The STEP standard supports multiple solid representations. Constructive Solid Geometry (CSG) representations and Boundary Represented Solids (BREPs) are available. Different representations are suitable for different purposes, applications, required complexity, and levels of detail. CSG representations are easy to use and normally give superior performance, but they cannot reproduce complex solids such as those used in CAD systems. BREP representations can handle more extended topologies and reproduce the most complex solids. All constructed solids can stream out their contents via appropriate methods and streaming operators. For all solids it is possible to estimate the geometrical volume and the surface area by invoking the methods: G4double GetCubicVolume() G4double GetSurfaceArea() which return an estimate of the solid volume and total area in internal units respectively. For elementary solids the functions compute the exact geometrical quantities, while for composite or complex solids an estimate is made using Monte Carlo techniques. For all solids it is also possible to generate pseudo-random points lying on their surfaces, by invoking the method G4ThreeVector GetPointOnSurface() const which returns the generated point in local coordinates relative to the solid. CSG solids are defined directly as three-dimensional primitives. They are described by a minimal set of parameters necessary to define the shape and size of the solid. CSG solids are Boxes, Tubes and their sections, Cones and their sections, Spheres, Wedges, and Toruses. Box: To create a box one can use the constructor: G4Box(const G4String& pName, G4double pX, G4double pY, G4double pZ) In the picture: pX = 30, pY = 40, pZ = 60 by giving the box a name and its half-lengths along the X, Y and Z axis: pX half length in X pY half length in Y pZ half length in Z This will create a box that extends from -pX to +pX in X, from -pY to +pY in Y, and from -pZ to +pZ in Z. For example to create a box that is 2 by 6 by 10 centimeters in full length, and called BoxA one should use the following code: G4Box* aBox = new G4Box("BoxA", 1.0*cm, 3.0*cm, 5.0*cm); Cylindrical Section or Tube: Similarly to create a cylindrical section or tube, one would use the constructor: G4Tubs(const G4String& pName, G4double pRMin, G4double pRMax, G4double pDz, G4double pSPhi, G4double pDPhi) In the picture: pRMin = 10, pRMax = 15, pDz = 20 giving its name pName and its parameters which are: pRMin Inner radius pRMax Outer radius pDz half length in z pSPhi the starting phi angle in radians pDPhi the angle of the segment in radians Cone or Conical section: Similarly to create a cone, or conical section, one would use the constructor G4Cons(const G4String& pName, G4double pRmin1, G4double pRmax1, G4double pRmin2, G4double pRmax2, G4double pDz, G4double pSPhi, G4double pDPhi) In the picture: pRmin1 = 5, pRmax1 = 10, pRmin2 = 20, pRmax2 = 25, pDz = 40, pSPhi = 0, pDPhi = 4/3*Pi giving its name pName, and its parameters which are: pRmin1 inside radius at -pDz pRmax1 outside radius at -pDz pRmin2 inside radius at +pDz pRmax2 outside radius at +pDz pDz half length in z pSPhi starting angle of the segment in radians pDPhi the angle of the segment in radians Parallelepiped: A parallelepiped is constructed using: G4Para(const G4String& pName, G4double dx, G4double dy, G4double dz, G4double alpha, G4double theta, G4double phi) In the picture: dx = 30, dy = 40, dz = 60 giving its name pName and its parameters which are: dx,dy,dz Half-length in x,y,z alpha Angle formed by the y axis and by the plane joining the centre of the faces parallel to the z-x plane at -dy and +dy theta Polar angle of the line joining the centres of the faces at -dz and +dz in z phi Azimuthal angle of the line joining the centres of the faces at -dz and +dz in z Trapezoid: To construct a trapezoid use: G4Trd(const G4String& pName, G4double dx1, G4double dx2, G4double dy1, G4double dy2, G4double dz) In the picture: dx1 = 30, dx2 = 10, dy1 = 40, dy2 = 15, dz = 60 to obtain a solid with name pName and parameters dx1 Half-length along x at the surface positioned at -dz dx2 Half-length along x at the surface positioned at +dz dy1 Half-length along y at the surface positioned at -dz dy2 Half-length along y at the surface positioned at +dz dz Half-length along z axis Generic Trapezoid: To build a generic trapezoid, the G4Trap class is provided. Here are the two costructors for a Right Angular Wedge and for the general trapezoid for it: G4Trap(const G4String& pName, G4double pZ, G4double pY, G4double pX, G4double pLTX) G4Trap(const G4String& pName, G4double pDz, G4double pTheta, G4double pPhi, G4double pDy1, G4double pDx1, G4double pDx2, G4double pAlp1, G4double pDy2, G4double pDx3, G4double pDx4, G4double pAlp2) In the picture: pDx1 = 30, pDx2 = 40, pDy1 = 40, pDx3 = 10, pDx4 = 14, pDy2 = 16, pDz = 60, pTheta = 20*Degree, pPhi = 5*Degree, pAlp1 = pAlp2 = 10*Degree to obtain a Right Angular Wedge with name pName and parameters: pZ Length along z pY Length along y pX Length along x at the wider side pLTX Length along x at the narrower side (plTX<=pX) or to obtain the general trapezoid (see the Software Reference Manual): pDx1 Half x length of the side at y=-pDy1 of the face at -pDz pDx2 Half x length of the side at y=+pDy1 of the face at -pDz pDz Half z length pTheta Polar angle of the line joining the centres of the faces at -/+pDz pPhi Azimuthal angle of the line joining the centre of the face at -pDz to the centre of the face at +pDz pDy1 Half y length at -pDz pDy2 Half y length at +pDz pDx3 Half x length of the side at y=-pDy2 of the face at +pDz pDx4 Half x length of the side at y=+pDy2 of the face at +pDz pAlp1 Angle with respect to the y axis from the centre of the side (lower endcap) pAlp2 Angle with respect to the y axis from the centre of the side (upper endcap) Note on pAlph1/2: the two angles have to be the same due to the planarity condition. Sphere or Spherical Shell Section: To build a sphere, or a spherical shell section, use: G4Sphere(const G4String& pName, G4double pRmin, G4double pRmax, G4double pSPhi, G4double pDPhi, G4double pSTheta, G4double pDTheta ) In the picture: pRmin = 100, pRmax = 120, pSPhi = 0*Degree, pDPhi = 180*Degree, pSTheta = 0 Degree, pDTheta = 180*Degree to obtain a solid with name pName and parameters: pRmin Inner radius pRmax Outer radius pSPhi Starting Phi angle of the segment in radians pDPhi Delta Phi angle of the segment in radians pSTheta Starting Theta angle of the segment in radians pDTheta Delta Theta angle of the segment in radians Full Solid Sphere: To build a full solid sphere use: G4Orb(const G4String& pName, G4double pRmax) In the picture: pRmax = 100 The Orb can be obtained from a Sphere with: pRmin = 0, pSPhi = 0, pDPhi = 2*Pi, pSTheta = 0, pDTheta = Pi pRmax Outer radius Torus: To build a torus use: G4Torus(const G4String& pName, G4double pRmin, G4double pRmax, G4double pRtor, G4double pSPhi, G4double pDPhi) In the picture: pRmin = 40, pRmax = 60, pRtor = 200, pSPhi = 0, pDPhi = 90*Degree to obtain a solid with name pName and parameters: pRmin Inside radius pRmax Outside radius pRtor Swept radius of torus pSPhi Starting Phi angle in radians (fSPhi+fDPhi<=2PI, fSPhi>-2PI) pDPhi Delta angle of the segment in radians In addition, the Geant4 Design Documentation shows in the Solids Class Diagram the complete list of CSG classes, and the STEP documentation contains a detailed EXPRESS description of each CSG solid. Specific CSG Solids Polycons: Polycons (PCON) are implemented in Geant4 through the G4Polycon class: G4Polycone(const G4String& pName, G4double phiStart, G4double phiTotal, G4int numZPlanes, const G4double zPlane[], const G4double rInner[], const G4double rOuter[]) G4Polycone(const G4String& pName, G4double phiStart, G4double phiTotal, G4int numRZ, const G4double r[], const G4double z[]) In the picture: phiStart = 1/4*Pi, phiTotal = 3/2*Pi, numZPlanes = 9, rInner = { 0, 0, 0, 0, 0, 0, 0, 0, 0}, rOuter = { 0, 10, 10, 5 , 5, 10 , 10 , 2, 2}, z = { 5, 7, 9, 11, 25, 27, 29, 31, 35 } where: phiStart Initial Phi starting angle phiTotal Total Phi angle numZPlanes Number of z planes numRZ Number of corners in r,z space zPlane Position of z planes rInner Tangent distance to inner surface rOuter Tangent distance to outer surface r r coordinate of corners z z coordinate of corners Polyhedra (PGON): Polyhedra (PGON) are implemented through G4Polyhedra: G4Polyhedra(const G4String& pName, G4double phiStart, G4double phiTotal, G4int numSide, G4int numZPlanes, const G4double zPlane[], const G4double rInner[], const G4double rOuter[] ) G4Polyhedra(const G4String& pName, G4double phiStart, G4double phiTotal, G4int numSide, G4int numRZ, const G4double r[], const G4double z[] ) In the picture: phiStart = -1/4*Pi, phiTotal= 5/4*Pi, numSide = 3, nunZPlanes = 7, rInner = { 0, 0, 0, 0, 0, 0, 0 }, rOuter = { 0, 15, 15, 4, 4, 10, 10 }, z = { 0, 5, 8, 13 , 30, 32, 35 } where: phiStart Initial Phi starting angle phiTotal Total Phi angle numSide Number of sides numZPlanes Number of z planes numRZ Number of corners in r,z space zPlane Position of z planes rInner Tangent distance to inner surface rOuter Tangent distance to outer surface r r coordinate of corners z z coordinate of corners Tube with an elliptical cross section: A tube with an elliptical cross section (ELTU) can be defined as follows: G4EllipticalTube(const G4String& pName, G4double Dx, G4double Dy, G4double Dz) The equation of the surface in x/y is 1.0 = (x/dx)**2 +(y/dy)**2 In the picture: Dx = 5, Dy = 10, Dz = 20 Dx Half length X Dy Half length Y Dz Half length Z General Ellipsoid: The general ellipsoid with possible cut in Z can be defined as follows: G4Ellipsoid(const G4String& pName, G4double pxSemiAxis, G4double pySemiAxis, G4double pzSemiAxis, G4double pzBottomCut=0, G4double pzTopCut=0) In the picture: pxSemiAxis = 10, pySemiAxis = 20, pzSemiAxis = 50, pzBottomCut = -10, pzTopCut = 40 A general (or triaxial) ellipsoid is a quadratic surface which is given in Cartesian coordinates by: 1.0 = (x/pxSemiAxis)**2 + (y/pySemiAxis)**2 + (z/pzSemiAxis)**2 where: pxSemiAxis Semiaxis in X pySemiAxis Semiaxis in Y pzSemiAxis Semiaxis in Z pzBottomCut lower cut plane level, z pzTopCut upper cut plane level, z Cone with Elliptical Cross Section: A cone with an elliptical cross section can be defined as follows: G4EllipticalCone(const G4String& pName, G4double pxSemiAxis, G4double pySemiAxis, G4double zMax, G4double pzTopCut) In the picture: pxSemiAxis = 30/75, pySemiAxis = 60/75, zMax = 50, pzTopCut = 25 where: pxSemiAxis Semiaxis in X pySemiAxis Semiaxis in Y zMax Height of elliptical cone pzTopCut upper cut plane level An elliptical cone of height zMax, semiaxis pxSemiAxis, and semiaxis pySemiAxis is given by the parametric equations: x = pxSemiAxis * ( zMax - u ) / u * Cos(v) y = pySemiAxis * ( zMax - u ) / u * Sin(v) z = u Where v is between 0 and 2*Pi, and u between 0 and h respectively. Paraboloid, a solid with parabolic profile: A solid with parabolic profile and possible cuts along the Z axis can be defined as follows: G4Paraboloid(const G4String& pName, G4double Dz, G4double R1, G4double R2) The equation for the solid is: rho**2 <= k1 * z + k2; -dz <= z <= dz r1**2 = k1 * (-dz) + k2 r2**2 = k1 * ( dz) + k2 In the picture: R1 = 20, R2 = 35, Dz = 20 Dz Half length Z R1 Radius at -Dz R2 Radius at +Dz greater than R1 Tube with Hyperbolic Profile: A tube with a hyperbolic profile (HYPE) can be defined as follows: G4Hype(const G4String& pName, G4double innerRadius, G4double outerRadius, G4double innerStereo, G4double outerStereo, G4double halfLenZ) In the picture: innerStereo = 0.7, outerStereo = 0.7, halfLenZ = 50, innerRadius = 20, outerRadius = 30 G4Hype is shaped with curved sides parallel to the z-axis, has a specified half-length along the z axis about which it is centred, and a given minimum and maximum radius. A minimum radius of 0 defines a filled Hype (with hyperbolic inner surface), i.e. inner radius = 0 AND inner stereo angle = 0. The inner and outer hyperbolic surfaces can have different stereo angles. A stereo angle of 0 gives a cylindrical surface: innerRadius Inner radius outerRadius Outer radius innerStereo Inner stereo angle in radians outerStereo Outer stereo angle in radians halfLenZ Half length in Z Tetrahedra: A tetrahedra solid can be defined as follows: G4Tet(const G4String& pName, G4ThreeVector anchor, G4ThreeVector p2, G4ThreeVector p3, G4ThreeVector p4, G4bool *degeneracyFlag=0) In the picture: anchor = {0, 0, sqrt(3)}, p2 = { 0, 2*sqrt(2/3), -1/sqrt(3) }, p3 = { -sqrt(2), -sqrt(2/3),-1/sqrt(3) }, p4 = { sqrt(2), -sqrt(2/3) , -1/sqrt(3) } The solid is defined by 4 points in space: anchor Anchor point p2 Point 2 p3 Point 3 p4 Point 4 degeneracyFlag Flag indicating degeneracy of points Extruded Polygon: The extrusion of an arbitrary polygon (extruded solid) with fixed outline in the defined Z sections can be defined as follows (in a general way, or as special construct with two Z sections): G4ExtrudedSolid(const G4String& pName, std::vector<G4TwoVector> polygon, std::vector<ZSection> zsections) G4ExtrudedSolid(const G4String& pName, std::vector<G4TwoVector> polygon, G4double hz, G4TwoVector off1, G4double scale1, G4TwoVector off2, G4double scale2) In the picture: poligon = {-30,-30},{-30,30},{30,30},{30,-30}, {15,-30},{15,15},{-15,15},{-15,-30} zsections = [-60,{0,30},0.8], [-15, {0,-30},1.], [10,{0,0},0.6], [60,{0,30},1.2] The z-sides of the solid are the scaled versions of the same polygon. polygon the vertices of the outlined polygon defined in clock-wise order zsections the z-sections defined by z position in increasing order hz Half length in Z off1, off2 Offset of the side in -hz and +hz respectively scale1, scale2 Scale of the side in -hz and +hz respectively Box Twisted: A box twisted along one axis can be defined as follows: G4TwistedBox(const G4String& pName, G4double twistedangle, G4double pDx, G4double pDy, G4double pDz) In the picture: twistedangle = 30*Degree, pDx = 30, pDy =40, pDz = 60 G4TwistedBox is a box twisted along the z-axis. The twist angle cannot be greater than 90 degrees: twistedangle Twist angle pDx Half x length pDy Half y length pDz Half z length Trapezoid Twisted along One Axis: trapezoid twisted along one axis can be defined as follows: G4TwistedTrap(const G4String& pName, G4double twistedangle, G4double pDxx1, G4double pDxx2, G4double pDy, G4double pDz) G4TwistedTrap(const G4String& pName, G4double twistedangle, G4double pDz, G4double pTheta, G4double pPhi, G4double pDy1, G4double pDx1, G4double pDx2, G4double pDy2, G4double pDx3, G4double pDx4, G4double pAlph) In the picture: pDx1 = 30, pDx2 = 40, pDy1 = 40, pDx3 = 10, pDx4 = 14, pDy2 = 16, pDz = 60, pTheta = 20*Degree, pDphi = 5*Degree, pAlph = 10*Degree, twistedangle = 30*Degree The first constructor of G4TwistedTrap produces a regular trapezoid twisted along the z-axis, where the caps of the trapezoid are of the same shape and size. The second constructor produces a generic trapezoid with polar, azimuthal and tilt angles. The twist angle cannot be greater than 90 degrees: twistedangle Twisted angle pDx1 Half x length at y=-pDy pDx2 Half x length at y=+pDy pDy Half y length pDz Half z length pTheta Polar angle of the line joining the centres of the faces at -/+pDz pDy1 Half y length at -pDz pDx1 Half x length at -pDz, y=-pDy1 pDx2 Half x length at -pDz, y=+pDy1 pDy2 Half y length at +pDz pDx3 Half x length at +pDz, y=-pDy2 pDx4 Half x length at +pDz, y=+pDy2 pAlph Angle with respect to the y axis from the centre of the side Twisted Trapezoid with x and y dimensions varying along z: A twisted trapezoid with the x and y dimensions varying along z can be defined as follows: G4TwistedTrd(const G4String& pName, G4double pDx1, G4double pDx2, G4double pDy1, G4double pDy2, G4double pDz, G4double twistedangle) In the picture: dx1 = 30, dx2 = 10, dy1 = 40, dy2 = 15, dz = 60, twistedangle = 30*Degree where: pDx1 Half x length at the surface positioned at -dz pDx2 Half x length at the surface positioned at +dz pDy1 Half y length at the surface positioned at -dz pDy2 Half y length at the surface positioned at +dz pDz Half z length twistedangle Twisted angle Tube Section Twisted along Its Axis: A tube section twisted along its axis can be defined as follows: G4TwistedTubs(const G4String& pName, G4double twistedangle, G4double endinnerrad, G4double endouterrad, G4double halfzlen, G4double dphi) In the picture: endinnerrad = 10, endouterrad = 15, halfzlen = 20, dphi = 90*Degree, twistedangle = 60*Degree G4TwistedTubs is a sort of twisted cylinder which, placed along the z-axis and divided into phi-segments is shaped like an hyperboloid, where each of its segmented pieces can be tilted with a stereo angle. It can have inner and outer surfaces with the same stereo angle: twistedangle Twisted angle endinnerrad Inner radius at endcap endouterrad Outer radius at endcap halfzlen Half z length dphi Phi angle of a segment Additional constructors are provided, allowing the shape to be specified either as: the number of segments in phi and the total angle for all segments, or a combination of the above constructors providing instead the inner and outer radii at z=0 with different z-lengths along negative and positive z-axis. Simple solids can be combined using Boolean operations. For example, a cylinder and a half-sphere can be combined with the union Boolean operation. Creating such a new Boolean solid, requires: Two solids A Boolean operation: union, intersection or subtraction. Optionally a transformation for the second solid. The solids used should be either CSG solids (for examples a box, a spherical shell, or a tube) or another Boolean solid: the product of a previous Boolean operation. An important purpose of Boolean solids is to allow the description of solids with peculiar shapes in a simple and intuitive way, still allowing an efficient geometrical navigation inside them. The solids used can actually be of any type. However, in order to fully support the export of a Geant4 solid model via STEP to CAD systems, we restrict the use of Boolean operations to this subset of solids. But this subset contains all the most interesting use cases. The tracking cost for navigating in a Boolean solid in the current implementation, is proportional to the number of constituent solids. So care must be taken to avoid extensive, unecessary use of Boolean solids in performance-critical areas of a geometry description, where each solid is created from Boolean combinations of many other solids. Examples of the creation of the simplest Boolean solids are given below: G4Box* box = new G4Box("Box",20*mm,30*mm,40*mm); G4Tubs* cyl = new G4Tubs("Cylinder",0,50*mm,50*mm,0,twopi); // r: 0 mm -> 50 mm // z: -50 mm -> 50 mm // phi: 0 -> 2 pi G4UnionSolid* union = new G4UnionSolid("Box+Cylinder", box, cyl); G4IntersectionSolid* intersection = new G4IntersectionSolid("Box*Cylinder", box, cyl); G4SubtractionSolid* subtraction = new G4SubtractionSolid("Box-Cylinder", box, cyl); where the union, intersection and subtraction of a box and cylinder are constructed. The more useful case where one of the solids is displaced from the origin of coordinates also exists. In this case the second solid is positioned relative to the coordinate system (and thus relative to the first). This can be done in two ways: Either by giving a rotation matrix and translation vector that are used to transform the coordinate system of the second solid to the coordinate system of the first solid. This is called the passive method. Or by creating a transformation that moves the second solid from its desired position to its standard position, e.g., a box's standard position is with its centre at the origin and sides parallel to the three axes. This is called the active method. In the first case, the translation is applied first to move the origin of coordinates. Then the rotation is used to rotate the coordinate system of the second solid to the coordinate system of the first. G4RotationMatrix* yRot = new G4RotationMatrix; // Rotates X and Z axes only yRot->rotateY(M_PI/4.*rad); // Rotates 45 degrees G4ThreeVector zTrans(0, 0, 50); G4UnionSolid* unionMoved = new G4UnionSolid("Box+CylinderMoved", box, cyl, yRot, zTrans); // // The new coordinate system of the cylinder is translated so that // its centre is at +50 on the original Z axis, and it is rotated // with its X axis halfway between the original X and Z axes. // Now we build the same solid using the alternative method // G4RotationMatrix invRot = *(yRot->invert()); G4Transform3D transform(invRot, zTrans); G4UnionSolid* unionMoved = new G4UnionSolid("Box+CylinderMoved", box, cyl, transform); Note that the first constructor that takes a pointer to the rotation-matrix (G4RotationMatrix*), does NOT copy it. Therefore once used a rotation-matrix to construct a Boolean solid, it must NOT be modified. In contrast, with the alternative method shown, a G4Transform3D is provided to the constructor by value, and its transformation is stored by the Boolean solid. The user may modify the G4Transform3D and eventually use it again. When positioning a volume associated to a Boolean solid, the relative center of coordinates considered for the positioning is the one related to the first of the two constituent solids. BREP solids are defined via the description of their boundaries. The boundaries can be made of planar and second order surfaces. Eventually these can be trimmed and have holes. The resulting solids, such as polygonal, polyconical solids are known as Elementary BREPS. In addition, the boundary surfaces can be made of Bezier surfaces and B-Splines, or of NURBS (Non-Uniform-Rational-B-Splines) surfaces. The resulting solids are Advanced BREPS. Currently, the implementation for surfaces generated by Beziers, B-Splines or NURBS is only at the level of prototype and not fully functional. Extensions in this area are foreseen in future. We have defined a few simple Elementary BREPS, that can be instantiated simply by a user in a manner similar to the construction of Constructed Solids (CSGs). We summarize their capabilities in the following section. Most BREPS Solids are however defined by creating each surface separately and tying them together. Specific BREP Solids: We have defined one polygonal and one polyconical shape using BREPS. The polycone provides a shape defined by a series of conical sections with the same axis, contiguous along it. The polyconical solid G4BREPSolidPCone is a shape defined by a set of inner and outer conical or cylindrical surface sections and two planes perpendicular to the Z axis. Each conical surface is defined by its radius at two different planes perpendicular to the Z-axis. Inner and outer conical surfaces are defined using common Z planes. G4BREPSolidPCone( const G4String& pName, G4double start_angle, G4double opening_angle, G4int num_z_planes, // sections, G4double z_start, const G4double z_values[], const G4double RMIN[], const G4double RMAX[] ) The conical sections do not need to fill 360 degrees, but can have a common start and opening angle. start_angle starting angle opening_angle opening angle num_z_planes number of planes perpendicular to the z-axis used. z_start starting value of z z_values z coordinates of each plane RMIN radius of inner cone at each plane RMAX radius of outer cone at each plane The polygonal solid G4BREPSolidPolyhedra is a shape defined by an inner and outer polygonal surface and two planes perpendicular to the Z axis. Each polygonal surface is created by linking a series of polygons created at different planes perpendicular to the Z-axis. All these polygons all have the same number of sides (sides) and are defined at the same Z planes for both inner and outer polygonal surfaces. The polygons do not need to fill 360 degrees, but have a start and opening angle. The constructor takes the following parameters: G4BREPSolidPolyhedra( const G4String& pName, G4double start_angle, G4double opening_angle, G4int sides, G4int num_z_planes, G4double z_start, const G4double z_values[], const G4double RMIN[], const G4double RMAX[] ) which in addition to its name have the following meaning: start_angle starting angle opening_angle opening angle sides number of sides of each polygon in the x-y plane num_z_planes number of planes perpendicular to the z-axis used. z_start starting value of z z_values z coordinates of each plane RMIN radius of inner polygon at each corner RMAX radius of outer polygon at each corner the shape is defined by the number of sides sides of the polygon in the plane perpendicular to the z-axis. In Geant4 it is also implemented a class G4TessellatedSolid which can be used to generate a generic solid defined by a number of facets (G4VFacet). Such constructs are especially important for conversion of complex geometrical shapes imported from CAD systems bounded with generic surfaces into an approximate description with facets of defined dimension (see ).

They can also be used to generate a solid bounded with a generic surface made of planar facets. It is important that the supplied facets shall form a fully enclose space to represent the solid. Two types of facet can be used for the construction of a G4TessellatedSolid: a triangular facet (G4TriangularFacet) and a quadrangular facet (G4QuadrangularFacet). An example on how to generate a simple tessellated shape is given below. // First declare a tessellated solid // G4TessellatedSolid solidTarget = new G4TessellatedSolid("Solid_name"); // Define the facets which form the solid // G4double targetSize = 10*cm ; G4TriangularFacet *facet1 = new G4TriangularFacet (G4ThreeVector(-targetSize,-targetSize, 0.0), G4ThreeVector(+targetSize,-targetSize, 0.0), G4ThreeVector( 0.0, 0.0,+targetSize), ABSOLUTE); G4TriangularFacet *facet2 = new G4TriangularFacet (G4ThreeVector(+targetSize,-targetSize, 0.0), G4ThreeVector(+targetSize,+targetSize, 0.0), G4ThreeVector( 0.0, 0.0,+targetSize), ABSOLUTE); G4TriangularFacet *facet3 = new G4TriangularFacet (G4ThreeVector(+targetSize,+targetSize, 0.0), G4ThreeVector(-targetSize,+targetSize, 0.0), G4ThreeVector( 0.0, 0.0,+targetSize), ABSOLUTE); G4TriangularFacet *facet4 = new G4TriangularFacet (G4ThreeVector(-targetSize,+targetSize, 0.0), G4ThreeVector(-targetSize,-targetSize, 0.0), G4ThreeVector( 0.0, 0.0,+targetSize), ABSOLUTE); G4QuadrangularFacet *facet5 = new G4QuadrangularFacet (G4ThreeVector(-targetSize,-targetSize, 0.0), G4ThreeVector(-targetSize,+targetSize, 0.0), G4ThreeVector(+targetSize,+targetSize, 0.0), G4ThreeVector(+targetSize,-targetSize, 0.0), ABSOLUTE); // Now add the facets to the solid // solidTarget->AddFacet((G4VFacet*) facet1); solidTarget->AddFacet((G4VFacet*) facet2); solidTarget->AddFacet((G4VFacet*) facet3); solidTarget->AddFacet((G4VFacet*) facet4); solidTarget->AddFacet((G4VFacet*) facet5); Finally declare the solid is complete // solidTarget->SetSolidClosed(true); The G4TriangularFacet class is used for the contruction of G4TessellatedSolid. It is defined by three vertices, which shall be supplied in anti-clockwise order looking from the outside of the solid where it belongs. Its constructor looks like: G4TriangularFacet ( const G4ThreeVector Pt0, const G4ThreeVector vt1, const G4ThreeVector vt2, G4FacetVertexType fType ) i.e., it takes 4 parameters to define the three vertices: G4FacetVertexType ABSOLUTE in which case Pt0, vt1 and vt2 are the three vertices in anti-clockwise order looking from the outside. G4FacetVertexType RELATIVE in which case the first vertex is Pt0, the second vertex is Pt0+vt1 and the third vertex is Pt0+vt2, all in anti-clockwise order when looking from the outside. The G4QuadrangularFacet class can be used for the contruction of G4TessellatedSolid as well. It is defined by four vertices, which shall be in the same plane and be supplied in anti-clockwise order looking from the outside of the solid where it belongs. Its constructor looks like: G4QuadrangularFacet ( const G4ThreeVector Pt0, const G4ThreeVector vt1, const G4ThreeVector vt2, const G4ThreeVector vt3, G4FacetVertexType fType ) i.e., it takes 5 parameters to define the four vertices: G4FacetVertexType ABSOLUTE in which case Pt0, vt1, vt2 and vt3 are the four vertices required in anti-clockwise order when looking from the outside. G4FacetVertexType RELATIVE in which case the first vertex is Pt0, the second vertex is Pt0+vt, the third vertex is Pt0+vt2 and the fourth vertex is Pt0+vt3, in anti-clockwise order when looking from the outside. Importing CAD models as tessellated shapes Tessellated solids can also be used to import geometrical models from CAD systems (see ). In order to do this, it is required to convert first the CAD shapes into tessellated surfaces. A way to do this is to save the shapes in the geometrical model as STEP files and convert them using a tool like STViewer or FASTRAD to tessellated (faceted surfaces) solids. This strategy allows to import any shape with some degree of approximation; the converted CAD models can then be imported through GDML (Geometry Description Markup Language) into Geant4 and be represented as G4TessellatedSolid shapes.