DOCSLIB.ORG

Projective Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Projective geometry Projective geometry is an extension of Euclidean geometry, endowed with many nice properties incurred by affixing an extra ‘line at infinity’. Certain theorems (such as Desargues’ and Pascal’s theorems) have projective geometry as their more natural setting, and the wealth of projective transformations can simplify problems in ordinary Euclidean geometry. The real projective plane In Euclidean geometry, we assign a coordinate pair x, y to each point in the plane. In projective geometry, we augment this with an extra coordinate, so three values are used to represent a point: x, y, z. Moreover, scalar multiples are considered equivalent; x, y, z and x, y, z represent the same point. R 0, 0, 0 is not part of the projective plane, but can be regarded as a ‘projector’, from which all points, lines, circles et cetera emanate. Since scalar multiples of points are considered equivalent, we can identify points in 2 (the real projective plane) with lines through the origin. Projective lines are identified with planes through the origin, and are of the form a x b y c z 0. Note that this equation is homogeneous : all terms are of first degree. In general, all algebraic curves are represented by homogeneous polynomials in x, y and z. In the diagram above, a triangle A B C is shown in the reference plane ( z 1). The point A is ‘projected’ from R along the orange line, intersecting the reference plane at A. Similarly, the ‘plane’ containing the yellow triangle represents the line B C . Horizontal ‘lines’ such as the blue one do not intersect the reference plane, so correspond to points ‘at infinity’. The horizontal ‘plane’ (parallel to the reference plane) through R represents the line at infinity ( z 0). Parallel lines on the projective plane can be considered to meet at a point on the line at infinity. 1. Prove that any two distinct lines intersect in precisely one point. x y z 2. Show that the equation of a line through points x1, y1, z1 and x2, y2, z2 is given by det x1 y1 z1 0. x2 y2 z2 You have shown that any two points share a common line, and any two lines share a common point. This suggests a fundamental interchangeability between lines and points, known as projective duality . We shall explore this later. 3. Let A B C and A! B! C! be two triangles. Let A B meet A' B' at C!! , and define A!! and B!! similarly. Show that A A !, B B ! and C C ! are concurrent if and only if A!! , B!! and C!! are collinear. [Desargues’ theorem] A B C P C' C'' A' B' B'' A'' If two triangles exhibit this relationship, they are said to be in perspective . The point P is the perspector , and the line A!! B!! C!! is the perspectrix . As Desargues’ theorem is the projective dual of its converse, you only need to prove the statement in one direction. Surprisingly, it is actually easier to prove Desargues in three dimensions (where the triangles are in different planes); the two-dimensional result then follows by projecting it onto the plane. 4. Let A B C be a triangle and X be a point inside the triangle. The lines A X , B X and C X meet the circle A B C again at P, Q and R, respectively. Choose a point U on X P which is between X and P. Suppose that the lines through U which are parallel to A B and C A meet X Q and X R at points V and W respectively. Prove that the points R, W, V and Q lie on a circle. [BMO2 2011, Question 1] Cross-ratio and harmonic ranges You may have encountered homogeneous projective coordinates before in the form of areal (or barycentric) coordinates. Even though qualitative properties such as collinearity and concurrency can be defined in terms of any projective coordinates, to compare distances they must first be normalised , i.e. projected onto the reference plane. Firstly, we define a vector, n, perpendicular to the reference plane. (For Cartesian coordinates, we generally allow z 1 to be the reference plane, so n 0, 0, 1 . For areal coordinates, x y z 1 is the reference plane, so n 1, 1, 1 . In general, if the reference plane is given by a x b y c z 1, the vector n a, b, c. For a given x vector x in the projective plane, it is normalised by the operation x - nVx 5. If A, B, C and D are collinear points represented by normalised vectors a, b, c and d, respectively, show that ab A B . C D cd P R S Q A B C D Suppose we have three points, A, B and C, which are collinear. We can select an arbitrary point P, and a further arbitrary point Q lying on the line P B . Let Q C meet P A at R, and Q A meet P C at S. Finally, let R S meet line A B C at D. 6. Prove that A B V C D 1, irrespectively of the locations of P and Q. (Hint: this can be done by applying two B C V D A similar theorems in quick succession.) It is a remarkable fact that the location of D does not depend on that of P and Q. B and D are described as projective harmonic conjugates with respect to the line segment A C . For any four collinear points, the quantity A B V C D A, C; B, D is known as the cross-ratio . If A, C; B, D 1, we say that they form a harmonic range . B C V D A ^ A, C; B, D 1 is equivalent to A and C being inverse points with respect to the circle on diameter B D , which is in turn equivalent to the circles on diameters A C and B D being orthogonal. [Equivalent definitions of harmonic range] There is another remarkable and useful fact concerning harmonic ranges. Let P be a point not on the line A B C D . Then any two of the following four properties implies the other two: O A P C ; 2 P A is an angle bisector of O B P D ; P C is the other angle bisector of O B P D ; A, C; B, D 1. P A B C D By Thales’ theorem, the first of these conditions is equivalent to P lying on the circle with diameter A C . Hence, if P lies on either of the intersection points of the (orthogonal) circles on diameters A C and B D , the four lines through P divide the plane into eight equal octants of angle . 4 Projective transformations Returning to the idea of representing points in the projective plane as three-element vectors, we can consider the group of operations represented by linear maps x - M x , where M is a non-singular matrix. These are known as projective transformations , or collineations . 7. Show that applying a projective transformation to a line results in another line, and thus that collinear points remain collinear. A generalisation is that degree- d algebraic curves are mapped to degree- d algebraic curves by projective transformations. Hence, conics are preserved. Moreover, it is possible to choose a projective transformation carefully to map a given conic to any other conic, with three real degrees of freedom remaining. 8. Show that applying a projective transformation with matrix M to a point x p q results in a point with M p M q normalised coordinates . n V M p n V M q 9. Hence show that the cross-ratio A, C; B, D of four collinear points is preserved under projective transformations. 10. For any four points A, B, C and D, no three of which are collinear, show that there exists a unique projective transformation mapping them to 1, 0, 0 , 0, 1, 0 , 0, 0, 1 and 1, 1, 1 , respectively. 11. Hence show that there exists a unique projective transformation mapping any four points (no three of which are collinear) to any other four points (no three of which are collinear). The last of these theorems enables one to simplify a projective problem by converting any quadrilateral into a parallelogram. This enables one to find all of the harmonic ranges in the complete quadrangle displayed below. Try to spot as many as you can! B C Y A D X Z 12. The diagonals of the quadrilateral A B C D meet at X . The circumcircles of A B X and C D X intersect again at Y ; the circumcircles of B C X and D A X intersect again at Z. The midpoints of the diagonals A C and B D are denoted M and N, respectively. Prove that M, N, X , Y and Z are concyclic. [Sherry Gong, Trinity 2012] Another configuration occurring in many instances is a quadrilateral with an inscribed conic. With a projective transformation, we can convert the quadrilateral into a parallelogram. To simplify matters even further, a (possibly imaginary) affine transformation is capable of turning the conic into a circle. The symmetry of the configuration implies that the diagonals of the quadrilateral are concurrent with the lines joining opposite points of tangency. When viewed as a complete quadrilateral (four lines in general position intersecting in six points), we have three of these concurrency points and plenty of harmonic ranges! The centre of the inscribed conic also happens to lie on a line passing through the three midpoints of pairs opposite vertices, by Newton’s theorem .
Recommended publications
  • Projective Geometry: a Short Introduction

    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.
  • Algebraic Geometry Michael Stoll

    Algebraic Geometry Michael Stoll

    Introductory Geometry Course No. 100 351 Fall 2005 Second Part: Algebraic Geometry Michael Stoll Contents 1. What Is Algebraic Geometry? 2 2. Affine Spaces and Algebraic Sets 3 3. Projective Spaces and Algebraic Sets 6 4. Projective Closure and Affine Patches 9 5. Morphisms and Rational Maps 11 6. Curves — Local Properties 14 7. B´ezout’sTheorem 18 2 1. What Is Algebraic Geometry? Linear Algebra can be seen (in parts at least) as the study of systems of linear equations. In geometric terms, this can be interpreted as the study of linear (or affine) subspaces of Cn (say). Algebraic Geometry generalizes this in a natural way be looking at systems of polynomial equations. Their geometric realizations (their solution sets in Cn, say) are called algebraic varieties. Many questions one can study in various parts of mathematics lead in a natural way to (systems of) polynomial equations, to which the methods of Algebraic Geometry can be applied. Algebraic Geometry provides a translation between algebra (solutions of equations) and geometry (points on algebraic varieties). The methods are mostly algebraic, but the geometry provides the intuition. Compared to Differential Geometry, in Algebraic Geometry we consider a rather restricted class of “manifolds” — those given by polynomial equations (we can allow “singularities”, however). For example, y = cos x defines a perfectly nice differentiable curve in the plane, but not an algebraic curve. In return, we can get stronger results, for example a criterion for the existence of solutions (in the complex numbers), or statements on the number of solutions (for example when intersecting two curves), or classification results.
  • The Trigonometry of Hyperbolic Tessellations

    The Trigonometry of Hyperbolic Tessellations

    Canad. Math. Bull. Vol. 40 (2), 1997 pp. 158±168 THE TRIGONOMETRY OF HYPERBOLIC TESSELLATIONS H. S. M. COXETER ABSTRACT. For positive integers p and q with (p 2)(q 2) Ù 4thereis,inthe hyperbolic plane, a group [p, q] generated by re¯ections in the three sides of a triangle ABC with angles ôÛp, ôÛq, ôÛ2. Hyperbolic trigonometry shows that the side AC has length †,wherecosh†≥cÛs,c≥cos ôÛq, s ≥ sin ôÛp. For a conformal drawing inside the unit circle with centre A, we may take the sides AB and AC to run straight along radii while BC appears as an arc of a circle orthogonal to the unit circle.p The circle containing this arc is found to have radius 1Û sinh †≥sÛz,wherez≥ c2 s2, while its centre is at distance 1Û tanh †≥cÛzfrom A. In the hyperbolic triangle ABC,the altitude from AB to the right-angled vertex C is ê, where sinh ê≥z. 1. Non-Euclidean planes. The real projective plane becomes non-Euclidean when we introduce the concept of orthogonality by specializing one polarity so as to be able to declare two lines to be orthogonal when they are conjugate in this `absolute' polarity. The geometry is elliptic or hyperbolic according to the nature of the polarity. The points and lines of the elliptic plane ([11], x6.9) are conveniently represented, on a sphere of unit radius, by the pairs of antipodal points (or the diameters that join them) and the great circles (or the planes that contain them). The general right-angled triangle ABC, like such a triangle on the sphere, has ®ve `parts': its sides a, b, c and its acute angles A and B.
  • Projective Geometry Lecture Notes

    Projective Geometry Lecture Notes

    Projective Geometry Lecture Notes Thomas Baird March 26, 2014 Contents 1 Introduction 2 2 Vector Spaces and Projective Spaces 4 2.1 Vector spaces and their duals . 4 2.1.1 Fields . 4 2.1.2 Vector spaces and subspaces . 5 2.1.3 Matrices . 7 2.1.4 Dual vector spaces . 7 2.2 Projective spaces and homogeneous coordinates . 8 2.2.1 Visualizing projective space . 8 2.2.2 Homogeneous coordinates . 13 2.3 Linear subspaces . 13 2.3.1 Two points determine a line . 14 2.3.2 Two planar lines intersect at a point . 14 2.4 Projective transformations and the Erlangen Program . 15 2.4.1 Erlangen Program . 16 2.4.2 Projective versus linear . 17 2.4.3 Examples of projective transformations . 18 2.4.4 Direct sums . 19 2.4.5 General position . 20 2.4.6 The Cross-Ratio . 22 2.5 Classical Theorems . 23 2.5.1 Desargues' Theorem . 23 2.5.2 Pappus' Theorem . 24 2.6 Duality . 26 3 Quadrics and Conics 28 3.1 Affine algebraic sets . 28 3.2 Projective algebraic sets . 30 3.3 Bilinear and quadratic forms . 31 3.3.1 Quadratic forms . 33 3.3.2 Change of basis . 33 1 3.3.3 Digression on the Hessian . 36 3.4 Quadrics and conics . 37 3.5 Parametrization of the conic . 40 3.5.1 Rational parametrization of the circle . 42 3.6 Polars . 44 3.7 Linear subspaces of quadrics and ruled surfaces . 46 3.8 Pencils of quadrics and degeneration . 47 4 Exterior Algebras 52 4.1 Multilinear algebra .
  • Perspectives on Projective Geometry • Jürgen Richter-Gebert

    Perspectives on Projective Geometry • Jürgen Richter-Gebert

    Perspectives on Projective Geometry • Jürgen Richter-Gebert Perspectives on Projective Geometry A Guided Tour Through Real and Complex Geometry 123 Jürgen Richter-Gebert TU München Zentrum Mathematik (M10) LS Geometrie Boltzmannstr. 3 85748 Garching Germany [email protected] ISBN 978-3-642-17285-4 e-ISBN 978-3-642-17286-1 DOI 10.1007/978-3-642-17286-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011921702 Mathematics Subject Classification (2010): 51A05, 51A25, 51M05, 51M10 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation,reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) About This Book Let no one ignorant of geometry enter here! Entrance to Plato’s academy Once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, “and what is the use of a book,” thought Alice, “without pictures or conversations?” Lewis Carroll, Alice’s Adventures in Wonderland Geometry is the mathematical discipline that deals with the interrelations of objects in the plane, in space, or even in higher dimensions.
  • Chapter 9 the Geometrical Calculus

    Chapter 9 the Geometrical Calculus

    Chapter 9 The Geometrical Calculus 1. At the beginning of the piece Erdmann entitled On the Universal Science or Philosophical Calculus, Leibniz, in the course of summing up his views on the importance of a good characteristic, indicates that algebra is not the true characteristic for geometry, and alludes to a “more profound analysis” that belongs to geometry alone, samples of which he claims to possess.1 What is this properly geometrical analysis, completely different from algebra? How can we represent geometrical facts directly, without the mediation of numbers? What, finally, are the samples of this new method that Leibniz has left us? The present chapter will attempt to answer these questions.2 An essay concerning this geometrical analysis is found attached to a letter to Huygens of 8 September 1679, which it accompanied. In this letter, Leibniz enumerates his various investigations of quadratures, the inverse method of tangents, the irrational roots of equations, and Diophantine arithmetical problems.3 He boasts of having perfected algebra with his discoveries—the principal of which was the infinitesimal calculus.4 He then adds: “But after all the progress I have made in these matters, I am no longer content with algebra, insofar as it gives neither the shortest nor the most elegant constructions in geometry. That is why... I think we still need another, properly geometrical linear analysis that will directly express for us situation, just as algebra expresses magnitude. I believe I have a method of doing this, and that we can represent figures and even 1 “Progress in the art of rational discovery depends for the most part on the completeness of the characteristic art.
  • Single View Geometry Camera Model & Orientation + Position Estimation

    Single View Geometry Camera Model & Orientation + Position Estimation

    Single View Geometry Camera model & Orientation + Position estimation What am I? Vanishing point Mapping from 3D to 2D Point & Line Goal: Homogeneous coordinates Point – represent coordinates in 2 dimensions with a 3-vector &x# &x# homogeneous coords $y! $ ! ''''''→$ ! %y" %$1"! The projective plane • Why do we need homogeneous coordinates? – represent points at infinity, homographies, perspective projection, multi-view relationships • What is the geometric intuition? – a point in the image is a ray in projective space y (sx,sy,s) (x,y,1) (0,0,0) z x image plane • Each point (x,y) on the plane is represented by a ray (sx,sy,s) – all points on the ray are equivalent: (x, y, 1) ≅ (sx, sy, s) Projective Lines Projective lines • What does a line in the image correspond to in projective space? • A line is a plane of rays through origin – all rays (x,y,z) satisfying: ax + by + cz = 0 ⎡x⎤ in vector notation : 0 a b c ⎢y⎥ = [ ]⎢ ⎥ ⎣⎢z⎦⎥ l p • A line is also represented as a homogeneous 3-vector l Line Representation • a line is • is the distance from the origin to the line • is the norm direction of the line • It can also be written as Example of Line Example of Line (2) 0.42*pi Homogeneous representation Line in Is represented by a point in : But correspondence of line to point is not unique We define set of equivalence class of vectors in R^3 - (0,0,0) As projective space Projective lines from two points Line passing through two points Two points: x Define a line l is the line passing two points Proof: Line passing through two points • More
  • Where Parallel Lines Meet

    Where Parallel Lines Meet

    Where Parallel Lines WhereMeet parallel lines meet ...a geometric love story Renzo Cavalieri Renzo Cavalieri Colorado State University MATH DAY 2009 University of Northern Colorado Oct 22, 2014 Renzo Cavalieri Where parallel lines meet... Renzo Cavalieri Where Parallel Lines Meet Geometry γ"!µετρια \Geos = earth" + \Metron = measure" Was developed by King Sesostris as a way to counter tax fraud! Renzo Cavalieri Where Parallel Lines Meet Abstraction Abstraction In order to study in broader generalityIn order to the study properties in broader of spaces, __________ mathematiciansgenerality the properties created concepts of spaces, thatmathematicians abstract and created generalize concepts our physicalthat abstract experience. and generalize our Forphysical example, experience. a line or a plane do not reallyFor example, exist in our a line world!or a plane do not really exist in our world! Renzo Cavalieri Where Parallel Lines Meet Renzo Cavalieri Where Parallel Lines Meet Euclid's Posulates 1 A straight line segment can be drawn joining any two points. 2 Any straight line segment can be extended indefinitely in a straight line. 3 Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4 All right angles are congruent. 5 Given a line m and a point P not belonging to m, there is a unique line through P that never intersects m. Renzo Cavalieri Where Parallel Lines Meet The parallel postulate The parallel postulate was first shaken by the following puzzle: A hunter gets off her pickup and walks south for one mile. Then she makes a 90◦ left turn and walks east for another mile.
  • PROJECTIVE GEOMETRY Contents 1. Basic Definitions 1 2. Axioms Of

    PROJECTIVE GEOMETRY Contents 1. Basic Definitions 1 2. Axioms Of

    PROJECTIVE GEOMETRY KRISTIN DEAN Abstract. This paper investigates the nature of finite geometries. It will focus on the finite geometries known as projective planes and conclude with the example of the Fano plane. Contents 1. Basic Definitions 1 2. Axioms of Projective Geometry 2 3. Linear Algebra with Geometries 3 4. Quotient Geometries 4 5. Finite Projective Spaces 5 6. The Fano Plane 7 References 8 1. Basic Definitions First, we must begin with a few basic definitions relating to geometries. A geometry can be thought of as a set of objects and a relation on those elements. Definition 1.1. A geometry is denoted G = (Ω,I), where Ω is a set and I a relation which is both symmetric and reflexive. The relation on a geometry is called an incidence relation. For example, consider the tradional Euclidean geometry. In this geometry, the objects of the set Ω are points and lines. A point is incident to a line if it lies on that line, and two lines are incident if they have all points in common - only when they are the same line. There is often this same natural division of the elements of Ω into different kinds such as the points and lines. Definition 1.2. Suppose G = (Ω,I) is a geometry. Then a flag of G is a set of elements of Ω which are mutually incident. If there is no element outside of the flag, F, which can be added and also be a flag, then F is called maximal. Definition 1.3. A geometry G = (Ω,I) has rank r if it can be partitioned into sets Ω1,..., Ωr such that every maximal flag contains exactly one element of each set.
  • Euclidean Versus Projective Geometry

    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
  • Research Article Traditional Houses and Projective Geometry: Building Numbers and Projective Coordinates

    Research Article Traditional Houses and Projective Geometry: Building Numbers and Projective Coordinates

    Hindawi Journal of Applied Mathematics Volume 2021, Article ID 9928900, 25 pages https://doi.org/10.1155/2021/9928900 Research Article Traditional Houses and Projective Geometry: Building Numbers and Projective Coordinates Wen-Haw Chen 1 and Ja’faruddin 1,2 1Department of Applied Mathematics, Tunghai University, Taichung 407224, Taiwan 2Department of Mathematics, Universitas Negeri Makassar, Makassar 90221, Indonesia Correspondence should be addressed to Ja’faruddin; [email protected] Received 6 March 2021; Accepted 27 July 2021; Published 1 September 2021 Academic Editor: Md Sazzad Hossien Chowdhury Copyright © 2021 Wen-Haw Chen and Ja’faruddin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The natural mathematical abilities of humans have advanced civilizations. These abilities have been demonstrated in cultural heritage, especially traditional houses, which display evidence of an intuitive mathematics ability. Tribes around the world have built traditional houses with unique styles. The present study involved the collection of data from documentation, observation, and interview. The observations of several traditional buildings in Indonesia were based on camera images, aerial camera images, and documentation techniques. We first analyzed the images of some sample of the traditional houses in Indonesia using projective geometry and simple house theory and then formulated the definitions of building numbers and projective coordinates. The sample of the traditional houses is divided into two categories which are stilt houses and nonstilt house. The present article presents 7 types of simple houses, 21 building numbers, and 9 projective coordinates.
  • A Gallery of Conics by Five Elements

    A Gallery of Conics by Five Elements

    Forum Geometricorum Volume 14 (2014) 295–348. FORUM GEOM ISSN 1534-1178 A Gallery of Conics by Five Elements Paris Pamfilos Abstract. This paper is a review on conics defined by five elements, i.e., either lines to which the conic is tangent or points through which the conic passes. The gallery contains all cases combining a number (n) of points and a number (5 − n) of lines and also allowing coincidences of some points with some lines. Points and/or lines at infinity are handled as particular cases. 1. Introduction In the following we review the construction of conics by five elements: points and lines, briefly denoted by (αP βT ), with α + β =5. In these it is required to construct a conic passing through α given points and tangent to β given lines. The six constructions, resulting by giving α, β the values 0 to 5 and considering the data to be in general position, are considered the most important ([18, p. 387]), and are to be found almost on every book about conics. It seems that constructions for which some coincidences are allowed have attracted less attention, though they are related to many interesting theorems of the geometry of conics. Adding to the six main cases those with the projectively different possible coincidences we land to 12 main constructions figuring on the first column of the classifying table below. The six added cases can be considered as limiting cases of the others, in which a point tends to coincide with another or with a line. The twelve main cases are the projectively inequivalent to which every other case can be reduced by means of a projectivity of the plane.