Filters








3,194 Hits in 9.9 sec

The Envelope of Lines Meeting a Fixed Line and Tangent to Two Spheres

G�bor Megyesi, Frank Sottile
2005 Discrete & Computational Geometry  
We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations consisting of a single line and three spheres for which there are infinitely many lines tangent  ...  to the three spheres that also meet the given line.  ...  The envelope of lines meeting a fixed line and tangent to two spheres Let ℓ be a line and S 1 and S 2 be spheres in R 3 .  ... 
doi:10.1007/s00454-005-1160-8 fatcat:qi32oc4gordrnbu4l6wgecoi64

The envelope of lines meeting a fixed line that are tangent to two spheres [article]

Gábor Megyesi
2005 arXiv   pre-print
We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations consisting of a single line and three spheres for which there are infinitely many lines tangent  ...  to the three spheres that also meet the given line.  ...  The envelope of lines meeting a fixed line and tangent to two spheres Let ℓ be a line and let S 1 and S 2 be spheres in R 3 .  ... 
arXiv:math/0304346v2 fatcat:yqxucnde4naodaccvemnl2ijba

Page 100 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 16, Issue 1 [page]

1915 American Mathematical Society. Transactions of the American Mathematical Society  
is a fixed 3-flat, the circles of S are all orthogonal to a fixed sphere oc, and the point-pairs of S' are inverse with respect tog.  ...  The circles of S are again tangent to a pair of curves S', as in the preceding case, but are now no longer orthogonal to a sphere; so S cannot be anallagmatic.  ... 

On the Differential Geometry of Ruled Surfaces in 4-Space and Cyclic Surfaces in 3-Space

Arthur Ranum
1915 Transactions of the American Mathematical Society  
Any two spheres A and 77 determine a pencil of spheres enveloping a circle a, just as two points in F4 determine a range of points lying on a line.  ...  A pencil of lines corresponds to a pencil of circles (lying on a sphere and passing through two points of the sphere).  ... 
doi:10.2307/1988748 fatcat:dfuresdu5bev3gmwpi2s2hfa5i

On the differential geometry of ruled surfaces in $4$-space and cyclic surfaces in $3$-space

Arthur Ranum
1915 Transactions of the American Mathematical Society  
Any two spheres A and 77 determine a pencil of spheres enveloping a circle a, just as two points in F4 determine a range of points lying on a line.  ...  A pencil of lines corresponds to a pencil of circles (lying on a sphere and passing through two points of the sphere).  ... 
doi:10.1090/s0002-9947-1915-1501002-6 fatcat:uwsoks63nvah3cdcvbonxiygvm

On the geometry of Dupin cyclides

V. Chandru, D. Dutta, C. M. Hoffmann
1989 The Visual Computer  
We present a unified perspective of their results and use them to devise effective algorithms for synthesizing cyclides.  ...  Abslracl In the 19th century, the French geometer Charles Pierre Dupin discovered a nonspherical surface with circular lines of curvature.  ...  If three spheres of this envelope are fixed and the definition 1 reapplied, a second envelope is obtained. All spheres of the second envelope are tangent to the fixed spheres of the first envelope.  ... 
doi:10.1007/bf01914786 fatcat:uspf3ntol5hrtaqzl3inu43zu4

Tetrahedra in Relation to Spheres and Quadrics

J. H. Grace
1918 Proceedings of the London Mathematical Society  
Then, since HK is a tangent to the conic <r\ the other tangents from H and K to <r' must meet on a.; but, if h and k are the other generators of 8' through H and K, these two tangents are the lines in  ...  It is now easy to deduce that no tangent plane •*•' to the envelope sphere can meet S in points that are inside S'.  ... 
doi:10.1112/plms/s2-17.1.259 fatcat:lttexxvh3zd7zjjtyvdzjivoc4

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.  ...  Indeed, the condition for a line to meet a fixed line (2) is linear in the Plücker coordinates, the condition for a line to be tangent to a sphere (5) is quadratic in the Plücker coordinates, and the set  ... 
arXiv:math/0610407v2 fatcat:dzae7umpyfcujm5ple4plpsovm

Page 155 of The American Mathematical Monthly Vol. 24, Issue 4 [page]

1917 The American Mathematical Monthly  
TO CERTAIN SYSTEMS OF SPHERES. 155 Let | be a fixed straight line, C any plane curve, ¢ a tangent to C meeting 1 in X and makiggfan angle @ with J; then C’, the transformed curve of C, is the envelope  ...  The method of transformation is as follows: Let a be a fixed plane, S any surface, and 8 a plane tangent to S intersecting a in the line 7 and making with a an angle 6.  ... 

On Speherical Curves. Part II

Harold Hilton
1905 Proceedings of the London Mathematical Society  
To save space I shall assume that the reader has access to this paper. t The number of circles through two given points normal to the curve = m ; the number of circles touching the curve, orthogonal to  ...  a given circle, and cutting another given oirole at a giveu angle -2m; the number of circles touching the curve and cutting two given omues at given angles = 4wi.  ...  Two points P, Q on a sphere are inverse with respect to a circle,/' when the line PQ passes through the pole of the plane of j with respect to the sphere.  ... 
doi:10.1112/plms/s2-2.1.150 fatcat:lbkgsmc33fdxzc2fczyprcfxmm

Page 12 of Mathematical Gazette Vol. 2, Issue 25 [page]

1901 Mathematical Gazette  
A variable straight line meets two fixed straight lines Ox, Oy in A and B, so that the sum or difference of OA, OB is constant : find the envelope of the Euler circle of the triangle OA B, and the locus  ...  Material sufficient to make a solid sphere of radius unity is divided at random into two parts, each of which is made into a solid sphere : show that the expectation of the sum of the radii is 3.  ... 

On the Principle of Duality in the Geometry of the Sphere

Arthur Ranum
1912 American Journal of Mathematics  
On a tangent p to a curve E any two The angle between the tangents a of its involutes I and I' cut off a and a' that can be drawn from a seginent AA' of constant length point P' of a curve E' to any two  ...  Hence: Twice the length of the arc of a curve The area exterior to a curve and lying between two of its points. between two of its tangents. 6. Now suppose the curves to be closed.  ... 
doi:10.2307/2370110 fatcat:4sy2th56sfafzf6vafhewgj5he

On quadratic two-parameter families of spheres and their envelopes

Martin Peternell, Boris Odehnal, Maria Lucia Sampoli
2008 Computer Aided Geometric Design  
In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R 3 .  ...  The four dimensional cyclographic model of the set of spheres in R 3 is an appropriate framework to show that a quadratic triangular Bézier patch in R 4 corresponds to a two-parameter family of spheres  ...  Moreover, part of this research has been carried out through a research stay of the first author at the University of Siena.  ... 
doi:10.1016/j.cagd.2007.10.007 fatcat:bgk4nwiggbaxhf32q54vkfu564

On Certain General Properties of Cones of the Second Degree

Charles Graves
1840 Proceedings of the Royal Irish Academy (1836-1869)  
If two fixed tangent arcs be drawn to a spherical conic, and any third tangent arc be drawn meeting them in two points, the arcs passing through these two points and through the pole of a cyclic arc will  ...  formed by the intersection of the cotne and sphere is called a spherical conic, and the two planes meet the surface of the sphere in two great circles which are called the cyclic arcs of the conic.  ... 
fatcat:nweufky3lnaglji4jmer3es4fm

Problems

1901 Mathematical Gazette  
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive.  ...  We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.  ...  A variable straight line meets two fixed straight lines Ox, Oy in A and B, so that the sum or difference of OA, OB is constant: find the envelope of the Euler circle of the triangle OAB, and the locus  ... 
doi:10.2307/3602926 fatcat:nfjnzoixuvbqheuoxx6fzu2yy4
« Previous Showing results 1 — 15 out of 3,194 results