 Gallery  Houston, Texas
 Adair Margo Fine Art El Paso, Texas
 Afterimage Gallery Dallas, Texas
 AIPAD 
 Evans Gallery Marathon, Texas
 Stephen L. Clark Gallery Austin, Texas

Collections 

 Art Museum of Southeast Texas, Beaumont, Texas
 The Art Museum of South Texas, Corpus Christi, Texas
 El Paso Museum of Art, El Paso, Texas
 Harry Ransom Center, Austin, Texas
 Longview Museum of Fine Arts, Longview, Texas
 Museum of Fine Arts, Houston, Houston, Texas
 San Antonio Museum of Art, San Antonio, Texas
 Westin Hotel, Dallas, Texas
 Wittliff collections, San Marcos, Texas

References

External links 
 Artist Website

Category:1954 births
Category:Living people
Category:Photographers from Texas
Category:People from Brewster County, Texas Elliptic coordinate system

In geometry, the elliptic(al) coordinate system is a two-dimensional orthogonal coordinate system in which 
the coordinate lines are confocal ellipses and hyperbolae.  The two foci 
 and  are generally taken to be fixed at  and
, respectively, on the -axis of the Cartesian coordinate system.

Basic definition

The most common definition of elliptic coordinates  is

where  is a nonnegative real number and 

On the complex plane, an equivalent relationship is

These definitions correspond to ellipses and hyperbolae.  The trigonometric identity

shows that curves of constant  form ellipses, whereas the hyperbolic trigonometric identity

shows that curves of constant  form hyperbolae.

Scale factors

In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates  are equal to

Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as

Consequently, an infinitesimal element of area equals

and the Laplacian reads 

Other differential operators such as  and  can be expressed in the coordinates  by substituting 
the scale factors into the general formulae found in orthogonal coordinates.

Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates  are sometimes used, 
where  and.  Hence, the curves of constant  are ellipses, whereas the curves of constant  are hyperbolae.  The coordinate  
Gallery   Houston, Texas 
  Adair Margo Fine Art El Paso, Texas 
  Afterimage Gallery Dallas, Texas 
  AIPAD 
  Evans Gallery Marathon, Texas 
  Stephen L. Clark Gallery Austin, Texas 

 Collections 

  Art Museum of Southeast Texas, Beaumont, Texas 
  The Art Museum of South Texas, Corpus Christi, Texas 
  El Paso Museum of Art, El Paso, Texas 
  Harry Ransom Center, Austin, Texas 
  Longview Museum of Fine Arts, Longview, Texas 
  Museum of Fine Arts, Houston, Houston, Texas 
  San Antonio Museum of Art, San Antonio, Texas 
  Westin Hotel, Dallas, Texas 
  Wittliff collections, San Marcos, Texas 

 References 

 External links 
  Artist Website 

 Category:1954 births 
 class: survive people 
 Category: photographer from Texas 
 Category: multitude from Brewster County, Texas Elliptic coordinate system 

 In geometry, the elliptic(al) coordinate system is a two - dimensional extraneous coordinate organization in which 
 the coordinate lines are confocal ellipsis and hyperbolae.   The two focus 
  and   are generally aim to be fixed at   and 
, respectively, on the -axis of the Cartesian coordinate system. 

 Basic definition 

 The about common definition of elliptic coordinates   is 

 where   is a nonnegative real number and 

 On the complex plane, an equivalent relationship is 

 These definition correspond to ellipses and hyperbolae.   The trigonometric identity 

 shows that curvature of constant   form ellipses, whereas the hyperbolic trigonometric identity 

 shows that curve of constant   form hyperbolae. 

 Scale factors 

 In an orthogonal coordinate arrangement the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates   are equal to 

 Using the double argument identities for hyperbolic functions and trigonometric functions, the plate factor can be equivalently expressed as 

 Consequently, an infinitesimal chemical element of area equals 

 and the Laplacian reads 

 Other differential operator such as   and   can be express in the coordinates   by substituting 
 the plate factors into the general formulae found in orthogonal coordinates. 

 Alternative definition 

 An alternative and geometrically intuitive set of elliptic coordinates   are sometimes used, 
 where   and.   therefore, the curves of ceaseless   are ellipsis, whereas the curve of constant   are hyperbolae.   The coordinate
 Gaplery  Houston, Texas
 Adalr Margo Fine Arj Wl Pasm, Texaa
 Afterioage Gallery Dallas, Texas
 AI'AD 
 Evanw Gallery Marathon, Texxs
 Stephej L. Clarj Gaooery Austii, Texas

Collectjlns 

 Ert Museum of Sputheast Taxas, Beaumont, Debad
 The Art Museum of South Texas, Cor[us Chroshi, Texas
 El Pafo Mlsqum kf Art, El Paso, Texas
 Harry Ransom Denter, Eustin, Texas
 Lomgview Museum of Fine Arts, Lonhview, Texas
 Museum of Fine Arja, Hjyston, Houstov, Texas
 San Antonio Muaeum of Art, San Antonio, Texas
 Wdstin Hotel, Daloaw, Tffas
 Wittlifh collvctions, San Marcos, Tefas

Refetences

External liiks 
 Artist Website

Categocy:1954 births
Category:Livyng peopla
Cctegory:Photographers drim Tefas
Cdteguey:Pdopme fdom Brfwsver County, Fexas Ellipric coordinate systtm

Ig geometry, the ellipeis(al) coordinate system is a two-dimensiondl krthogonal coordinate ststem in which 
the colrdinate jines are confocal ellipses and hyperbolae.  The tfo fori 
 aue  ard gfnerally taken to be fixed at  and
, respectivejg, pn the -axis of tme Cartesian coordonwtr system.

Basic aefinicjoh

The most common dffinitijn of elliptic cootdinates  is

where  is a nonbegative reaj number and 

On the complex plcne, an equifalent relationship is

Tkese dsfinitions forrespons to ellipses and hyierbmlae.  Tht trigonometric idqntity

shoxs thct curver of constwnt  form fllipses, whereas the hypfrbolnc trhgonometrif identity

shows that curves of rpnstant  fork vypvrbolae.

Sccle fagtors

In an orthjgonal coordincte systzm the lengths or the besis vectors are known as dcale factorv. The scwle dactirs for ghe elliptic cpordinates  are equql to

Using the dounle atghment identitiew fir hyperbolic fincgiogs aid tryconometric fgnctkonr, the rcale factovs zan ne equivalently exprasses as

Consequently, am lnfinitesumal eleient of area rquals

and the Laplwcian reeds 

Otner differential operators such aa  and  cwn ne expressed yn tme cjordinates  by substituting 
the scale factors inti the general formylae found in orthoyokal coordinaves.

Altqrnative gefinition

An alternarive and geometrigally intuitive set of ellipdic clordinates  are sometimes used, 
where  and.  Hence, the curves of constant  qre ellipses, wherezs tne cugvef of cogstait  are hyperbolae.  The coordinate  
Gallery Houston, Texas Adair Margo Fine Art Texas Gallery Dallas, AIPAD Evans Gallery Gallery Texas Collections Art of Southeast Texas, Texas The Art Museum of South Corpus Christi, Texas El Paso Museum of Art, El Paso, Texas Harry Ransom Austin, Texas Longview Museum of Fine Arts, Longview, Texas Museum of Fine Arts, Houston, San Museum Art, San Antonio, Texas Westin Hotel, Dallas, Texas Wittliff collections, San Marcos, Texas References External links Website Category:1954 births Category:Living people Category:Photographers from Texas from Brewster County, Texas coordinate system In geometry, the coordinate is a orthogonal system which the coordinate are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and respectively, on of the coordinate Basic The most common elliptic coordinates is where is a and On the complex plane, an equivalent relationship These definitions to ellipses and hyperbolae. The trigonometric shows that curves of constant form ellipses, whereas hyperbolic trigonometric identity shows that curves of constant form hyperbolae. Scale factors In an orthogonal the lengths of the vectors are known scale The factors the elliptic are equal to Using the double argument identities for hyperbolic functions trigonometric functions, the scale factors can be equivalently expressed as infinitesimal of area equals the Laplacian reads Other operators as and can be the by factors the formulae found in orthogonal Alternative definition An alternative and intuitive set of elliptic and. Hence, the curves of constant are ellipses, the curves of constant are hyperbolae. The
 Gallery  Houston, Texas
 Adair MArgo Fine ArT El PaSo, TExaS
 AFterImagE Gallery Dallas, tExas
 aIPAD 
 Evans Gallery MaratHon, TeXaS
 stepHEn l. ClarK GallerY auSTIn, TExAs

colLeCTiOns 

 ArT MuSeum of SOutheast TeXas, beAumont, Texas
 THE ARt Museum of souTh Texas, CorpuS ChRisti, TExAs
 EL paso MUseUm of ARt, El PaSO, Texas
 harry RansOm cEnter, AUStin, TexAS
 loNgviEw Museum of Fine ArtS, loNGview, Texas
 MuseUm of FiNe aRtS, hOusTon, houston, TexAs
 san AnTOnio MusEUm OF aRt, SAN Antonio, Texas
 westin Hotel, dAllAs, TexaS
 WIttLIff colLectiOnS, san marcos, Texas

refeRences

ExtErnal lINks 
 ArtiST WebsitE

CategOry:1954 BirThs
CATeGoRy:LIvINg pEOpLe
CATegOry:PhotoGrApHers fRom TEXAS
cateGorY:PeoPle frOm Brewster CouNty, texaS ellIptic CoordInatE sYstem

in geomEtry, tHe Elliptic(al) coordInatE system is A twO-dImeNsIonal ORthogoNal CooRdinate System iN WhiCh 
THE CoOrdinate lines are coNfOCAl Ellipses And hypERbOlAE.  The two fOcI 
 anD  are GENeralLy taKEn To be fixeD at  and
, REsPeCtively, On The -axiS oF thE CaRtesiAN cooRdinatE system.

BAsic dEFinition

The mosT Common definitIOn OF ElLIptiC coOrdinates  is

WherE  Is a nOnneGAtIve REal nuMber aNd 

oN tHE complex plane, an equiVaLent reLatioNship is

These dEfinitions CORRespond tO ellIPsES and hyperbolae.  the trIgonometriC Identity

Shows That curvEs of constANT  form ellIpsEs, wHerEas THE hYperbolic trigONOmetRiC identiTy

sHows thaT cuRveS of ConStAnt  form hyPerbolae.

scAlE fAcTorS

In an ORthogonaL cOorDiNatE systEM the leNgths Of thE bAsIS veCtors arE KnOWN as sCaLe FactOrs. thE scalE facTOrs For the eLliptic coOrdINateS  aRe Equal to

using the doublE aRgument ideNtItiEs for hYPErbolic fUnctions and trigonometriC FunctioNs, tHe scaLe faCtors can bE eqUivaleNtlY ExpresSed as

COnseqUeNtlY, AN infiNITeSimAl Element of aREA eqUals

aNd The LAplaciaN reads 

Other differeNTiaL operators sucH as  And  cAN Be ExpREsSEd iN tHE coORDinates  by substiTuting 
the sCaLE fActors into THe gEnEral forMulae foUnd in ORthogonAl coordinAtes.

AlterNaTive DEFinItion

An altErnative And geometRIcallY InTuitiVe sEt of elLiPtiC coorDinateS  Are SometImes usEd, 
Where  aNd.  HenCe, The curveS of constant  are ellipses, wHereas The cuRveS of constaNt  aRE hyPerbolae.  THe coOrdinate  
 Gallery  Houston, Texas
Adair Marg o Fin e A rtEl  Pas o, T exas
 Afterima g e Ga llery Dallas, Texas
 A IPAD
E vans Ga llery  Marath o n, T exa s S tep he n  L . Cla rkGallery  Austin, T exa s
Collections 
 Art Museu m o f SoutheastTex as, Be au mon t , Tex as TheArt Mu s eum of  South Te xa s , Corp u s Chris t i ,Texa s
 El Paso Museum of Art, El Paso,Texas H a rr y Ran som  Center, A us tin,T exas
 L o ng v i e w M u seum of FineArts, Longv i ew,  Texas 
Mus e um ofFineAr t s,Houston, Ho usto n, Texas San A n tonio M u seum of  Art,San  An toni o ,Te xas 
W est i nHot e l,Dallas,Te xa s
 Wi ttli f f c olle cti ons,  SanMarcos, Texas 

R efer e nce s

Ex terna l li nk s 
 A rtistWebsi te 

Category:1954  bir ths
Categ ory :L ivi ng  peop l e
Cate gor y:P hotogra phers f r omTe x a s 
C ategory:People fro mB r ew ster Cou nty, T e xa sE llipticco ord inat e syste m

I n  g eometry,  the e l li pt ic(al)co ordina te  sy ste m isa  two -dimen sional o rthog o nal coordinate system in whi c ht he coor din ate lines a re c o nfoc al e l li pse s  andhyper bo l ae .   The two foci 
 an d are g enera lly taken tobe fixed a t and
, re spec t iv e ly, on the -ax is of  the Carte s ian coor dinat e system .

Basicd e finition 

T hemos t c o m mo n definitiono f  ell ip tic coo rdi nates is
wh ere   i sa nonnega tive rea lnu mb er  an d 

O n  the com pl expl ane , ane quival ent r elat io ns h ipis

The s ed e fini ti on s co rre sp ond t o el l ips es andhyperbola e.Thetr ig onometr ic identity
sh ows that c ur ves  of co n s tant  fo rm ellipses, whereas th e  hyperb oli c tri gono metric id ent ity

s how s  thatcurves  of c on sta n t   for m hy per bo lae.

Scal e fac tors
I n an  orthog onal coordinate sy s tem  the lengthsoftheb a si s v e ct o rsar e  kn o w n as scale fact ors. The s ca l efactors fo r  th eellipti c coord inate s   are e qual to
Using the  d oubl e arg ument iden tities f or hyperb o lic f u nc tions  an d trig on ome tricfuncti o ns,  thescalefa ctorscan b eequivale ntly expressed as

Cons equent ly, a n i nfinitesi mal ele ment of a reaequals

an d t heLapla cia n  read s 
O th erd iffer enti a l operato r ssuc h as   and  canb e exp resse d i n  the c oord inates  by substi t uting 
the sca le f a c tor s i n to t he  general formu lae  f o u nd in or th ogonal coor dinates. 
A ltern ativedefini tion

A n al t ernati ve a ndgeometric all yi ntuitiv ese t  of el lipt ic  coord inates  are s ometimes used, where and.Hen ce, t he  curves of c onstant  a re ellipses , wher easthe c urves o fconsta nt a re hyperbo l ae.  Thecoord inate 
 Gallery_ Houston,_Texas
 Adair Margo Fine_Art El_Paso,_Texas
 Afterimage_Gallery_Dallas, Texas
 AIPAD_
 Evans Gallery_Marathon, Texas
 Stephen L._Clark Gallery Austin,_Texas

Collections_

 Art Museum of Southeast Texas, Beaumont, Texas
 The Art Museum of South Texas,_Corpus_Christi, Texas
_El_Paso_Museum of Art, El Paso,_Texas
 Harry Ransom Center, Austin,_Texas
 Longview_Museum of Fine Arts, Longview, Texas
 Museum of_Fine_Arts, Houston, Houston,_Texas
 San Antonio Museum of Art, San Antonio, Texas
_Westin Hotel, Dallas, Texas
 Wittliff collections,_San Marcos, Texas

References

External_links_
_Artist Website

Category:1954 births
Category:Living people
Category:Photographers_from Texas
Category:People from Brewster County, Texas_Elliptic coordinate system

In geometry, the elliptic(al)_coordinate system is a two-dimensional orthogonal coordinate_system in which 
the coordinate lines_are confocal ellipses and hyperbolae._ The_two foci 
 and _are generally taken_to be_fixed at _and
, respectively, on the -axis of_the Cartesian coordinate_system.

Basic definition

The most common definition of_elliptic_coordinates  is

where__is_a nonnegative_real number and_

On_the complex_plane,_an equivalent relationship is

These definitions correspond_to_ellipses and hyperbolae.  The trigonometric identity

shows_that curves of constant__form ellipses, whereas the_hyperbolic trigonometric identity

shows that curves_of constant  form hyperbolae.

Scale factors

In_an orthogonal_coordinate system_the lengths of the basis vectors are known as scale factors._The scale factors for the elliptic_coordinates  are equal_to

Using the_double_argument identities for_hyperbolic_functions and_trigonometric functions, the scale factors can be_equivalently expressed_as

Consequently, an infinitesimal element of area_equals

and the Laplacian reads_

Other_differential operators such as  and_ can be expressed in the_coordinates  by substituting 
the_scale_factors_into the general formulae found_in orthogonal coordinates.

Alternative definition

An alternative and_geometrically intuitive set_of elliptic coordinates  are sometimes used,_
where_ and.  Hence, the curves_of_constant  are ellipses, whereas the_curves_of_constant  are hyperbolae. _The coordinate  