Complete Quadrilateral Research Articles

Isoptic Point of the Complete Quadrangle

In this paper, we study the complete quadrangle. We started this investigation in a few of our previous papers. In those papers and here, the rectangular coordinates are used to enable us to prove the properties of the rich geometry of a quadrangle using the same method. Now, we are focused on the isoptic point of the complete quadrangle ABCD, which is the inverse point to A′,B′,C′, and D′ with respect to circumscribed circles of the triangles BCD, ACD, ABD, and ABC, respectively, where A′,B′,C′, and D′ are isogonal points to A,B,C, and D with respect to these triangles. In studying the properties of the quadrangle regarding its isoptic point, some new results are obtained as well.

On Some Properties of a Complete Quadrangle

In this paper, we study the properties of a complete quadrangle in the Euclidean plane. The proofs are based on using rectangular coordinates symmetrically on four vertices and four parameters a,b,c,d. Here, many properties of the complete quadrangle known from earlier research are proved using the same method, and some new results are given.

Side-side-angle triangle congruence axiom and the complete quadrilaterals

<abstract><p>Among the triangle congruence axioms, the side-side-angle (SsA) axiom states that two triangles are congruent if and only if two pairs of corresponding sides and the angles opposite the longer sides are equal. We construct two triangle sequences in which the items satisfy a modified condition. We require that the opposite angles of the shorter sides be equal. The locus of the intersection points of other sides of triangles is derived to be a hyperbola, and in a generalized form defined by a complete quadrilateral, it is a conic section.</p></abstract>

Circles Related to a Complete Quadrangle

U radu dajemo pregled nekih svojstava potpunog četverovrha ABCD u euklidskoj ravnini. Proučavamo kružnice s promjerima AB, AC, AD, BC, BD, CD, kao i nožišne trokute i nožišne kružnice točaka A, B, C, D s obzirom na trokute BCD, ACD, ABD, ABC redom navedene. Svi prikazani rezultati su poznati iz literature, ali ih ovdje dokazujemo koristeći istu metodu.

The Shape of the (15,3)-Arc of PG(2,7)

Marcugini et al. proved, by computer-based proof, the unicity of the maximum (k,3)-arc in PG(2,7). In this paper, we show how the (15,3)-Arc in PG(2,7) may be described using only geometrical properties. The description we provide, believing it is novel, relies on the union of a conic and a complete external quadrangle.

A Rarity in Geometry

We study the locus C of all points in the plane whose pedal points on the six sides of a complete quadrangle lie on a conic. In the Euclidean plane, it turns out that C is an algebraic curve of degree 7 and genus 5 and not of degree 12 as it could be expected. Septic curves occur rather seldom in geometry which motivates a detailed study of this particular curve. We look at its singularities, focal points, and those points on C whose pedal conics degenerate. Then, we show that the septic curve occurs as the locus curve for a more general question. Further, we describe those cases where C degenerates or is of degree less than 7 depending on the shape of the initial quadrilateral

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

We prove the equisingular rigidity of the singular Hirzebruch–Kummer coverings X(n, $$\mathcal {L}$$ ) of the projective plane branched on line configurations $$\mathcal {L}$$ , satisfying some technical condition. In the case, $$\mathcal {L}=$$ the complete quadrangle, we give explicit equations of the Hirzebruch–Kummer covering $$S_n(=$$ the minimal desingularisation of $$X (n, \mathcal {L}))$$ in a product of four Fermat curves of degree n. Since $$S_n$$ is the $$(\mathbb {Z}/n)^5$$ covering of the Del Pezzo surface $$Y_5$$ of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of $$Y_5$$ in $$(\mathbb {P}^1)^4$$ .We describe more generally determinantal equations for all Del Pezzo surfaces of degree $$9-k \le 6$$ as subvarieties of the k-fold product of the projective line.

On rigid compact complex surfaces and manifolds

This article investigates the subject of rigid compact complex manifolds. First of all we investigate the different notions of rigidity (local rigidity, global rigidity, infinitesimal rigidity, etale rigidity and strong rigidity) and the relations among them. Only for curves these notions coincide and the only rigid curve is the projective line. For surfaces we prove that a rigid surface which is not minimal of general type is either a Del Pezzo surface of degree ≥5 or an Inoue surface. We give examples of rigid manifolds of dimension n≥3 and Kodaira dimensions 0, and 2≤k≤n. Our main theorem is that the Hirzebruch Kummer coverings of exponent n≥4 branched on a complete quadrangle are infinitesimally rigid. Moreover, we pose a number of questions.

Method to solve intrinsic geometric parameters of a paracatadioptric camera using 3D lines

The paracatadioptric projection of a 3D line is a conic. In this paper, the constraint relation of three conics formed by three 3D lines according to the paracatadioptric projection is divided into three situations. In the first, three groups of vanishing points in orthogonal directions are formed by the principle point coordinates and six intersection points of the three conics. In the second, six groups of vanishing points in orthogonal directions are determined by common diameters on the unit sphere and tangents at intersection points of the three conics. In the third, three groups of circular points are formed by each side of a complete quadrangle formed by two circular points and the intersection points of the three conics. These three methods can solve the camera intrinsic parameters effectively and accurately, as verified by simulation and real data applications.

Linear Determination of a Camera's Intrinsic Parameters Using Two Intersecting Circles

An approach using two intersecting circles is proposed as a linear approach for determining a camera's intrinsic parameters. The two intersecting coplanar circles have four intersection points in the projective plane: two real points and two circular points. In the image plane, the diagonal triangle - on which the image of the four intersection points composes a complete quadrangle - is a self-polar triangle for the projection curves of the circles. The vertex of the self-polar triangle is the null space of the degenerate conic formed by the image of the four intersection points. By solving the three vertices of the self-polar triangle using the image coordinates of the two real intersection points, the degenerate conic can be obtained. The image of the two circular points is then computed from the intersection points of the degenerate conic. Using the image of the circular points from the three images of the same planar pattern with different directions, the intrinsic parameters can be linearly determined.

Central Points of the Complete Quadrangle

Generalizing the classical geometry of the triangle in the Euclidean plane E, we define a central point of an n-gon as a symmetric function En→ E which commutes with all similarities. We first review various geometrical characterizations of some well-known central points of the quadrangle (n = 4) and show how a look at their mutual positions produces a morphologic classification (cyclic, trapezoidal, orthogonal etc.). From a basis of four central points, full information on the quadrangle can be retrieved. This generalizes a problem first faced by Euler for the triangle. Reconstructing a quadrangle from its central points is a geometric analogue of solving an algebraic equation of degree 4: here the diagonal triangle plays the role of a Lagrange resolvent and the determination of loci for the central points replaces the examination of discriminants for real roots.

Kegelschnittbüschel und Inversion

The conjugacy mapping rel. to a complete quadrangle in a Pappian projective plane of characteristic ≠2 is constructed by using a bijection of the line set onto the bundle of conics through the diagonal points of the quadrangle. The inversion with center O of the inversion circle going through the point P in the Euclidean plane proves to be the product of the reflection at OP and the affine restriction of the conjugacy mapping rel. to the quadrangle having P as one of its vertices and O together with the circular points at infinity as diagonal points.

Eine Variante des Fano-Axioms fü angeordnete Ebenen

An ordered plane is an incidence structure (\((E,\mathcal{G})\)) with an order function ω, which satisfies the axioms (G), (V) and (S), but no continuation--axiom is required. Points a, b ∈ E are said to be in distinct sides of a line \(G \in \mathcal{G}\) iff \(\omega (G,a,b) = 1\) and in the same side if \(\omega (G,a,b) = 1\) , respectively. For any lines \(A = \overline {pa}\) , \(B = \overline {pb}\) and \(C = \overline {pc}\) we prove that if b,c are in the same side of line A and a,c are in the same side of B , then a and b are in distinct sides of C. As conclusions we deduce that ω is harmonic and that in each complete quadrangle the intersection points of the diagonals are never collinear, which is known as the axiom of Fano. So the Fano-axiom holds in each ordered plane, and also in those with boundary points.

On the Nine Point Conic Associated with a Complete Quadrangle

(1971). On the Nine Point Conic Associated with a Complete Quadrangle. Mathematics Magazine: Vol. 44, No. 5, pp. 261-266.

On the Nine Point Conic Associated with a Complete Quadrangle

On the Nine Point Conic Associated with a Complete Quadrangle

Profile Analysis--A Geologically Oriented Gravity Interpretation: ABSTRACT

A byproduct of the examination of three different analytic methods of reducing gravity data to a form useful to the geologists is the presentation of the local gravitational field of one complete quadrangle in the Jurassic trend of Mississippi. The major salt dome minima are qualitatively confirmed by all three methods, although the quantitative effects vary. The more subtle effects of the Jurassic features are more susceptible to distortion by the process of removing the influence of regional density changes. The least distorting method of regional removal is shown to be the interlocking profile network. This technique obtains the most definitive resolution of the local gravitational effects of deep, low-volume structures. End_of_Article - Last_Page 2035------------

On a Complete Quadrangle Inscribed in a Conic

On a Complete Quadrangle Inscribed in a Conic

On a Complete Quadrangle Inscribed in a Conic

On a Complete Quadrangle Inscribed in a Conic

Collegiate Articles: Graduate Training not Required for Reading: The Complete Quadrilateral

Collegiate Articles: Graduate Training not Required for Reading: The Complete Quadrilateral

Points Connected with the Complete Quadrilateral

1. The stimulus that provoked this article was the following proof of the fact that the middle points of the diagonals of a complete quadrilateral are collinear.