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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
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
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. ...
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
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
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
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
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
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. ...
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
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 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
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
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
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 email@example.com. ... 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
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