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### 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 .  ...

### 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 .  ...

### 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).  ...

### 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).  ...

### 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.  ...

### 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'.  ...

### 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  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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  ...
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