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

2002
*
arXiv
*
pre-print

many

arXiv:math/0206044v1
fatcat:lfi3lylinfhvhdcnvcovun7phi
*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. ...##
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Common Transversals and Tangents to Two Lines and Two Quadrics in P

2003
*
Discrete & Computational Geometry
*

many

doi:10.1007/s00454-003-0789-4
fatcat:pdny6lgoz5djpetc3fdspgserq
*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. ...##
###
Quadric splines

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*. ...

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

2006
*
Journal of combinatorial theory. Series A
*

It was shown by Thas

doi:10.1016/j.jcta.2005.03.004
fatcat:jprgevs7z5e2hf7zoalrl5c5o4
*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*...##
###
Families of sections of quadrics and classical geometries

2000
*
Functional analysis and its applications
*

Conversely, any

doi:10.1007/bf02482419
fatcat:wo4gbmqxfbbxlopy5b6zrq222u
*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. ...##
###
Line problems in nonlinear computational geometry
[article]

2007
*
arXiv
*
pre-print

The main part of this survey is recent work on a core algebraic problem--studying the

arXiv:math/0610407v2
fatcat:dzae7umpyfcujm5ple4plpsovm
*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. ...##
###
Real k-flats tangent to quadrics in R^n

2005
*
arXiv
*
pre-print

For each k between 1

arXiv:math/0206099v4
fatcat:byekh3a7qzcetckrid6pxvwada
*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. ...##
###
A new look at the Kummer surface

1967
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

Each

doi:10.4153/cjm-1967-087-5
fatcat:fuqnuvao3zctrbh4yj7lmuumwu
*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 ...##
###
An Enumerative Geometry Framework for Algorithmic Line Problems in $\mathbb R^3$

2002
*
SIAM journal on computing (Print)
*

For this purpose, we study the

doi:10.1137/s009753970038050x
fatcat:zokfljtnxzhvxnlcjnqhfj4bxe
*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. ...##
###
Real $k$-flats tangent to quadrics in $\mathbb {R}^n$

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

doi:10.1090/s0002-9939-05-07880-9
fatcat:czxdt7njajhrderscsdbq5vg6a
*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. ...##
###
Problem.-To Fit Together Two or More Quadrics so That Their Intersections Shall Be Plane

1876
*
The Analyst
*

The points

doi:10.2307/2635804
fatcat:52zsa4kehjfcli2sncp77wvg6q
*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*? ...##
###
Ovoidal packings of P G ( 3 , q ) for even q

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. ...

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

2008
*
IEEE Transactions on Information Theory
*

We study the functional codes C2(X) defined on projective varieties X,

doi:10.1109/tit.2007.913450
fatcat:3uwufzqv5besfdgwdtdhcfdfxq
*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*. ...##
###
Codes Defined By Forms Of Degree 2 On Quadric Surfaces
[article]

2005
*
arXiv
*
pre-print

We study the functional codes C_2(X) defined on projective varieties X,

arXiv:math/0511679v1
fatcat:z56ksjibv5ej3c77blotm5b7re
*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*. ...##
###
Circle range transversals of circle ranges in a plane: A problem of simple construction

1907
*
Rendiconti del circolo matematico di Palermo
*

All of them are

doi:10.1007/bf03015066
fatcat:dg3vbyac4jfydfupdr3disam3i
*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. ...
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