A Twisted Trapezoidal Shape for Geant4

A Twisted TrapezoidalShape for Geant4Based on „Stereo Mini-jet Cells in aCylindrical Drift Chamber“(hep-ex/0303014v1, K. Hoshina et al.)Oliver Link, EP-SFTApr 05

G4VTwistedFactedBase class: G4VSolid, Similar to G4TwistedTubs- DistanceToIn-DistanceToOut-Inside ..- Constructor calls- G4TwistedTrapAlphaSide- G4TwistedTrapParallelSide- G4TwistedTrapBoxSide as special casefor a twisted box- G4FlatTrapSide (for the endcaps)Nov 2003 Oliver Link, EP-SFT 2

⎛⎜⎜( w(u,ϕ)+ Δx⎜(w(u,ϕ)+ Δx⎜⎜⎜⎝G4TwistedTrapAlphaSideSurface EquationϕΔϕϕΔϕ)cosϕ− ( u + Δy)sinϕ+ ( u + ΔyThe resulting solutioncontains terms in Sin(ϕ)and Cos(ϕ) which areapproximated with Padéexpansions.Polynom: 7 th order.ϕΔϕϕΔϕ)sinϕ=)cosϕ=u ∈[−Lϕ=Δϕϕ ∈[−1212w(u)=ppxypzΔφ,++ tv+ tv+ tv12b(ϕ),+xyz⎞⎟⎟⎟⎟⎟⎟⎠Δφ]12b(ϕ)b(ϕ)]αa(ϕ)d(ϕ)2 Free parameters (ϕ,u)a(ϕ)d(ϕ)− a(ϕ)⎡d(ϕ)− a(ϕ)⎤+− u⎢− tanα2 4⎥⎣ 2b(ϕ)⎦Nov 2003 Oliver Link, EP-SFT 3

Planarity conditionAn untwisted trapezoidn( Δx,Δy)requires planar surfaces.β 2b 2a 1 αβ 1 d 2b 1a 2rd 1αPlanarity is equivalent toβ1= β 2d11α− ab11= α=d22−b2a2Nov 2003 Oliver Link, EP-SFT 4

G4TwistedBoxfDxfDyG4TwistedBox(Name,Δϕ,fDx,fDy,fDz)Eg: 70*deg,20*cm,30*cm,80*cmNov 2003 Oliver Link, EP-SFT 5

G4TwistedTrdfDx 2fDx 1fDyfDy 21G4TwistedTrap(Name,fDx 1,fDx 2,fDy 1,fDy 2,fDz,Δϕ)Eg: 15*cm,25*cm,30*cm,20*cm,80*cm,70*degNov 2003 Oliver Link, EP-SFT 6

G4TwistedTrapTwisted trapezoid with equal sized endcapsand no tilt angle.Twist angle ab/2a/2d/2G4TwistedTrap(“Trap“,70*deg,15*cm,25*cm,30*cm,80*cm)d/2 a/2 b/2L/2Nov 2003 Oliver Link, EP-SFT 7

G4TwistedTrapGeneral Twisted trapezoid withdifferent sized endcaps and and tilt angle.)fDx 4fDx 2fDy 1αfDx 3fDy 2n r ( ϑ,ϕfDzαfDx 1-fDzG4TwistedTrap(Name,Δϕ,fDz,θ,ϕ,fDx 1,fDx 2,fDy 1,fDx 3,fDx 4,fDy 2,α)Nov 2003 Oliver Link, EP-SFT 8

Solved ProblemsThe visualisation showed cracks in the surface andwrongly tracked events. This issus was successfullysolved by a– division by zero check (added special case)– new surface-point finder– introducing G4JTPolynomialSolver– special treatment for events parallel to the surfacebeforeafterNov 2003 Oliver Link, EP-SFT 9

Two intersectionsNov 2003 Oliver Link, EP-SFT 10

Testing the SolidWith traditional tools• test10: successful• SolidsChecker:1 event „Track stuck“ out of 100 M... and with a new testing tool• SurfaceChecker: successfulNov 2003 Oliver Link, EP-SFT 11

SurfaceChecker• Generate a randomparticle position P and arandom point X true on thesurface of the solid.• Ask G4 for theintersection point giventhe point P and itsdirection v (select theintersection closest to Xtrue).X reconst.X trueX reconst.The distance δ between X true andX reconst. gives us informationabout the goodness of thereconstruction.δv rPNov 2003 Oliver Link, EP-SFT 12

Results I0.10.090.080.070.06boxconeorbspheretorustubetwistedboxtwistedtrap2twistedtrap3twistedtraptwistedtrdtwistedtubs0.050.040.030.020.01010 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1 10 10 2 10 3 10 4 10 5dist (mm)Nov 2003 Oliver Link, EP-SFT 13δ

Results II0.10.090.080.070.06boxconeorbspheretorustubetwistedboxtwistedtrap2twistedtrap3twistedtraptwistedtrdtwistedtubs0.050.040.030.020.01010 -12 10 -11 10 -10 10 -9dist (mm)Nov 2003 Oliver Link, EP-SFT 14δ

Conclusions• G4TwistedBox with equal endcaps is replacedby the new generic version.• G4TwistedTrd with rectangular endcaps ofdifferent size (but no tilt angle) added.• G4TwistedTrap with trapezoidal endcaps ofdifferent size and tilt angle added. The oldversion with equal endcaps is covered by thegeneric version.• G4JTPolynomialSolver to solve the high orderpolynoms is added to HEPNumerics.Nov 2003 Oliver Link, EP-SFT 15