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

2005
*
Discrete & Computational Geometry
*

We study

doi:10.1007/s00454-005-1160-8
fatcat:qi32oc4gordrnbu4l6wgecoi64
*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]

2005
*
arXiv
*
pre-print

We study

arXiv:math/0304346v2
fatcat:yqxucnde4naodaccvemnl2ijba
*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

1915
*
Transactions of the American Mathematical Society
*

Any

doi:10.2307/1988748
fatcat:dfuresdu5bev3gmwpi2s2hfa5i
*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

1915
*
Transactions of the American Mathematical Society
*

Any

doi:10.1090/s0002-9947-1915-1501002-6
fatcat:uwsoks63nvah3cdcvbonxiygvm
*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

1989
*
The Visual Computer
*

We present

doi:10.1007/bf01914786
fatcat:uspf3ntol5hrtaqzl3inu43zu4
*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

1918
*
Proceedings of the London Mathematical Society
*

Then, since HK is

doi:10.1112/plms/s2-17.1.259
fatcat:lttexxvh3zd7zjjtyvdzjivoc4
*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]

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

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

1912
*
American Journal of Mathematics
*

On

doi:10.2307/2370110
fatcat:4sy2th56sfafzf6vafhewgj5he
*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

2008
*
Computer Aided Geometric Design
*

In

doi:10.1016/j.cagd.2007.10.007
fatcat:bgk4nwiggbaxhf32q54vkfu564
*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

1840
*
Proceedings of the Royal Irish Academy (1836-1869)
*

If

fatcat:nweufky3lnaglji4jmer3es4fm
*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

doi:10.2307/3602926
fatcat:nfjnzoixuvbqheuoxx6fzu2yy4
*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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