DOCSLIB.ORG

Geometry (Lines) SOL 4.14?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry (Lines) SOL 4.14? Geometry (Lines) lines, segments, rays, points, angles, intersecting, parallel, & perpendicular Point “A point is an exact location in space. It has no length or width.” Points have names; represented with a Capital Letter. – Example: A a Lines “A line is a collection of points going on and on infinitely in both directions. It has no endpoints.” A straight line that continues forever It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified because it has arrows on the ends. It is named by “a single lower case letter”. – Example: “line a” A B C D Line Segment “A line segment is part of a line. It has two endpoints and includes all the points between those endpoints. “ A straight line that stops It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified by points at the ends It is named by the Capital Letter End Points – Example: “line segment AB” or “line segment AD” C B A Ray “A ray is part of a line. It has one endpoint and continues on and on in one direction.” A straight line that stops on one end and keeps going on the other. It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified by a point at one end and an arrow at the other. It can be named by saying the endpoint first and then say the name of one other point on the ray. – Example: “Ray AC” or “Ray AB” C Angles A B Two Rays That Have the Same Endpoint Form an Angle. This Endpoint Is Called the Vertex. Angles Are Found Wherever Lines and Line Segments Intersect. C A Angles A B An Angle Can Be Named in Three Different Ways by Using – Three Letters to Name, in This Order, Example: “Angle BAC” A Point on One Ray, The Vertex, and 1 A Point on the Other Ray; – One Letter at the Vertex; “Angle A” – Or a Number Written Inside the Rays of the Angle. Example: “Angle 1” C Angles A B There are 3 types of angles – Acute Angle: Smaller than 90 degree opening C A B – Obtuse Angle: Larger than 90 degree opening C A B – Right Angle: 90 degree opening C A B Intersecting Lines “Intersecting lines are lines that cross and have one point in common.” Example: “Line AC intersects Line DE at Point B” D C B A E Perpendicular Lines “Perpendicular lines are special intersecting lines that form right angles (square corners) where they intersect.” Parallel Lines “Parallel lines are lines that lie on the same flat surface (plane) and never cross. Parallel lines are always the same distance apart and do not share any points.” Example: “Line AB is Parallel to Line CD” C D A B Name That Line! 1. 3. a A B C D A C 5. 2. A B 4. C B A Name That Angle! 7. 6. C C A B A B 8. C A B Name The Type of Lines! 11. 9. D C B A E 10. Now, Draw Your Own Lines. Point Intersecting Lines Line Perpendicular Line Segment Lines Ray Parallel Lines Angles – Acute – Obtuse – Right .
Load more

Recommended publications
  • Unsteadiness in Flow Over a Flat Plate at Angle-Of-Attack at Low Reynolds Numbers∗
    Unsteadiness in Flow over a Flat Plate at Angle-of-Attack at Low Reynolds Numbers∗ Kunihiko Taira,† William B. Dickson,‡ Tim Colonius,§ Michael H. Dickinson¶ California Institute of Technology, Pasadena, California, 91125 Clarence W. Rowleyk Princeton University, Princeton, New Jersey, 08544 Flow over an impulsively started low-aspect-ratio flat plate at angle-of-attack is inves- tigated for a Reynolds number of 300. Numerical simulations, validated by a companion experiment, are performed to study the influence of aspect ratio, angle of attack, and plan- form geometry on the interaction of the leading-edge and tip vortices and resulting lift and drag coefficients. Aspect ratio is found to significantly influence the wake pattern and the force experienced by the plate. For large aspect ratio plates, leading-edge vortices evolved into hairpin vortices that eventually detached from the plate, interacting with the tip vor- tices in a complex manner. Separation of the leading-edge vortex is delayed to some extent by having convective transport of the spanwise vorticity as observed in flow over elliptic, semicircular, and delta-shaped planforms. The time at which lift achieves its maximum is observed to be fairly constant over different aspect ratios, angles of attack, and planform geometries during the initial transient. Preliminary results are also presented for flow over plates with steady actuation near the leading edge. I. Introduction In recent years, flows around flapping insect wings have been investigated and, in particular, it has been observed that the leading-edge vortex (LEV) formed during stroke reversal remains stably attached throughout the wing stroke, greatly enhancing lift.1, 2 Such LEVs are found to be stable over a wide range of angles of attack and for Reynolds number over a range of Re ≈ O(102) − O(104).
    [Show full text]
  • Points, Lines, and Planes a Point Is a Position in Space. a Point Has No Length Or Width Or Thickness
    Points, Lines, and Planes A Point is a position in space. A point has no length or width or thickness. A point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. A A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. It does not have a beginning or end. A B C D A line consists of infinitely many points. The four points A, B, C, D are all on the same line. Postulate: Two points determine a line. We name a line by using any two points on the line, so the above line can be named as any of the following: ! ! ! ! ! AB BC AC AD CD Any three or more points that are on the same line are called colinear points. In the above, points A; B; C; D are all colinear. A Ray is part of a line that has a beginning point, and extends indefinitely to one direction. A B C D A ray is named by using its beginning point with another point it contains. −! −! −−! −−! In the above, ray AB is the same ray as AC or AD. But ray BD is not the same −−! ray as AD. A (line) segment is a finite part of a line between two points, called its end points. A segment has a finite length. A B C D B C In the above, segment AD is not the same as segment BC Segment Addition Postulate: In a line segment, if points A; B; C are colinear and point B is between point A and point C, then AB + BC = AC You may look at the plus sign, +, as adding the length of the segments as numbers.
    [Show full text]
  • The Mass of an Asymptotically Flat Manifold
    The Mass of an Asymptotically Flat Manifold ROBERT BARTNIK Australian National University Abstract We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic operators with rough coefficients on weighted Sobolev spaces are summarised. Some relations between the mass. xalar curvature and harmonic maps are described and the positive mass theorem for ,c-dimensional spin manifolds is proved. Introduction Suppose that (M,g) is an asymptotically flat 3-manifold. In general relativity the mass of M is given by where glJ, denotes the partial derivative and dS‘ is the normal surface element to S,, the sphere at infinity. This expression is generally attributed to [3]; for a recent review of this and other expressions for the mass and other physically interesting quantities see (41. However, in all these works the definition depends on the choice of coordinates at infinity and it is certainly not clear whether or not this dependence is spurious. Since it is physically quite reasonable to assume that a frame at infinity (“observer”) is given, this point has not received much attention in the literature (but see [15]). It is our purpose in this paper to show that, under appropriate decay conditions on the metric, (0.1) generalises to n-dimensions for n 2 3 and gives an invariant of the metric structure (M,g). The decay conditions roughly stated (for n = 3) are lg - 61 = o(r-’12), lag1 = o(f312), etc, and thus include the usual falloff conditions.
    [Show full text]
  • Barycentric Coordinates in Tetrahedrons
    Computational Geometry Lab: BARYCENTRIC COORDINATES IN TETRAHEDRONS John Burkardt Information Technology Department Virginia Tech http://people.sc.fsu.edu/∼jburkardt/presentations/cg lab barycentric tetrahedrons.pdf December 23, 2010 1 Introduction to this Lab In this lab we consider the barycentric coordinate system for a tetrahedron. This coordinate system will give us a common map that can be used to relate all tetrahedrons. This will give us a unified way of dealing with the computational tasks that must take into account the varying geometry of tetrahedrons. We start our exploration by asking whether we can come up with an arithmetic formula that measures the position of a point relative to a plane. The formula we arrive at will actually return a signed distance which is positive for points on one side of the plane, and negative on the other side. The extra information that the sign of the distance gives us is enough to help us answer the next question, which is whether we can determine if a point is inside a tetrahedron. This turns out to be simple if we apply the signed distance formula in the right way. The distance function turns out to have another crucial use. Given any face of the tetrahedron, we know that the tetrahedral point that is furthest from that face is the vertex opposite to the face. Therefore, any other point in the tetrahedron has a relative distance to the face that is strictly less than 1. For a given point, if we compute all four possible relative distances, we come up with our crucial barycentric coordinate system.
    [Show full text]
  • Combinatorial Properties of Complex Hyperplane Arrangements
    Combinatorial Properties of Complex Hyperplane Arrangements by Ben Helford A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics Northern Arizona University December 2006 Approved: Michael Falk, Ph.D., Chair N´andor Sieben, Ph.D. Stephen E. Wilson, Ph.D. Abstract Combinatorial Properties of Complex Hyperplane Arrangements Ben Helford Let A be a complex hyperplane arrangement, and M(A) be the com- [ plement, that is C` − H. It has been proven already that the co- H∈A homology algebra of M(A) is determined by its underlying matroid. In fact, in the case where A is the complexification of a real arrangement, one can determine the homotopy type by its oriented matroid. This thesis uses complex oriented matroids, a tool developed recently by D Biss, in order to combinatorially determine topological properties of M(A) for A an arbitrary complex hyperplane arrangement. The first chapter outlines basic properties of hyperplane arrange- ments, including defining the underlying matroid and, in the real case, the oriented matroid. The second chapter examines topological proper- ties of complex arrangements, including the Orlik-Solomon algebra and Salvetti’s theorem. The third chapter introduces complex oriented ma- troids and proves some ways that this codifies topological properties of complex arrangements. Finally, the fourth chapter examines the differ- ence between complex hyperplane arrangements and real 2-arrangements, and examines the problem of formulating the cone/decone theorem at the level of posets. ii Acknowledgements It almost goes without saying in my mind that this thesis could not be possible without the direction and expertise provided by Dr.
    [Show full text]
  • A Theory of Multi-Layer Flat Refractive Geometry
    MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com A Theory of Multi-Layer Flat Refractive Geometry Agrawal, A.; Ramalingam, S.; Taguchi, Y., Chari, V. TR2012-047 June 2012 Abstract Flat refractive geometry corresponds to a perspective camera looking through single/multiple parallel flat refractive mediums. We show that the underlying geometry of rays corresponds to an axial camera. This realization, while missing from previous works, leads us to develop a general theory of calibrating such systems using 2D-3D correspondences. The pose of 3D points is assumed to be unknown and is also recovered. Calibration can be done even using a single image of a plane. We show that the unknown orientation of the refracting layers corresponds to the underlying axis, and can be obtained independently of the number of layers, their distances from the camera and their refractive indices. Interestingly, the axis estimation can be mapped to the classical essential matrix computation and 5-point algorithm [15] can be used. After computing the axis, the thicknesses of layers can be obtained linearly when refractive indices are known, and we derive analytical solutions when they are unknown. We also derive the analytical forward projection (AFP) equations to compute the projection of a 3D point via multiple flat refractions, which allows non-linear refinement by minimizing the reprojection error. For two refractions, AFP is either 4th or 12th degree equation depending on the refractive indices. We analyze ambiguities due to small field of view, stability under noise, and show how a two layer system can be well approximated as a single layer system.
    [Show full text]
  • Topology Vs. Geometry
    Topology vs. Geometry Imagine a surface made of thin, easily stretchable rubber. Bend, stretch, twist, and deform this surface any way you want (just don't tear it). As you deform the surface, it will change in many ways, but some aspects of its nature will stay the same. For example, the surface at the far left, deformed as it is, is still recognizable as a sort of sphere, whereas the surface at the far right is recognizable as a deformed two-holed doughnut. The aspect of a surface's nature which is unaffected by deformation is called the topology of the surface. Thus the two surfaces on the left have the same topology, as do the two on the right, but the sphere and the two-holed doughnut surface have different topologies: no matter how you try, you can never deform one to look like the other. (Remember, cutting a surface open and regluing it to resemble the other is not allowed.) // \\ A surface's geometry consists of those properties which do change when the surface is deformed. Curvature is the most important geometric property. Other geometric properties include area, distance and angle. An eggshell and a ping-pong ball have the same topology, but they have different geometries. (In this and subsequent examples, the reader should idealize objects like eggshells and ping-pong balls as being truly two-dimensional, thus ignoring any thickness the real objects may possess.) EXERCISE 1: Which of the surfaces below have the same topology? If we physically glue the top edge of a square to its bottom edge and its left edge to its right edge, we get a doughnut surface.
    [Show full text]
  • 12 Notation and List of Symbols
    12 Notation and List of Symbols Alphabetic ℵ0, ℵ1, ℵ2,... cardinality of infinite sets ,r,s,... lines are usually denoted by lowercase italics Latin letters A,B,C,... points are usually denoted by uppercase Latin letters π,ρ,σ,... planes and surfaces are usually denoted by lowercase Greek letters N positive integers; natural numbers R real numbers C complex numbers 5 real part of a complex number 6 imaginary part of a complex number R2 Euclidean 2-dimensional plane (or space) P2 projective 2-dimensional plane (or space) RN Euclidean N-dimensional space PN projective N-dimensional space Symbols General ∈ belongs to such that ∃ there exists ∀ for all ⊂ subset of ∪ set union ∩ set intersection A. Inselberg, Parallel Coordinates, DOI 10.1007/978-0-387-68628-8, © Springer Science+Business Media, LLC 2009 532 Notation and List of Symbols ∅ empty st (null set) 2A power set of A, the set of all subsets of A ip inflection point of a curve n principal normal vector to a space curve nˆ unit normal vector to a surface t vector tangent to a space curve b binormal vector τ torsion (measures “non-planarity” of a space curve) × cross product → mapping symbol A → B correspondence (also used for duality) ⇒ implies ⇔ if and only if <, > less than, greater than |a| absolute value of a |b| length of vector b norm, length L1 L1 norm L L norm, Euclidean distance 2 2 summation Jp set of integers modulo p with the operations + and × ≈ approximately equal = not equal parallel ⊥ perpendicular, orthogonal (,,) homogeneous coordinates — ordered triple representing
    [Show full text]
  • Flat and Wireframe Shading, a Unity Tutorial
    Catlike Coding Unity C# Tutorials Flat and Wireframe Shading derivatives and geometry Use screen-space derivatives to find triangle normals. Do the same via a geometry shader. Use generated barycentric coordinates to create a wireframe. Make the wires fixed-width and configurable. This tutorial covers how to add support for flat shading and showing the wireframe of a mesh. It uses advanced rendering techniques and assumes you're familiar with the material covered in the Rendering series. This tutorial is made with Unity 2017.1.0. Exposing the triangles. 1 Flat Shading Meshes consist of triangles, which are flat by definition. We use surface normal vectors to add the illusion of curvature. This makes it possible to create meshes that represent seemingly smooth surfaces. However, sometimes you actually want to display flat triangles, either for style or to better see the mesh's topology. To make the triangles appear as flat as they really are, we have to use the surface normals of the actual triangles. It will give meshes a faceted appearance, known as flat shading. This can be done by making the normal vectors of a triangle's three vertices equal to the triangle's normal vector. This makes it impossible to share vertices between triangles, because then they would share normals as well. So we end up with more mesh data. It would be convenient if we could keep sharing vertices. Also, it would be nice if we could use a flat-shading material with any mesh, overriding its original normals, if any. Besides flat shading, it can also be useful or stylish to show a mesh's wireframe.
    [Show full text]
  • Positivity Theorems for Hyperplane Arrangements Via Intersection Theory
    Positivity theorems for hyperplane arrangements via intersection theory Nicholas Proudfoot University of Oregon Arrangements and Flats Let V be a finite dimensional vector space, \ and A a finite set of hyperplanes in V with H = f0g. H2A Definition A flat F ⊂ V is an intersection of some hyperplanes. Example V • 1 flat of dimension 2 (V itself) • 3 flats of dimension 1 (the lines) • 1 flat of dimension 0 (the origin) 1 Arrangements and Flats Example 6 Suppose V = C and A consists of 8 generic hyperplanes. 8 • 0 = 1 flat of dimension 6 (V itself) 8 • 1 = 8 flats of dimension 5 (the hyperplanes) 8 • 2 = 28 flats of dimension 4 8 • 3 = 56 flats of dimension 3 8 • 4 = 70 flats of dimension 2 8 • 5 = 56 flats of dimension 1 • 1 flat of dimension 0 (the origin) 2 Arrangements and Flats Example 7 7 Suppose V = C =C∆ = (z1;:::; z7) 2 C C · (1;:::; 1) 7 and A consists of the 2 hyperplanes Hij := fzi = zj g. H14 \ H46 \ H37 = (z1;:::; z7) j z1 = z4 = z6; z3 = z7 =C∆ is a flat of dimension 3. flats $ partitions of the set f1;:::; 7g H14 \ H46 \ H37 $ f1; 4; 6g t f3; 7g t f2g t f7g H14 $ f1; 4g t f2g t f3g t f5g t f6g t f7g V $ f1g t f2g t f3g t f4g t f5g t f6g t f7g f0g $ f1;:::; 7g flats of dimension k $ partitions into k + 1 parts 3 The Top-Heavy conjecture Theorem (Top-Heavy conjecture) 1 If k ≤ 2 dim V , then # of flats of codimension k ≤ # of flats of dimension k: Furthermore, the same statement holds for matroids (combinatorial abstractions of hyperplane arrangements for which we can still make sense of flats).
    [Show full text]
  • Flat Geometry on Riemann Surfaces and Teichm¨Uller Dynamics in Moduli Spaces
    Flat geometry on Riemann surfaces and Teichm¨uller dynamics in moduli spaces Anton Zorich Obergurgl, December 15, 2008 From Billiards to Flat Surfaces 2 Billiardinapolygon ................................. ...............................3 Closedtrajectories................................. ................................4 Challenge.......................................... ..............................5 Motivationforstudyingbilliards ..................... ..................................6 Gasoftwomolecules.................................. .............................7 Unfoldingbilliardtrajectories ...................... ...................................8 Flatsurfaces....................................... ...............................9 Rationalpolygons................................... .............................. 10 Billiardinarectangle ............................... ............................... 11 Unfoldingrationalbilliards ......................... ................................. 13 Flatpretzel........................................ .............................. 14 Surfaceswhich are more flat thanthe others. ............................. 15 Very flat surfaces 16 Very flat surfaces: construction from a polygon. ............................... 17 Propertiesofveryflatsurfaces........................ ............................... 19 Conicalsingularity ................................. ............................... 20 Familiesofflatsurfaces.............................. .............................. 21 Familyofflattori...................................
    [Show full text]
  • 2 Basics of General Relativity 15
    2 BASICS OF GENERAL RELATIVITY 15 2 Basics of general relativity 2.1 Principles behind general relativity The theory of general relativity completed by Albert Einstein in 1915 (and nearly simultaneously by David Hilbert) is the current theory of gravity. General relativity replaced the previous theory of gravity, Newtonian gravity, which can be understood as a limit of general relativity in the case of isolated systems, slow motions and weak fields. General relativity has been extensively tested during the past century, and no deviations have been found, with the possible exception of the accelerated expansion of the universe, which is however usually explained by introducing new matter rather than changing the laws of gravity [1]. We will not go through the details of general relativity, but we try to give some rough idea of what the theory is like, and introduce a few concepts and definitions that we are going to need. The principle behind special relativity is that space and time together form four-dimensional spacetime. The essence of general relativity is that gravity is a manifestation of the curvature of spacetime. While in Newton's theory gravity acts directly as a force between two bodies1, in Einstein's theory the gravitational interac- tion is mediated by spacetime. In other words, gravity is an aspect of the geometry of spacetime. Matter curves spacetime, and this curvature affects the motion of other matter (as well as the motion of the matter generating the curvature). This can be summarised as the dictum \matter tells spacetime how to curve, spacetime tells matter how to move" [2].
    [Show full text]