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Common transversals and tangents to two lines and two quadrics in P^3 [article]

Gábor Megyesi , Thorsten Theobald
2002 arXiv   pre-print
many lines simultaneously transversal to the two lines and tangent to the two spheres?  ...  complete geometric description of the set of (second) quadrics for which the 2 lines and 2 quadrics have infinitely many transversals and tangents: In the nine-dimensional projective space P^9 of quadrics  ...  Acknowledgements We thank Dan Grayson and Mike Stillman, who, as editors for the book "Computations in Algebraic Geometry with Macaulay 2" introduced the typography used in the Appendix.  ... 
arXiv:math/0206044v1 fatcat:lfi3lylinfhvhdcnvcovun7phi

Common Transversals and Tangents to Two Lines and Two Quadrics in P

G�bor Megyesi, Frank Sottile, Thorsten Theobald
2003 Discrete & Computational Geometry  
many lines simultaneously transversal to the two lines and tangent to the two spheres?  ...  complete geometric description of the set of (second) quadrics for which the 2 lines and 2 quadrics have infinitely many transversals and tangents: In the nine-dimensional projective space P 9 of quadrics  ...  Acknowledgements We thank Dan Grayson and Mike Stillman, who, as editors for the book "Computations in Algebraic Geometry with Macaulay 2" introduced the typography used in the Appendix.  ... 
doi:10.1007/s00454-003-0789-4 fatcat:pdny6lgoz5djpetc3fdspgserq

Quadric splines

Claudia Bangert, Hartmut Prautzsch
1999 Computer Aided Geometric Design  
In particular, Dahmen 1989] and Guo 1991] used triangular segments of quadrics to build tangent plane continuous surfaces interpolating the vertices of a triangular net with prescribed normals.  ...  While Dahmen's and Guo's approach is completely algebraic, the objective o f this paper is to derive their quadric splines solely geometrically in projective space.  ...  Following Dahmen and Guo we c hoose the six planes separating the six quadric segments so as to meet in some so called transversal line L through p .  ... 
doi:10.1016/s0167-8396(98)00047-8 fatcat:kfu5uh5izjeihnqhznikxiwuda

Flock generalized quadrangles and tetradic sets of elliptic quadrics of PG(3,q)

S.G. Barwick, Matthew R. Brown, Tim Penttila
2006 Journal of combinatorial theory. Series A  
It was shown by Thas in 1987 that to such flocks correspond generalized quadrangles of order (q 2 , q), previously constructed algebraically by Kantor (q odd) and Payne (q even).  ...  In this paper we characterise these sets of elliptic quadrics by a simple property, construct the generalized quadrangle synthetically from the properties of the set and strengthen the main theorem of  ...  Hence by Lemma 14 the two elliptic quadrics are in a common transversal, a contradiction since by Lemma 15 the can only have two common tangent planes, but we know that each element of C * is tangent to  ... 
doi:10.1016/j.jcta.2005.03.004 fatcat:jprgevs7z5e2hf7zoalrl5c5o4

Families of sections of quadrics and classical geometries

A. G. Khovanskii
2000 Functional analysis and its applications  
Conversely, any quadric tangent to the plane z = 0 at zero is given by an equation of the form z = B 2 + L 1 z + cz 2 , where B 2 and L 1 are homogeneous polynomials in x and y of degrees one and two,  ...  A nondegenerate quadric intersects a tangent plane to itself by a pair of lines, real or complex. We can readily verify the following assertion. Lemma 2.  ...  In this case, it is locally rectifiable in a neighborhood of the quadric point if and only if all transversal planes pass through a common straight line transversal to the surface at the quadric point.  ... 
doi:10.1007/bf02482419 fatcat:wo4gbmqxfbbxlopy5b6zrq222u

Line problems in nonlinear computational geometry [article]

Frank Sottile, Thorsten Theobald
2007 arXiv   pre-print
The main part of this survey is recent work on a core algebraic problem--studying the lines tangent to k spheres that also meet 4-k fixed lines.  ...  We give an example of four disjoint spheres with 12 common real tangents.  ...  Suppose the lines ℓ 1 and ℓ 2 meet in a point p and span a plane H. Then there are two families of transversals to the two lines; lines that pass through p and lines contained in H.  ... 
arXiv:math/0610407v2 fatcat:dzae7umpyfcujm5ple4plpsovm

Real k-flats tangent to quadrics in R^n

Frank Sottile, Thorsten Theobald
2005 arXiv   pre-print
For each k between 1 and n-2 there are 2^d_k,n· #_k,n (a priori complex) k-planes in P^n tangent to d_k,n general quadratic hypersurfaces in P^n.  ...  ., for each k between 1 and n-2 there exists a configuration of d_k,n real quadrics in (affine) real space R^n so that all the mutually tangent k-flats are real.  ...  The two common transversal lines are given by x 2 = x 4 = 0 and by x 1 = x 4 = 0.  ... 
arXiv:math/0206099v4 fatcat:byekh3a7qzcetckrid6pxvwada

A new look at the Kummer surface

W. L. Edge
1967 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Each quadric of 31 is shown, in §5, to contain 32 tangent planes of F.  ...  All the quadrics of 31 have a common self-polar simplex 2, and 31 can, in a double-infinity of ways, be based on a quadric 12i and two quadrics that 12i reciprocates into each other.  ...  Two lines on F that have, on F, a common transversal map two nodes on the same conic or two conies through the same node ; if two lines on F do have a common transversal, they have two: if two nodes of  ... 
doi:10.4153/cjm-1967-087-5 fatcat:fuqnuvao3zctrbh4yj7lmuumwu

An Enumerative Geometry Framework for Algorithmic Line Problems in $\mathbb R^3$

Thorsten Theobald
2002 SIAM journal on computing (Print)  
For this purpose, we study the common tangent lines/transversals to k balls of arbitrary radii and 4 − k lines in R 3 .  ...  Our results extend the results of Macdonald, Pach, and Theobald who investigated common tangents to four unit balls in R 3 [Discrete  ...  The author would like to thank Abhi Dattasharma for his useful comments and the anonymous referees for their suggestions and for pointing out the issue of envelopes.  ... 
doi:10.1137/s009753970038050x fatcat:zokfljtnxzhvxnlcjnqhfj4bxe

Real $k$-flats tangent to quadrics in $\mathbb {R}^n$

Frank Sottile, Thorsten Theobald
2005 Proceedings of the American Mathematical Society  
For each 1 ≤ k ≤ n − 2 there are 2 d k,n ·# k,n (a priori complex) k-planes in P n tangent to d k,n general quadratic hypersurfaces in P n .  ...  ., for 1 ≤ k ≤ n − 2 there exists a configuration of d k,n real quadrics in (affine) real space R n so that all the mutually tangent k-flats are real.  ...  The two common transversal lines are given by x 2 = x 4 = 0 and by x 1 = x 4 = 0.  ... 
doi:10.1090/s0002-9939-05-07880-9 fatcat:czxdt7njajhrderscsdbq5vg6a

Problem.-To Fit Together Two or More Quadrics so That Their Intersections Shall Be Plane

E. W. Hyde
1876 The Analyst  
The points in which these two plane curves intersect will be at the extremities of a chord common to the two quadrics, and at these two points common tangent planes to the two surfaces may be drawn.  ...  If the tangents touch GKHL on opposite sides of GH, as in the case of BF and QR, the point P?  ... 
doi:10.2307/2635804 fatcat:52zsa4kehjfcli2sncp77wvg6q

Ovoidal packings of P G ( 3 , q ) for even q

Bhaskar Bagchi, N.S. Narasimha Sastry
2013 Discrete Mathematics  
lines, and they consist of the transversals to the pair xy, π x ∩ π y of skew lines.  ...  There is a similar (but more complicated) result for the common tangent lines to q ovoids in P G(3, q) which are mutually tangent at a common point and share a common tangent plane through this point.  ...  Acknowledgement: The authors thank Hendrik Van Maldeghem for helpful discussion with one of us, regarding Theorem 2.2 and Lemma 3.1; and also A. Bruen and I.  ... 
doi:10.1016/j.disc.2013.05.021 fatcat:nl7lqgllpneyxmr4g3wzwzotmu

Codes Defined by Forms of Degree $2$ on Quadric Surfaces

FrÉdÉric A. B. Edoukou
2008 IEEE Transactions on Information Theory  
We study the functional codes C2(X) defined on projective varieties X, in the case where X ⊂ P 3 is a 1-degenerate quadric or a non-degenerate quadric (hyperbolic or elliptic).  ...  One result states that the codes C2(X) defined on the elliptic quadrics are good codes according to the table of A. E. Brouwer.  ...  In [5] p.14 we have Table I of the quadrics in P G (3, q) . A regulus is the set of transversals of three skew lines in P 3 . It consists of q + 1 skew lines.  ... 
doi:10.1109/tit.2007.913450 fatcat:3uwufzqv5besfdgwdtdhcfdfxq

Codes Defined By Forms Of Degree 2 On Quadric Surfaces [article]

Frederic A. B. Edoukou
2005 arXiv   pre-print
We study the functional codes C_2(X) defined on projective varieties X, in the case where X⊂P^3 is a 1-degenerate quadric or a non-degenerate quadric (hyperbolic or elliptic).  ...  One result states that the codes C_2(X) defined on the elliptic quadrics are good codes according to the table of A. E. Brouwer.  ...  In [5] p.14 we have Table I of the quadrics in P G (3, q) . A regulus is the set of transversals of three skew lines in P 3 . It consists of q + 1 skew lines.  ... 
arXiv:math/0511679v1 fatcat:z56ksjibv5ej3c77blotm5b7re

Circle range transversals of circle ranges in a plane: A problem of simple construction

C. J. Keyser
1907 Rendiconti del circolo matematico di Palermo  
All of them are tangent to the radical axes of t~ and p2 ; any three of them are tangent to the radical axis of one of the given ranges r,, r=, r3, r4; any two of them are tangent to two of the four r's  ...  The condition that two ranges p and p' shall have a circle in common is t r r P t t (I8) re(p, p') ~P,2P34 -~-P,3P4~ -~-P,4P=, Of-P23P,4 "~-P4=P,3 +P34P,= --o.  ... 
doi:10.1007/bf03015066 fatcat:dg3vbyac4jfydfupdr3disam3i
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