DOCSLIB.ORG

Compositions of Linear Transformations and Their Fixed Points Transformation Geometry Transformations of the Plane Transformatio

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Compositions of Linear Transformations and Their Fixed Points Transformation Geometry Transformations of the Plane Transformatio Transformation Geometry Transformations of the Plane Transformations via Coordinates Instead of f (P)=P′, we could write f (x, y)=(x ′, y ′), where P has • A transformation of the plane is a function f to each point P in the coordinates (x, y) and P′ has coordinates (x ′, y ′). This is equivalent to plane another point P′ = f (P) so that having two functions f and f giving the x ′ and y ′ in terms of x and y : Compositions of Linear Transformations and • f is one-to-one, i.e., 1 2 The main idea behind the introduction of transformations in geometry is x ′ f x y Their Fixed Points to provide the means for comparing two figures. P �= Q ⇒ f (P) �= f (Q) = 1( , ) f : ′ • Two figures are congruent if there is a special transformation of the y = f2(x, y) plane (an isometry) taking the first figure onto the second. Sara Schleissmann Example. • Two figures are similar is there is a special transformation of the plane Q (a similarity) taking the first figure onto the second. x ′ =3x +2y − 1 State University of New York Q' f : ′ College at Oneonta Transformations or “motions”of the plane are used in Euclid’s P y = −x +4y +2 P' [email protected] “Elements”in Proposition I.4 (the SAS Postulate). Euclid assumes that if ′ ′ ′ two triangles satisfy the SAS Hypothesis, then one can “apply”one The image of the point P(2, 3) is the point P (x , y ), where x ′ =3· 2+2· 3 − 1 = 11 and y ′ = −2+4· 3+2=12. Student Research and Creative Activity Day triangle on top of the other to get a perfect match, thus proving the • f is onto, i.e., each point R in the plane has a preimage under f , The preimage of the point R(6, 9) is the point S(x, y) such that March 26, 2014 congruence of the given triangles. that is there exists a point S so that f (S)=R. 6=3x +2y − 1 --1 S=f (R) R 9=−x +4y +2 The only solution of the system above is x =1, y =2 S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points If, on the other hand P is not on either of the previous lines, then let’s • Linear transformations preserve noncollinearity. ←→ Linear Transformations look at line PC. Fixed Points of Transformations To prove this, let A, B, and C be three arbitrary, noncollinear points. We ←→ ←→ ←→ ←→ • If a transformation preserves collinearity, that is, if it maps any three a) If PC is not parallel to AB, then PC must intersect AB at Q. It would will show that A′ = f (A), B′ = f (B), C ′ = f (C) are noncollinear as well. collinear points to three collinear points, then the transformation is said ′ ←−→′ ′ ′ • A transformation f of the plane is said to have A as a fixed point if ′ ′ ′ then follow that f (Q)=Q belongs to A B = l and since C ∈ l, it to be linear. By contradiction, suppose that A , B , C are collinear, so they belong to f (A)=A. ′ ←−→′ ′ a line l. Now let P, different from A,B,C, be an arbitrary point in the would then follow that P = f (P) belongs to Q C = l. • If a given transformation fixes any point of the plane, then the C' plane. We will prove that P will always map to some point on line l. This transformation is called the identity mapping. B' f B' C will contradict being onto. B Example 1. The linear transformation f ←→ ←→ ←→ Q' If P lies on either of the lines AB, AC , or BC, then since f preserves Q P' B ←−→ ←−→ ←−→ P ′ ′ ′ ′ ′ ′ ′ C ′ collinearity, P = f (P) belongs to line A B = A C = B C = l. C' x = x +2y A A' ′ A A' y =3y ←→ ←→ • In terms of coordinates, the transformation B B' b) If If PC is parallel to AB, then let D ∈ AC be an arbitrary point. Line has (0, 0) as a fixed point. In fact, any point on the x−axis is a fixed ←→ ←→ ′ point since (x, 0) goes to (x +2· 0, 3 · 0) = (x, 0). ′ PD will then intersect line AB at some point T . Then, points T = f (T ) x = f1(x, y) P' Question. Can a linear transformation of the form P and D′ = f (D) must belong to line l and so P′ = f (P) must belong to l. ′ y = f2(x, y) C C' x′ = ax + by B' B ′ is linear if and only if the functions f1 and f2 are linear in x and y. That is A P y = cx + dy A' P' ′ C x = a1x + b1y + c1 C' D A D' ever have exactly one other fixed point besides (0, 0)? ′ T A' y = a2x + b2y + c2 T' S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points Linear Algebra Approach Geometry Approach • Linear transformations send midpoints to midpoints. Proof. Let AC be an arbitrary segment and let M be its midpoint. Let Finding the fixed points of the given linear transformation is equivalent to ′ ′ • If a linear transformation fixes two points A, B, then any point on the Now let’s show that l � m . A′, C ′, M′ be the images of A, C, M via the linear transformation f . We finding the solutions of the system ′ ′ ′ ′ ′ segment AB is fixed. Indeed, by contradiction, suppose that l ∩ m = P . Since P ∈ l , it will prove that M′ is the midpoint of A′C ′. To do so, construct a ′ To prove this, we need some additional results. follows that P must come from some point on l. Similarly, since parallelogram ⋄ABCD. Since parallel lines go to to parallel lines, the x = ax + by ′ ′ ′ • Linear transformations send parallel lines to parallel lines. P ∈ m , it follows that P must come from some point on m. But then, quadrilateral ⋄A′B′C ′D′ is also a parallelogram with diagonals A′C ′ and ′ ′ −1 ′ y = cx + dy, Proof. Indeed, suppose l � m and let’s first show that l �= m . f (P ) ∈ l ∩ m, which is impossible since l ∩ m = ∅. B′D′ meeting at the midpoint N of A′C ′ and B′D′. But M lying on both AC and BD maps to M′ on both A′C ′ and B′D′. So M′ = N =midpoint l'=m' or equivalently l l' m' ′ ′ m A' l of A C . (a − 1)x + by =0 A m f C' D D' C' cx +(d − 1)y =0. f P' C B B' Such a system has solutions other than (0, 0) iff C M f = a − 1 b ′ ′ N M' det =0. By contradiction, suppose l = m . Let A, B ∈ l and C ∈ m. Since l � m, cd− 1 it follows that A, B, C are noncollinear. But then A B' A′ = f (A) ∈ l ′, B′ = f (B) ∈ l ′, C ′ = f (C) ∈ m′ = l ′ are collinear. This B A' If this is the case, then, in fact, the system has an entire “line worth”of contradicts the fact that linear transformations preserve noncollinearity. solutions. S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points S. Schleissmann Compositions and Fixed Points Back to the proof More fixed points Isometries: Special Linear Transformations • If a linear transformation fixes three noncollinear points, then it fixes • If a linear transformation f fixes points A and B, then f fixes any point every point of the plane and, hence, it is the identity transformation. on the segment AB. Observation. The same result can be proven by means of coordinates. Proof. Let A, B, and C be three arbitrary, noncollinear fixed points and Proof. Let P ∈ AB be arbitrary. Since f is linear, it takes the midpoint Definition. A linear transformation f is an isometry if and only if for any For example, consider the transformation f given by ′ ′ let P be an arbitrary point in the plane. Construct △ABC. If ′ ′ M1 of AB to the midpoint of A B = AB. But this means that M1 is two points P and Q with P = f (P) and Q = f (Q) we have that P ∈△ABC, then P is fixed based on the previous result. If P ∈△/ ABC, ′ ′ fixed. Now, without loss of generality assume P ∈ AM1 and look at the PQ = P Q . x ′ =3x +2y − 1 then construct lines l and m passing through P and intersecting the sides midpoint M2 of AM1. As before, M2 is fixed. If P ∈ AM2, we consider ′ of △ABC. y = −x +4y +2 M3 as the midpoint of AM2. If, on the other hand, P ∈ M1M2, we let M3 Q' Q be the midpoint of M1M2. A l f and the points A(2, 3) and B(−4, −3). The midpoint of AB is M(−1, 0). m ′ ′ ′ P2 f A A B B M M P1 P sends to (11, 12), to (−19, −6), and to (−4, 3). As we B ′ ′ ′ A M2 P M3 M1 can see, M turns out to be the midpoint of A B . P4 The same kind of argument can be used for any linear transformation f P P' B P and any two points A(x1, y1), B(x2, y2). 3 C This way, we construct an entire sequence of (fixed) midpoints {Mn}n≥1 converging to P.
Load more

Recommended publications
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Robot Vision: Projective Geometry
    Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations
    [Show full text]
  • Transformation Geometry — Math 331
    Transformation Geometry — Math 331 March 8, 2004 Conjugation of Isometries Definition. If f and g are invertible affine transformations of Rn, the conjugate of f by g is the transformation g ◦ f ◦ g−1. When the context is clear, one sometimes abbreviates notation for conjugation by writing g · f = g ◦ f ◦ g−1 . One views this as g “acting on” f. That is, an affine transformation of Rn not only describes a motion of Rn but also a motion on the set of affine transformations of Rn: the action of the set of affine transformations on itself by conjugation. Theorem 1. If f(x) = Ux + v is an affine transformation of Rn and τ(x) = x + a is translation by the vector a, then f · τ is translation by the vector Ua. Proof. f −1(x) = U −1x − U −1v, and, therefore y = τ ◦ f −1(x) = U −1x − U −1v + a. Then (f · τ)(x) = Uy + v = x + Ua. Warning. Although when a translation is conjugated by an arbitrary affine transformation, the result is a translation, it is not true that the result of conjugating an isometry by an affine transformation is always an isometry. For example, if f is the simple linear transformation (x, y) 7→ (2x, y) of R2 and ρ is rotation through the angle π/2 about the origin, i.e., (x, y) 7→ (−y, x), then f · ρ is the linear map (x, y) 7→ (−2y, x/2), which is not an isometry. On the other hand, it is clear that the result of conjugating an isometry by an isometry is an isometry.
    [Show full text]
  • The Dual Theorem Concerning Aubert Line
    The Dual Theorem concerning Aubert Line Professor Ion Patrascu, National College "Buzeşti Brothers" Craiova - Romania Professor Florentin Smarandache, University of New Mexico, Gallup, USA In this article we introduce the concept of Bobillier transversal of a triangle with respect to a point in its plan; we prove the Aubert Theorem about the collinearity of the orthocenters in the triangles determined by the sides and the diagonals of a complete quadrilateral, and we obtain the Dual Theorem of this Theorem. Theorem 1 (E. Bobillier) Let 퐴퐵퐶 be a triangle and 푀 a point in the plane of the triangle so that the perpendiculars taken in 푀, and 푀퐴, 푀퐵, 푀퐶 respectively, intersect the sides 퐵퐶, 퐶퐴 and 퐴퐵 at 퐴푚, 퐵푚 and 퐶푚. Then the points 퐴푚, 퐵푚 and 퐶푚 are collinear. 퐴푚퐵 Proof We note that = 퐴푚퐶 aria (퐵푀퐴푚) (see Fig. 1). aria (퐶푀퐴푚) 1 Area (퐵푀퐴푚) = ∙ 퐵푀 ∙ 푀퐴푚 ∙ 2 sin(퐵푀퐴푚̂ ). 1 Area (퐶푀퐴푚) = ∙ 퐶푀 ∙ 푀퐴푚 ∙ 2 sin(퐶푀퐴푚̂ ). Since 1 3휋 푚(퐶푀퐴푚̂ ) = − 푚(퐴푀퐶̂ ), 2 it explains that sin(퐶푀퐴푚̂ ) = − cos(퐴푀퐶̂ ); 휋 sin(퐵푀퐴푚̂ ) = sin (퐴푀퐵̂ − ) = − cos(퐴푀퐵̂ ). 2 Therefore: 퐴푚퐵 푀퐵 ∙ cos(퐴푀퐵̂ ) = (1). 퐴푚퐶 푀퐶 ∙ cos(퐴푀퐶̂ ) In the same way, we find that: 퐵푚퐶 푀퐶 cos(퐵푀퐶̂ ) = ∙ (2); 퐵푚퐴 푀퐴 cos(퐴푀퐵̂ ) 퐶푚퐴 푀퐴 cos(퐴푀퐶̂ ) = ∙ (3). 퐶푚퐵 푀퐵 cos(퐵푀퐶̂ ) The relations (1), (2), (3), and the reciprocal Theorem of Menelaus lead to the collinearity of points 퐴푚, 퐵푚, 퐶푚. Note Bobillier's Theorem can be obtained – by converting the duality with respect to a circle – from the theorem relative to the concurrency of the heights of a triangle.
    [Show full text]
  • The Generalization of Miquels Theorem
    THE GENERALIZATION OF MIQUEL’S THEOREM ANDERSON R. VARGAS Abstract. This papper aims to present and demonstrate Clifford’s version for a generalization of Miquel’s theorem with the use of Euclidean geometry arguments only. 1. Introduction At the end of his article, Clifford [1] gives some developments that generalize the three circles version of Miquel’s theorem and he does give a synthetic proof to this generalization using arguments of projective geometry. The series of propositions given by Clifford are in the following theorem: Theorem 1.1. (i) Given three straight lines, a circle may be drawn through their intersections. (ii) Given four straight lines, the four circles so determined meet in a point. (iii) Given five straight lines, the five points so found lie on a circle. (iv) Given six straight lines, the six circles so determined meet in a point. That can keep going on indefinitely, that is, if n ≥ 2, 2n straight lines determine 2n circles all meeting in a point, and for 2n +1 straight lines the 2n +1 points so found lie on the same circle. Remark 1.2. Note that in the set of given straight lines, there is neither a pair of parallel straight lines nor a subset with three straight lines that intersect in one point. That is being considered all along the work, without further ado. arXiv:1812.04175v1 [math.HO] 11 Dec 2018 In order to prove this generalization, we are going to use some theorems proposed by Miquel [3] and some basic lemmas about a bunch of circles and their intersections, and we will follow the idea proposed by Lebesgue[2] in a proof by induction.
    [Show full text]
  • II.3 : Measurement Axioms
    I I.3 : Measurement axioms A large — probably dominant — part of elementary Euclidean geometry deals with questions about linear and angular measurements , so it is clear that we must discuss the basic properties of these concepts . Linear measurement In elementary geometry courses , linear measurement is often viewed using the lengths of line segments . We shall view it somewhat differently as giving the (shortest) distance 2 3 between two points . Not surprisingly , in RRR and RRR we define this distance by the usual Pythagorean formula already given in Section I.1. In the synthetic approach , one may view distance as a primitive concept given formally by a function d ; given two points X, Y in the plane or space , the distance d (X, Y) is assumed to be a nonnegative real number , and we further assume that distances have the following standard properties : Axiom D – 1 : The distance d (X, Y) is equal to zero if and only if X = Y. Axiom D – 2 : For all X and Y we have d (X, Y) = d (Y, X). Obviously it is also necessary to have some relationships between an abstract notion of distance and the other undefined concepts introduced thus far ; namely , lines and planes . Intuitively we expect a geometrical line to behave just like the real number line , and the following axiom formulated by G. D. Birkhoff (1884 – 1944) makes this idea precise : Axiom D – 3 ( Ruler Postulate ) : If L is an arbitrary line , then there is a 1 – 1 correspondence between the points of L and the real numbers RRR such that if the points X and Y on L correspond to the real numbers x and y respectively , then we have d (X, Y) = |x – y |.
    [Show full text]
  • Linear Features in Photogrammetry
    Linear Features in Photogrammetry by Ayman Habib Andinet Asmamaw Devin Kelley Manja May Report No. 450 Geodetic Science and Surveying Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University Columbus, Ohio 43210-1275 January 2000 Linear Features in Photogrammetry By: Ayman Habib Andinet Asmamaw Devin Kelley Manja May Report No. 450 Geodetic Science and Surveying Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University Columbus, Ohio 43210-1275 January, 2000 ABSTRACT This research addresses the task of including points as well as linear features in photogrammetric applications. Straight lines in object space can be utilized to perform aerial triangulation. Irregular linear features (natural lines) in object space can be utilized to perform single photo resection and automatic relative orientation. When working with primitives, it is important to develop appropriate representations in image and object space. These representations must accommodate for the perspective projection relating the two spaces. There are various options for representing linear features in the above applications. These options have been explored, and an optimal representation has been chosen. An aerial triangulation technique that utilizes points and straight lines for frame and linear array scanners has been implemented. For this task, the MSAT (Multi Sensor Aerial Triangulation) software, developed at the Ohio State University, has been extended to handle straight lines. The MSAT software accommodates for frame and linear array scanners. In this research, natural lines were utilized to perform single photo resection and automatic relative orientation. In single photo resection, the problem is approached with no knowledge of the correspondence of natural lines between image space and object space.
    [Show full text]
  • 1.3 Collinear, Betweenness, and Assumptions Date: Goals: How Are Diagrams Interpreted in the Study of Geometry
    1.3 Collinear, Betweenness, and Assumptions Date: Goals: How are diagrams interpreted in the study of geometry Collinear/Noncollinear: Three or more points that lie on the same line are collinear. Points that do not lie on the same line are noncollinear. Two points are always collinear. For a point to be between two other points, all three points must be collinear. Example 1: Use the diagram to the right to answer the following questions. a. Are points A, B, and C collinear? A Yes, B is between A and C. b. Is point B between A and D? B No, the three points are noncollinear. c. Name 3 noncollinear points. AEC, , and E D C d. Are points A and D collinear? Yes, two points are always collinear. The Triangle Inequality: For any three points, there are only two possibilies: 1. They are collinear. One point must be between the other two. Two of the distances must add to the third. 2. They are noncollinear. The three points will form a triangle. Any two of the lengths will add to be larger than the third. Example 2: If AB=12, BC=13, and AC=25, can we determine if the points are collinear? Yes, the lengths 12 and 13 add to 25, therefore the points are collinear. How to Interpret a Diagram: You Should Assume: You Should Not Assume: Straight lines and angles Right Angles Collinearity of points Congruent Segments Betweenness of points Congruent Angles Relative position of points Relative sizes of segments and angles Example 3: Use the diagram to the right to answer the following questions.
    [Show full text]
  • A Simple Way to Test for Collinearity in Spin Symmetry Broken Wave Functions: General Theory and Application to Generalized Hartree Fock
    A simple way to test for collinearity in spin symmetry broken wave functions: general theory and application to Generalized Hartree Fock David W. Small and Eric J. Sundstrom and Martin Head-Gordon Department of Chemistry, University of California, Berkeley, California 94720 and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720 (Dated: February 17, 2015) Abstract We introduce a necessary and sufficient condition for an arbitrary wavefunction to be collinear, i.e. its spin is quantized along some axis. It may be used to obtain a cheap and simple computa- tional procedure to test for collinearity in electronic structure theory calculations. We adapt the procedure for Generalized Hartree Fock (GHF), and use it to study two dissociation pathways in CO2. For these dissociation processes, the GHF wave functions transform from low-spin Unre- stricted Hartree Fock (UHF) type states to noncollinear GHF states and on to high-spin UHF type states, phenomena that are succinctly illustrated by the constituents of the collinearity test. This complements earlier GHF work on this molecule. 1 I. INTRODUCTION Electronic structure practitioners have long relied on spin-unrestricted Hartree Fock (UHF) and Density Functional Theory (DFT) for open-shell and strongly correlated sys- tems. A key reason for this is that the latter's multiradical nature is partially accommo- dated by unrestricted methods: the paired spin-up and spin-down electrons are allowed to separate, while maintaining their spin alignment along a three-dimensional axis, i.e. their spin collinearity. But, considering the great potential for diversity in multitradical systems, it is not a stretch to concede that this approach is not always reasonable.
    [Show full text]
  • Dissertation Hyperovals, Laguerre Planes And
    DISSERTATION HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY Submitted by Luke Bayens Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Spring 2013 Doctoral Committee: Advisor: Tim Penttila Jeff Achter Willem Bohm Chris Peterson ABSTRACT HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY In 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries { we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups { indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting tran- sitively on flags, with an additional hypothesis { a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order.
    [Show full text]
  • Euclidean Versus Projective Geometry
    Projective Geometry Projective Geometry Euclidean versus Projective Geometry n Euclidean geometry describes shapes “as they are” – Properties of objects that are unchanged by rigid motions » Lengths » Angles » Parallelism n Projective geometry describes objects “as they appear” – Lengths, angles, parallelism become “distorted” when we look at objects – Mathematical model for how images of the 3D world are formed. Projective Geometry Overview n Tools of algebraic geometry n Informal description of projective geometry in a plane n Descriptions of lines and points n Points at infinity and line at infinity n Projective transformations, projectivity matrix n Example of application n Special projectivities: affine transforms, similarities, Euclidean transforms n Cross-ratio invariance for points, lines, planes Projective Geometry Tools of Algebraic Geometry 1 n Plane passing through origin and perpendicular to vector n = (a,b,c) is locus of points x = ( x 1 , x 2 , x 3 ) such that n · x = 0 => a x1 + b x2 + c x3 = 0 n Plane through origin is completely defined by (a,b,c) x3 x = (x1, x2 , x3 ) x2 O x1 n = (a,b,c) Projective Geometry Tools of Algebraic Geometry 2 n A vector parallel to intersection of 2 planes ( a , b , c ) and (a',b',c') is obtained by cross-product (a'',b'',c'') = (a,b,c)´(a',b',c') (a'',b'',c'') O (a,b,c) (a',b',c') Projective Geometry Tools of Algebraic Geometry 3 n Plane passing through two points x and x’ is defined by (a,b,c) = x´ x' x = (x1, x2 , x3 ) x'= (x1 ', x2 ', x3 ') O (a,b,c) Projective Geometry Projective Geometry
    [Show full text]
  • Augmented Reality Mobile Learning System: Study to Improve Psts’ Understanding of Mathematical Development
    Short Paper—Augmented Reality Mobile Learning System: Study Study to Improve PSTs’… Augmented Reality Mobile Learning System: Study to Improve PSTs’ Understanding of Mathematical Development https://doi.org/10.3991/ijim.v14i09.12909 Mohammad Faizal Amir(), Novia Ariyanti, Najih Anwar Universitas Muhammadiyah Sidoarjo, Sidoarjo, Indonesia [email protected] Erik Valentino STKIP Bina Insan Mandiri Surabaya, Surabaya, Indonesia Dian Septi Nur Afifah STKIP PGRI Tulungagung, Tulungagung, Indonesia Abstract—Augmented Reality research in mobile learning systems so far have not especially to improve Preservice Students Teachers' (PSTs) geometry understanding in mathematical development. Studies conducted in this study ba- sically use a case study. This research aims to develop a mobile augmented reality system for PSTs learning so that it can be used to improve PSTs understanding of mathematics development includes doing, image-making, an image having, property noticing, formalizing, observing, structuring and inventing. In this de- velopment, PSTs can understand the understanding of the geometry transfor- mation is a translation, reflection, rotation, and dilatation. Keywords—Mobile systems, augmented reality, understanding mathematics, geometry transformation. 1 Introduction In PISA 2015 mathematics assessed as minor domains, providing the opportunity to make comparisons of student performance over time. The results of preliminary field studies, mathematical understanding of geometry Preservice Student Teachers (PSTs) 20 of 36 in Universitas Muhammadiyah Sidoarjo have difficulty in solving mathemat- ical understanding of geometry. The development of learners by understanding participants [1] includes doing, im- age-making, an image having, property noticing, formalizing, observing, structuring, and inventing, linking concepts, ideas, facts, or mathematics procedures. Results of re- search by [2], [6] shows the importance of understanding the development problems of geometry PSTs that many of the teachers or prospective elementary school teachers do iJIM ‒ Vol.
    [Show full text]