DOCSLIB.ORG

10 Conics and Perspectivities

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
10 Conics and Perspectivities 10 Conics and perspectivities Some people have got a mental horizon of radius zero and call it their point of view. Attributed to David Hilbert In the last chapter we treated conics more or less as isolated objects. We defined points on them and lines tangent to them. Now we want to investigate various geometric and algebraic properties of conics. In particular, we will see how we can treat conics on the level of bracket algebra. 10.1 Conic through five points We start by calculating a conic through a given set of points. For this consider the quadratic equation that defines a conic. a x2 + b y2 + c z2 + d xy + e xz + f yz = 0. · · · · · · This equation has six parameters a, . , f 1. Multiplying all of them simultane- ously by the same non-zero scalar leads to the same conic. Thus the parameter vector (a, . , f) behaves like a vector of homogeneous coordinates. Counting degrees of freedom shows that in general it will take five points to uniquely determine a conic. To find the parameters for a conic through five points pi = (xi, yi, zi); i = 1, . , 5 we simply have to solve the following linear system of equations: 1 Compared to Section 9.1 we have relabeled the parameters and put the factor of 2 of the mixed terms into the parameters 164 10 Conics and perspectivities 2 2 2 a x1 y1 z1 x1y1 x1z1 y1z1 0 2 2 2 b x2 y2 z2 x2y2 x2z2 y2z2 0 2 2 2 c x3 y3 z3 x3y3 x3z3 y3z3 = 0 2 2 2 · d x4 y4 z4 x4y4 x4z4 y4z4 0 2 2 2 e x y z x5y5 x5z5 y5z5 0 5 5 5 f If this system has a full rank of 5 then there is an (up to scalar multiple) unique solution (a, . , f) that defines the corresponding conic. If more than 3 points are simultaneously collinear of if two points coincide the rank of the system may be lower than 5. This corresponds to the situation that there are more than one conic passing through the given set of points. This method of determining the parameter vector (a, . , f) is mathematically elegant how- ever it is computationally expensive. We first have to calculate the squared parameters and then have to solve a 5 times 6 system of equalities. There is also another way to calculate such a conic more or less directly. This way will also give us additional structural insight into the geometry and underlying algebra of a conic. In preparation we have to understand how to calculate a degenerate conic that consists of two lines with homogeneous coor- dinates g and h. A conic must be represented by a quadratic form pT AP p = 0 that vanishes if p is on either of the lines. The (non-symmetrized) matrix of such a quadratic form is simply given by A = ghT . This can be easily seen since the quadratic form pT Ap = pT (ghT )p = (pT g)(hT p) = p, g p, h ! "! " vanish if one of the two scalar products on right side vanish. This in turn corresponds geometrically to the situation in which p is on g or on h. Assume that the line g is spanned by two points labeled 1 and 2 and that line h is spanned by two points labeled 3 and 4. Then we have g = 1 2 and h = 3 4. The quadratic form becomes × × p, 1 2 p, 3 4 = 0. ! × "! × " We may as well express this term as the product of two determinants [p, 1, 2][p, 3, 4] = 0. Each factor describes a linear condition on the point p. The product calculates the conjunction between the two expressions. Now, assume that we want to describe the set of conics that passes through for points 1, . , 4 in general position. Clearly, there are many conics that satisfy this condition. The corresponding system of linear equations con- sists of four equations in six variables. Hence the solution space will be two- dimensional. One of these two degrees of freedom goes into the homogeneity of 10.1 Conic through five points 165 1 4 PSfrag replacements PSfrag replacements 1 2 4 3 2 3 Fig. 10.1. Bundles of conics though four points. Three degenerate special cases. the conic parameters. Therefore we have a bundle of geometric solutions with one degree of freedom. Figure 10.1 (left) illustrates such a bundle of conics. Among these conics there are three degenerate conics, each of them passing through a pair of lines spanned by the four points. In Figure 10.1 (right) these pairs of lines are marked by identical colors. They correspond to the following four quadratic forms: [p, 1, 2][p, 3, 4] = 0, [p, 1, 3][p, 2, 4] = 0, [p, 1, 4][p, 2, 3] = 0. A linear combination of two of these forms (say the last two) λ[p, 1, 3][p, 2, 4] + µ[p, 1, 4][p, 2, 3] = 0 generates again a quadratic form. The set of points p satisfying this equation forms again a conic. This conic passes through all four points 1, . , 4 since both summands vanish on these points. If λ and µ run trough all possible values we obtain all the conics in the bundle through the four points. Applying the technique of Plu¨cker’s µ (compare Section 6.3) we can adjust these values 1 4 PSfrag replacements q 3 2 Fig. 10.2. Constructing a conic through five points. 166 10 Conics and perspectivities such that the resulting conic passes through another given point q. For this we have to simply choose λ = [q, 1, 4][q, 2, 3]; µ = [q, 1, 3][q, 2, 4]. − The resulting conic equation can be written as [q, 1, 4][q, 2, 3][p, 1, 3][p, 2, 4] [q, 1, 3][q, 2, 4][p, 1, 4][p, 2, 3] = 0. − Observe that this equation is a multi-homogeneous bracket polynomial that is quadratic in each of the six involved points. Figure 10.2 illustrates the situa- tion. We can also interpret it as a bracket condition encoding the (projectively invariant) property that six points 1, . , 4, p, q are on a conic (compare Sec- tion 6.4 and Section 7.3). We will come back to this interpretation in the next Section. Before this we will give the procedure for calculating calculate the sym- metric matrix for the conic through the five points 1, 2, 3, 4, q. We give it as a kind of simple computer program: 1: g := 1 3; 1 × 2: g := 2 4; 2 × 3: h := 1 3; 1 × 4: h := 2 4; 2 × T 5: G := g1g2 ; T 6: H := h1h2 ; 7: M := qT HqG qT GqH; − 8: A := M + M T ; The matrix A assigned in the last line of the program contains the sym- metrized matrix. 10.2 Conics and cross ratios Let us come back to the equation [q, 1, 4][q, 2, 3][p, 1, 3][p, 2, 4] [q, 1, 3][q, 2, 4][p, 1, 4][p, 2, 3] = 0, ( ) − ∗ which characterizes whether six points are on a conic. First observe that this equation is highly symmetric. For each bracket in one term its complement (the bracket consisting of the other three letters) is in the other term. The symmetry becomes a bit more transparent if we rewrite the equation with new points labels: 10.2 Conics and cross ratios 167 2 3 4 1 PSfrag replacements p q Fig. 10.3. Four points on a conic seen from other points of a conic. [A, B, C][A, Y, Z][X, B, Z][X, Y, C] [A, B, Z][A, Y, C][X, B, C][X, Y, Z] = 0. − There is another important observation that we can make by rewriting equation ( ). We assume that the conic is non-degenerate and that none of the determ∗inants vanishes. In this case we can rewrite ( ) to the form ∗ [q, 1, 4][q, 2, 3] [p, 1, 4][p, 2, 3] = . [q, 1, 3][q, 2, 4] [p, 1, 3][p, 2, 4] Both sides of the equation represent cross ratios. The left side is a cross ratio of the lines p1, p2, p3, p4 the right side of the equation is a cross ratio of the lines q1, q2, q3, q4. We abbreviate [q, 1, 4][q, 2, 3] (1, 2; 3, 4) := . q [q, 1, 3][q, 2, 4] This is the cross ratio of 1, 2, 3, 4 as “seen from” point q. Thus equation ( ) may be restated as ∗ (1, 2; 3, 4)q = (1, 2; 3, 4)p. Point p and point q see the points 1, 2, 3, 4 under the same cross ratio. The situation is shown in Figure 10.3 We summarize this in a theorem: Theorem 10.1. Let 1, 2, 3, 4, p be five points on a conic such that p is distinct from the other four points. Then the cross ratio (1, 2; 3, 4)p is independent of the special choice of p. We will later on see that this theorem is very closely related to the so called exterior angle theorem for circles which states that in a circle a fixed secant is seen from an arbitrary point on the circle under the same angle (modulo π). 168 10 Conics and perspectivities The last theorem enables us to speak of the cross ratio of four points on a fixed conic as long as no more than two of the points (1, 2, 3, 4) coincide and we can speak of a cross ratio at all.
Load more

Recommended publications
  • Desargues' Theorem and Perspectivities
    Desargues' theorem and perspectivities A file of the Geometrikon gallery by Paris Pamfilos The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me - both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension. Paul Halmos, I Want to be a Mathematician, p.3 Contents 1 Desargues’ theorem1 2 Perspective triangles3 3 Desargues’ theorem, special cases4 4 Sides passing through collinear points5 5 A case handled with projective coordinates6 6 Space perspectivity7 7 Perspectivity as a projective transformation9 8 The case of the trilinear polar 11 9 The case of conjugate triangles 13 1 Desargues’ theorem The theorem of Desargues represents the geometric foundation of “photography” and “per- spectivity”, used by painters and designers in order to represent in paper objects of the space. Two triangles are called “perspective relative to a point” or “point perspective”, when we can label them ABC and A0B0C0 in such a way, that lines fAA0; BB0;CC0g pass through a 1 Desargues’ theorem 2 common point P: Points fA; A0g are then called “homologous” and similarly points fB; B0g and fC;C0g. The point P is then called “perspectivity center” of the two triangles. The A'' B'' C' C A' P A B B' C'' Figure 1: “Desargues’ configuration”: Theorem of Desargues two triangles are called “perspective relative to a line” or “line perspective” when we can label them ABC and A0B0C0 , in such a way (See Figure 1), that the points of intersection of their sides C00 = ¹AB; A0B0º; A00 = ¹BC; B0C0º and B00 = ¹CA;C0 A0º are contained in the same line ": Sides AB and A0B0 are then called “homologous”, and similarly the side pairs ¹BC; B0C0º and ¹CA;C0 A0º: Line " is called “perspectivity axis” of the two triangles.
    [Show full text]
  • Precise Image Registration and Occlusion Detection
    Precise Image Registration and Occlusion Detection A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Vinod Khare, B. Tech. Graduate Program in Civil Engineering The Ohio State University 2011 Master's Examination Committee: Asst. Prof. Alper Yilmaz, Advisor Prof. Ron Li Prof. Carolyn Merry c Copyright by Vinod Khare 2011 Abstract Image registration and mosaicking is a fundamental problem in computer vision. The large number of approaches developed to achieve this end can be largely divided into two categories - direct methods and feature-based methods. Direct methods work by shifting or warping the images relative to each other and looking at how much the pixels agree. Feature based methods work by estimating parametric transformation between two images using point correspondences. In this work, we extend the standard feature-based approach to multiple images and adopt the photogrammetric process to improve the accuracy of registration. In particular, we use a multi-head camera mount providing multiple non-overlapping images per time epoch and use multiple epochs, which increases the number of images to be considered during the estimation process. The existence of a dominant scene plane in 3-space, visible in all the images acquired from the multi-head platform formulated in a bundle block adjustment framework in the image space, provides precise registration between the images. We also develop an appearance-based method for detecting potential occluders in the scene. Our method builds upon existing appearance-based approaches and extends it to multiple views.
    [Show full text]
  • Set Theory. • Sets Have Elements, Written X ∈ X, and Subsets, Written a ⊆ X. • the Empty Set ∅ Has No Elements
    Set theory. • Sets have elements, written x 2 X, and subsets, written A ⊆ X. • The empty set ? has no elements. • A function f : X ! Y takes an element x 2 X and returns an element f(x) 2 Y . The set X is its domain, and Y is its codomain. Every set X has an identity function idX defined by idX (x) = x. • The composite of functions f : X ! Y and g : Y ! Z is the function g ◦ f defined by (g ◦ f)(x) = g(f(x)). • The function f : X ! Y is a injective or an injection if, for every y 2 Y , there is at most one x 2 X such that f(x) = y (resp. surjective, surjection, at least; bijective, bijection, exactly). In other words, f is a bijection if and only if it is both injective and surjective. Being bijective is equivalent to the existence of an inverse function f −1 such −1 −1 that f ◦ f = idY and f ◦ f = idX . • An equivalence relation on a set X is a relation ∼ such that { (reflexivity) for every x 2 X, x ∼ x; { (symmetry) for every x; y 2 X, if x ∼ y, then y ∼ x; and { (transitivity) for every x; y; z 2 X, if x ∼ y and y ∼ z, then x ∼ z. The equivalence class of x 2 X under the equivalence relation ∼ is [x] = fy 2 X : x ∼ yg: The distinct equivalence classes form a partition of X, i.e., every element of X is con- tained in a unique equivalence class. Incidence geometry.
    [Show full text]
  • Conic Homographies and Bitangent Pencils
    Forum Geometricorum b Volume 9 (2009) 229–257. b b FORUM GEOM ISSN 1534-1178 Conic Homographies and Bitangent Pencils Paris Pamfilos Abstract. Conic homographies are homographies of the projective plane pre- serving a given conic. They are naturally associated with bitangent pencils of conics, which are pencils containing a double line. Here we study this connec- tion and relate these pencils to various groups of homographies associated with a conic. A detailed analysis of the automorphisms of a given pencil specializes to the description of affinities preserving a conic. While the algebraic structure of the groups involved is simple, it seems that a geometric study of the vari- ous questions is lacking or has not been given much attention. In this respect the article reviews several well known results but also adds some new points of view and results, leading to a detailed description of the group of homographies preserving a bitangent pencil and, as a consequence, also the group of affinities preserving an affine conic. 1. Introduction Deviating somewhat from the standard definition I call bitangent the pencils P of conics which are defined in the projective plane through equations of the form αc + βe2 = 0. Here c(x,y,z) = 0 and e(x,y,z) = 0 are the equations in homogeneous coordi- nates of a non-degenerate conic and a line and α,β are arbitrary, no simultaneously zero, real numbers. To be short I use the same symbol for the set and an equation representing it. Thus c denotes the set of points of a conic and c = 0 denotes an equation representing this set in some system of homogeneous coordinates.
    [Show full text]
  • Foundations of Projective Geometry
    Foundations of Projective Geometry Robin Hartshorne 1967 ii Preface These notes arose from a one-semester course in the foundations of projective geometry, given at Harvard in the fall term of 1966–1967. We have approached the subject simultaneously from two different directions. In the purely synthetic treatment, we start from axioms and build the abstract theory from there. For example, we have included the synthetic proof of the fundamental theorem for projectivities on a line, using Pappus’ Axiom. On the other hand we have the real projective plane as a model, and use methods of Euclidean geometry or analytic geometry to see what is true in that case. These two approaches are carried along independently, until the first is specialized by the introduction of more axioms, and the second is generalized by working over an arbitrary field or division ring, to the point where they coincide in Chapter 7, with the introduction of coordinates in an abstract projective plane. Throughout the course there is special emphasis on the various groups of transformations which arise in projective geometry. Thus the reader is intro- duced to group theory in a practical context. We do not assume any previous knowledge of algebra, but do recommend a reading assignment in abstract group theory, such as [4]. There is a small list of problems at the end of the notes, which should be taken in regular doses along with the text. There is also a small bibliography, mentioning various works referred to in the preparation of these notes. However, I am most indebted to Oscar Zariski, who taught me the same course eleven years ago.
    [Show full text]
  • Differential Spatial Resection-Pose Estimation Using a Single Local Image Feature
    Differential Spatial Resection - Pose Estimation Using a Single Local Image Feature Kevin K¨oser and Reinhard Koch Institute of Computer Science Christian-Albrechts-University of Kiel 24098 Kiel, Germany {koeser,rk}@mip.informatik.uni-kiel.de Abstract. Robust local image features have been used successfully in robot localization and camera pose estimation; region tracking using affine warps is considered state of the art also for many years. Although such correspondences provide a warp of the local image region and are quite powerful, in direct pose estimation they are so far only consid- ered as points and therefore three of them are required to construct a camera pose. In this contribution we show how it is possible to directly compute a pose based upon one such feature, given the plane in space where it lies. This differential correspondence concept exploits the tex- ture warp and has recently gained attention in estimation of conjugate rotations. The approach can also be considered as the limiting case of the well-known spatial resection problem when the three 3D points ap- proach each other infinitesimally close. We show that the differential correspondence is more powerful than conic correspondences while its exploitation requires nothing more complicated than the roots of a third order polynomial. We give a detailed sensitivity analysis, a comparison against state-of-the-art pose estimators and demonstrate real-world ap- plicability of the algorithm based on automatic region recognition. 1 Introduction Since the first description of spatial resection from 3 points by Grunert[7] in 1841, many people have worked on pose estimation or the so called P3P problem [5,31,6,8].
    [Show full text]
  • Foundations of Projective Geometry
    CHAPTER 15 Foundations of Projective Geometry What a delightful thing this perspective is! — Paolo Uccello (1379-1475) Italian Painter and Mathemati- cian 15.1 AXIOMS OF PROJECTIVE GEOMETRY In section 9.3 of Chapter 9 we covered the four basic axioms of Projective geometry: P1 Given two distinct points there is a unique line incident on these points. P2 Given two distinct lines, there is at least one point incident on these lines. P3 There exist three non-collinear points. P4 Every line has at least three distinct points incident on it. We then introduced the notions of triangles and quadrangles and saw that there was a finite projective plane with 7 points and 7 lines that had a peculiar property in relation to quadrangles. The odd quadrangle behavior turns out to absent in most of the classical models of Projective geometry. If we want to rule out this behavior we need a fifth axiom: Axiom P5: (Fano’s Axiom) The three diagonal points of a complete quadrangle are not collinear. 171 172 Exploring Geometry - Web Chapters This axiom is named for the Italian mathematician Gino Fano (1871– 1952). Fano discovered the 7 point projective plane, which is now called the Fano plane. The Fano plane is the simplest figure which satisfies Ax- ioms P1–P4, but which has a quadrangle with collinear diagonal points. (For more detail on this topic, review section 9.3.4.) The basic set of four axioms is not strong enough to prove one of the classical theorems in Projective geometry —Desargues’ Theorem. Theorem 15.1.
    [Show full text]
  • CHAP05 Perspectivities and Projectivities
    5. PERSPECTIVITIES AND PROJECTIVITIES §5.1. Perspectivities If h and k are distinct projective lines and C is a projective point on neither, then the perspectivity from h to k, with centre C, is the map f(P) = CP ∩ k. C Q P h k f(P) f(Q) Theorem 1: Cross ratios are preserved by perspectivities. Proof: Let f: h → k be a perspectivity with centre C. Let P, Q, R, S be four points on h with P, Q, R distinct and S ≠ P. Let P ′ = f(P), Q ′ = f(Q), R ′ = f(R) and S ′ = f(S). S R Q P P′ Q′ R′ S′ k C If P = 〈p〉 then, by the Collinearity Lemma, there exists a vector q and a scalar λ such that Q = 〈q〉, R = 〈p + q〉, S = 〈λp + q〉 where λ = ℜ(P, Q; R, S). Also there exists c such that C = 〈c〉; P′ = 〈p + c〉 . Now C, Q, Q ′ are collinear so Q ′ = 〈αq + βc〉 for scalars α, β. Since Q ′ ≠ C, α ≠ 0 so without loss of generality we may take α = 1 and Q ′ = 〈q + βc〉. Since R ′ = CR ∩ P ′Q ′ and (p + c) +(q + βc) = (p + q) + (1 + β)c, we have R ′ = 〈(p + c) +(q + βc)〉 . Since S ′ = P ′ Q ′ ∩ CS and λ (p + c) +(q + βc) = (λp + q) + (λ + β)c, we have S ′ = 〈λ (p + c) + (q + βc〉. 63 So P ′ = 〈p + c〉, Q ′ = 〈q + βc〉, R ′ = 〈(p + c) + (q + βc)〉, S ′ = 〈λ (p + c) + (q + βc)〉. Hence ℜ( P ′, Q ′; R ′, S ′) = λ = ℜ(P, Q; R, S).
    [Show full text]
  • Real-Time Embedded Panoramic Imaging for Spherical Camera System
    Real-time Embedded Panoramic Imaging for Spherical Camera System Main Uddin-Al-Hasan This thesis is presented as part of Degree of Bachelor of Science in Electrical Engineering Blekinge Institute of Technology September 2013 Blekinge Institute of Technology School of Engineering Department of Electrical Engineering Supervisor: Dr. Siamak Khatibi Examiner: Dr. Sven Johansson 2 3 Abstract Panoramas or stitched images are used in topographical mapping, panoramic 3D reconstruction, deep space exploration image processing, medical image processing, multimedia broadcasting, system automation, photography and other numerous fields. Generating real-time panoramic images in small embedded computer is of particular importance being lighter, smaller and mobile imaging system. Moreover, this type of lightweight panoramic imaging system is used for different types of industrial or home inspection. A real-time handheld panorama imaging system is developed using embedded real- time Linux as software module and Gumstix Overo and PandaBoard ES as hardware module. The proposed algorithm takes 62.6602 milliseconds to generate a panorama frame from three images using a homography matrix. Hence, the proposed algorithm is capable of generating panorama video with 15.95909365 frames per second. However, the algorithm is capable to be much speedier with more optimal homography matrix. During the development, Ångström Linux and Ubuntu Linux are used as the operating system with Gumstix Overo and PandaBoard ES respectively. The real-time kernel patch is used to configure the non-real- time Linux distribution for real-time operation. The serial communication software tools C- Kermit, Minicom are used for terminal emulation between development computer and small embedded computer. The software framework of the system consist UVC driver, V4L/V4L2 API, OpenCV API, FFMPEG API, GStreamer, x264, Cmake, Make software packages.
    [Show full text]
  • Projectivities with Fixed Points on Every Line of the Plane
    PROJECTIVITIES WITH FIXED POINTS ON EVERY LINE OF THE PLANE REINHOLD BAER The system of fixed elements of a projectivity contains with any two points the line connecting them and with any two lines their in­ tersection. It is, therefore, in its structure very much like a subplane of the plane under consideration ; and thus one may expect the struc­ ture of the projectivity to be dominated by the structure of the system of its fixed elements, provided this system is not "too small." To substantiate this we propose to investigate in this note a class of projectivities which we term quasi-perspectivities. They are charac­ terized by the property that every line carries a fixed point or, equiva- lently, that every point is on some fixed line. Every perspectivity is a quasi-perspectivity, and a quasi-perspec- tivity is not a perspectivity if, and only if, the system of its fixed elements is a projective subplane. Involutions are quasi-perspectivi­ ties too, and if the Theorem of Pappus is valid in the plane under consideration, then every quasi-perspectivity is a perspectivity or an involution. But already in the projective plane over the field of real quaternions there exist quasi-perspectivities which are neither per- spectivities nor involutions, and we give a complete survey of the quasi-perspectivities in Desarguesian projective planes. Our results become particularly striking in the case of finite pro­ jective planes. If every line in such a plane carries n + 1 points, then we may show that there do not exist projectivities possessing exactly n fixed points, that a projectivity is a quasi-perspectivity if, and only if, the number of its fixed points is at least w + 1, and that it is a perspectivity if, and only if, the number of its fixed points is w+1 or n+2.
    [Show full text]
  • Projective Geometry in a Plane
    Projective Geometry in a Plane Fundamental Concepts Undefined Concepts: Point, line, and incidence Axiom 1. Any two points P, Q lie on exactly one line, denoted PQ. Axiom 2. Any two lines l, m intersect in at least one point, denoted l•m. Definition. A quadrangle is a set of four points, no three of which are collinear. Axiom 3. A quadrangle exists. Axiom 4. If PQRS is a quadrangle, then its diagonal points PQ•RS, PR•QS, and PS•QR are not collinear. Definition. A range on l is a set of points lying on the line l. Definition. A pencil at P is a set of lines passing through the point P. Definition. An elementary projectivity is the correspondence between lines in a pencil and points in a range given by incidence: s p q r S R Q P Figure 1. An elementary projectivity. Definition. A projectivity is a finite composition of elementary projectivities, written as PQRS P’Q’R’S’, PQRS pqrs, pqrs p’q’r’s’, or pqrs PQRS. Axiom 5. If a projectivity fixes three points on a line, it fixes every point on the line. Definition. A perspectivity, denoted , is a composition of two projectivities. If lines in a pencil at P relate the two ranges in a perspectivity (see Figure 2), the ranges are said to be perspective from center P. If the ranges lie on lines l and m, this may also be called a projection of points on l to points on m from P (or vice-versa) P l C D A B D' m C' B' A' .
    [Show full text]
  • Theorems of Points and Planes in Three-Dimensional Projective Space
    J. Aust. Math. Soc. 88 (2010), 75–92 doi:10.1017/S1446788708080981 THEOREMS OF POINTS AND PLANES IN THREE-DIMENSIONAL PROJECTIVE SPACE DAVID G. GLYNN (Received 15 July 2007; accepted 4 April 2008) Communicated by L. M. Batten Abstract We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n − 1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal. 2000 Mathematics subject classification: primary 51A20; secondary 51A45, 05B25, 05B35. Keywords and phrases: geometry, configuration, projective space, theorem, minimal energy electrons, forbidden minor, Fano, Pappus, Desargues, Möbius. 1. Introduction DEFINITION 1.1. A (combinatorial) nk configuration is an incidence structure .P; B; I / with a set P of n points and a set B of n blocks (which may be considered to be subsets of P), such that each block contains k of the points, and each point is on k of the blocks. A point p 2 P is on a block b 2 B if and only if pI b, where I is the incidence relation.
    [Show full text]