FK4-based B1950.0 system to the FK5-based J2000.0 system or to the International Celestial Reference System (ICRS). For more than 50 years, the definition of the Galactic coordinate system has remained unchanged from this IAU1958 version. On the basis of deep and all-sky catalogs, the position of the Galactic plane can be revised and updated definitions of the Galactic coordinate systems can be proposed. Aims. We re-determine the position of the Galactic plane based on modern large catalogs, such as the Two-Micron All-Sky Survey (2MASS) and the SPECFIND v2.0. This paper also aims to propose a possible definition of the optimal Galactic coordinate system by adopting the ICRS position of the Sgr A* at the Galactic center. -
An Introduction to Topology the Classification Theorem for Surfaces by E
An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5. -
Point of Concurrency the Three Perpendicular Bisectors of a Triangle Intersect at a Single Point
3.1 Special Segments and Centers of Classifications of Triangles: Triangles By Side: 1. Equilateral: A triangle with three I CAN... congruent sides. Define and recognize perpendicular bisectors, angle bisectors, medians, 2. Isosceles: A triangle with at least and altitudes. two congruent sides. 3. Scalene: A triangle with three sides Define and recognize points of having different lengths. (no sides are concurrency. congruent) Jul 249:36 AM Jul 249:36 AM Classifications of Triangles: Special Segments and Centers in Triangles By angle A Perpendicular Bisector is a segment or line 1. Acute: A triangle with three acute that passes through the midpoint of a side and angles. is perpendicular to that side. 2. Obtuse: A triangle with one obtuse angle. 3. Right: A triangle with one right angle 4. Equiangular: A triangle with three congruent angles Jul 249:36 AM Jul 249:36 AM Point of Concurrency The three perpendicular bisectors of a triangle intersect at a single point. Two lines intersect at a point. The point of When three or more lines intersect at the concurrency of the same point, it is called a "Point of perpendicular Concurrency." bisectors is called the circumcenter. Jul 249:36 AM Jul 249:36 AM 1 Circumcenter Properties An angle bisector is a segment that divides 1. The circumcenter is an angle into two congruent angles. the center of the circumscribed circle. BD is an angle bisector. 2. The circumcenter is equidistant to each of the triangles vertices. m∠ABD= m∠DBC Jul 249:36 AM Jul 249:36 AM The three angle bisectors of a triangle Incenter properties intersect at a single point. -
Basic Principles of Celestial Navigation James A
Basic principles of celestial navigation James A. Van Allena) Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242 ͑Received 16 January 2004; accepted 10 June 2004͒ Celestial navigation is a technique for determining one’s geographic position by the observation of identified stars, identified planets, the Sun, and the Moon. This subject has a multitude of refinements which, although valuable to a professional navigator, tend to obscure the basic principles. I describe these principles, give an analytical solution of the classical two-star-sight problem without any dependence on prior knowledge of position, and include several examples. Some approximations and simplifications are made in the interest of clarity. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1778391͔ I. INTRODUCTION longitude ⌳ is between 0° and 360°, although often it is convenient to take the longitude westward of the prime me- Celestial navigation is a technique for determining one’s ridian to be between 0° and Ϫ180°. The longitude of P also geographic position by the observation of identified stars, can be specified by the plane angle in the equatorial plane identified planets, the Sun, and the Moon. Its basic principles whose vertex is at O with one radial line through the point at are a combination of rudimentary astronomical knowledge 1–3 which the meridian through P intersects the equatorial plane and spherical trigonometry. and the other radial line through the point G at which the Anyone who has been on a ship that is remote from any prime meridian intersects the equatorial plane ͑see Fig. -
Constellations with Prominent Stars That Can Be Found Near the Meridian at 10 Pm on January 15
ONSTELLATIONS C Altitude Ruler The rotation of the Earth on its axis causes the stars to rise and set each evening. In addition, the orbit of the Earth around the Sun places different regions of the sky in our Horizon night-time view. The PLANISPHERE is an extremely useful tool for finding stars and 10 constellation in the sky, depicting not only what is currently in the sky but it also allows the 20 prediction of the rising and setting times of various celestial objects. 30 THE LAYOUT OF THE PLANISPHERE 40 50 The outer circumference of the dark blue circular disk (which is called the star wheel) you’ll notice that the wheel is divided into the 12 months, and that each month is divided into 60 individual dates. The star wheel rotates about the brass fastener, which represents the 70 North Celestial Pole. The frame of the planisphere has times along the outer edge. 80 Holding the planisphere on the southern corner you'll see "midnight" at the top. Moving Zenith counterclockwise, notice how the hours progress, through 1 AM, 2 AM, and so on through "noon" at the bottom. The hours then proceed through the afternoon and evening (1 PM, 2 PM, etc.) back toward midnight. Once you have the wheel set properly for the correct time and day, the displayed part represents what you see if you stand with the star and planet locator held directly over your head with the brass fastener toward the north. (Notice that the compass directions are also written on the corners of the frame.) Of course, you don't have to actually stand that way to make use of the Star and Planet Locator--this is just a description to help you understand what is displayed. -
Equatorial and Cartesian Coordinates • Consider the Unit Sphere (“Unit”: I.E
Coordinate Transforms Equatorial and Cartesian Coordinates • Consider the unit sphere (“unit”: i.e. declination the distance from the center of the (δ) sphere to its surface is r = 1) • Then the equatorial coordinates Equator can be transformed into Cartesian coordinates: right ascension (α) – x = cos(α) cos(δ) – y = sin(α) cos(δ) z x – z = sin(δ) y • It can be much easier to use Cartesian coordinates for some manipulations of geometry in the sky Equatorial and Cartesian Coordinates • Consider the unit sphere (“unit”: i.e. the distance y x = Rcosα from the center of the y = Rsinα α R sphere to its surface is r = 1) x Right • Then the equatorial Ascension (α) coordinates can be transformed into Cartesian coordinates: declination (δ) – x = cos(α)cos(δ) z r = 1 – y = sin(α)cos(δ) δ R = rcosδ R – z = sin(δ) z = rsinδ Precession • Because the Earth is not a perfect sphere, it wobbles as it spins around its axis • This effect is known as precession • The equatorial coordinate system relies on the idea that the Earth rotates such that only Right Ascension, and not declination, is a time-dependent coordinate The effects of Precession • Currently, the star Polaris is the North Star (it lies roughly above the Earth’s North Pole at δ = 90oN) • But, over the course of about 26,000 years a variety of different points in the sky will truly be at δ = 90oN • The declination coordinate is time-dependent albeit on very long timescales • A precise astronomical coordinate system must account for this effect Equatorial coordinates and equinoxes • To account -
Median and Altitude of a Triangle Goal: • to Use Properties of the Medians
Median and Altitude of a Triangle Goal: • To use properties of the medians of a triangle. • To use properties of the altitudes of a triangle. Median of a Triangle Median of a Triangle – a segment whose endpoints are the vertex of a triangle and the midpoint of the opposite side. Vertex Median Median of an Obtuse Triangle A D Point of concurrency “P” or centroid E P C B F Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. A D If P is the centroid of ABC, then AP=2 AF E 3 P C CP=22CE and BP= BD B F 33 Example - Medians of a Triangle P is the centroid of ABC. PF 5 Find AF and AP A D E P C B F 5 Median of an Acute Triangle A Point of concurrency “P” or centroid E D P C B F Median of a Right Triangle A F Point of concurrency “P” or centroid E P C B D The three medians of an obtuse, acute, and a right triangle always meet inside the triangle. Altitude of a Triangle Altitude of a triangle – the perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side A altitude C B Altitude of an Acute Triangle A Point of concurrency “P” or orthocenter P C B The point of concurrency called the orthocenter lies inside the triangle. Altitude of a Right Triangle The two legs are the altitudes A The point of concurrency called the orthocenter lies on the triangle. -
Earth-Centred Universe
Earth-centred Universe The fixed stars appear on the celestial sphere Earth rotates in one sidereal day The solar day is longer by about 4 minutes → scattered sunlight obscures the stars by day The constellations are historical → learn to recognise: Ursa Major, Ursa Minor, Cassiopeia, Pegasus, Auriga, Gemini, Orion, Taurus Sun’s Motion in the Sky The Sun moves West to East against the background of Stars Stars Stars stars Us Us Us Sun Sun Sun z z z Start 1 sidereal day later 1 solar day later Compared to the stars, the Sun takes on average 3 min 56.5 sec extra to go round once The Sun does not travel quite at a constant speed, making the actual length of a solar day vary throughout the year Pleiades Stars near the Sun Sun Above the atmosphere: stars seen near the Sun by the SOHO probe Shield Sun in Taurus Image: Hyades http://sohowww.nascom.nasa.g ov//data/realtime/javagif/gifs/20 070525_0042_c3.gif Constellations Figures courtesy: K & K From The Beauty of the Heavens by C. F. Blunt (1842) The Celestial Sphere The celestial sphere rotates anti-clockwise looking north → Its fixed points are the north celestial pole and the south celestial pole All the stars on the celestial equator are above the Earth’s equator How high in the sky is the pole star? It is as high as your latitude on the Earth Motion of the Sky (animated ) Courtesy: K & K Pole Star above the Horizon To north celestial pole Zenith The latitude of Northern horizon Aberdeen is the angle at 57º the centre of the Earth A Earth shown in the diagram as 57° 57º Equator Centre The pole star is the same angle above the northern horizon as your latitude. -
And Are Lines on Sphere B That Contain Point Q
11-5 Spherical Geometry Name each of the following on sphere B. 3. a triangle SOLUTION: are examples of triangles on sphere B. 1. two lines containing point Q SOLUTION: and are lines on sphere B that contain point Q. ANSWER: 4. two segments on the same great circle SOLUTION: are segments on the same great circle. ANSWER: and 2. a segment containing point L SOLUTION: is a segment on sphere B that contains point L. ANSWER: SPORTS Determine whether figure X on each of the spheres shown is a line in spherical geometry. 5. Refer to the image on Page 829. SOLUTION: Notice that figure X does not go through the pole of ANSWER: the sphere. Therefore, figure X is not a great circle and so not a line in spherical geometry. ANSWER: no eSolutions Manual - Powered by Cognero Page 1 11-5 Spherical Geometry 6. Refer to the image on Page 829. 8. Perpendicular lines intersect at one point. SOLUTION: SOLUTION: Notice that the figure X passes through the center of Perpendicular great circles intersect at two points. the ball and is a great circle, so it is a line in spherical geometry. ANSWER: yes ANSWER: PERSEVERANC Determine whether the Perpendicular great circles intersect at two points. following postulate or property of plane Euclidean geometry has a corresponding Name two lines containing point M, a segment statement in spherical geometry. If so, write the containing point S, and a triangle in each of the corresponding statement. If not, explain your following spheres. reasoning. 7. The points on any line or line segment can be put into one-to-one correspondence with real numbers. -
The Sky Tonight
MARCH POUTŪ-TE-RANGI HIGHLIGHTS Conjunction of Saturn and the Moon A conjunction is when two astronomical objects appear close in the sky as seen THE- SKY TONIGHT- - from Earth. The planets, along with the TE AHUA O TE RAKI I TENEI PO Sun and the Moon, appear to travel across Brightest Stars our sky roughly following a path called the At this time of the year, we can see the ecliptic. Each body travels at its own speed, three brightest stars in the night sky. sometimes entering ‘retrograde’ where they The brightness of a star, as seen from seem to move backwards for a period of time Earth, is measured as its apparent (though the backwards motion is only from magnitude. Pictured on the cover is our vantage point, and in fact the planets Sirius, the brightest star in our night sky, are still orbiting the Sun normally). which is 8.6 light-years away. Sometimes these celestial bodies will cross With an apparent magnitude of −1.46, paths along the ecliptic line and occupy the this star can be found in the constellation same space in our sky, though they are still Canis Major, high in the northern sky. millions of kilometres away from each other. Sirius is actually a binary star system, consisting of Sirius A which is twice the On March 19, the Moon and Saturn will be size of the Sun, and a faint white dwarf in conjunction. While the unaided eye will companion named Sirius B. only see Saturn as a bright star-like object (Saturn is the eighth brightest object in our Sirius is almost twice as bright as the night sky), a telescope can offer a spectacular second brightest star in the night sky, view of the ringed planet close to our Moon.