//last change: 2007/07/06 (Oliver Labs) /////////////////////////////////////////////////////////////////////////////// version="$Id: surfex.lib,v 1.9 2009-01-14 16:07:05 Singular Exp $"; category="Visualization"; info=" LIBRARY: surfex.lib Procedures for visualizing and rotating surfaces. @* It is still an alpha version (see http://www.AlgebraicSurface.net) AUTHOR: Oliver Labs This library uses the program surf (written by Stefan Endrass and others) and surfex (written by Oliver Labs and others, mainly Stephan Holzer). NOTE: This library requires the program surfex, surf and java to be installed. The software is used for producing raytraced images of surfaces. You can download @code{surfex} from http://www.surfex.AlgebraicSurface.net surfex is a front-end for surf which aims to be easier to use than the original tool. SEE ALSO: surf_lib PROCEDURES: plotRotated(poly,coord); Plot the surface given by the polynomial p with the coordinates coords(list) plotRot(poly); Similar to plotRotated, but guesses automatically which coordinates should be used plotRotatedList(varieties, coords); Plot the varieties given by the list varieties with the coordinates coords plotRotatedDirect(varieties); Plot the varieties given by the list varietiesList plotRotatedListFromSpecifyList(varietiesList); Plot the varieties given by the list varietiesList "; LIB "solve.lib"; LIB "primdec.lib"; LIB "sing.lib"; LIB "surf.lib"; /////////////////////////////////////////////////////////// // // the main procedures: // proc plotRot(poly p, list #) " USAGE: plotRot(poly p, list #) Similar to plotRotated, but guesses automatically which coordinates should be used. It opens the external program surfex for drawing the surface given by p, seen as a surface in the real affine space with coordinates coords. ASSUME: The basering is of characteristic zero and without parameters. " { list coords = list(); if(num_vars_id(p)==3) { execute("coords = "+string_of_vars(p)+";"); } else { if(num_vars_id(p)<3) { if(nvars(basering)==3) { execute("coords = "+varstr(basering)+";"); } else { if(nvars(basering)<3) { "Could not guess the coordinates because the number of variables in the basering is smaller than 3!"; "Please use plotRotated() instead of plotRot() and specify the coordinates explicitly."; return(0); } else { "Could not guess the coordinates because the number of variables in the polynomial is smaller than 3 and the number of variables in the basering is greater than three!"; "Please use plotRotated() instead of plotRot() and specify the coordinates explicitly."; return(0); } } } else { "Could not guess the coordinates because the number of variables in the polynomial is greater than 3!"; "Please use plotRotated() instead of plotRot() and specify the coordinates explicitly."; return(0); } } return(plotRotatedList(list(p), coords, #)); } example { "Example:"; echo=2; // More variables in the basering, but only 3 variables in the polynomial: ring r1 = 0, (w,x,y,z), dp; poly cayley_cubic = x^3+y^3+z^3+1^3-1/4*(x+y+z+1)^3; plotRot(cayley_cubic); // Three variables in the basering, but fewer variables in the polynomial: ring r2 = 0, (x,y,z), dp; plotRot(x^2+y^2-1); plotRot(y^2+z^2-1); // A cubic surface with a solitary point: // Use the additional parameter 3 to ask singular // to compute the singular locus before calling surfex. ring r3 = 0, (x,y,z), dp; poly kn_10 = x^3-3*x*y^2+z^3+3*x^2+3*y^2+z^2; plotRot(kn_10, 3); // The swallowtail: // a surface with a real solitary curve sticking out of the surface. // Use the additional parameter 3 to ask singular // to compute the singular locus before calling surfex. poly swallowtail = -4*y^2*z^3-16*x*z^4+27*y^4+144*x*y^2*z+128*x^2*z^2-256*x^3; } proc plotRotated(poly p, list coords, list #) " USAGE: plotRotated(poly p, list coords, list #) This opens the external program surfex for drawing the surface given by p, seen as a surface in the real affine space with coordinates coords. ASSUME: coords is a list of three variables. The basering is of characteristic zero and without parameters. " { return(plotRotatedList(list(p), coords, #)); } example { "Example:"; echo=2; // An easy example: a surface with four conical nodes. ring r = 0, (x,y,z), dp; poly cayley_cubic = x^3+y^3+z^3+1^3-1/4*(x+y+z+1)^3; // plotRotated(cayley_cubic, list(x,y,z)); // A difficult example: a surface with a one-dimensional real component! poly whitney_umbrella = x^2*z-y^2; // The Whitney Umbrella without its handle: plotRotated(whitney_umbrella, list(x,y,z)); // The Whitney Umbrella together with its handle: plotRotated(whitney_umbrella, list(x,y,z), 2); } proc plotRotatedList(list varieties, list coords, list #) " USAGE: plotRotatedList(list varieties, list coords, list #) This opens the external program surfex for drawing the surfaces given by varieties, seen as a surface in the real affine space with coordinates coords. ASSUME: coords is a list of three variables, varieties is a list of ideals describing the varieties to be shown. The basering is of characteristic zero and without parameters. " { def oring = basering; int plotquality = 0; if(size(#)>0) { plotquality = #[1]; } list varietiesList = list(list(), list(), list(), list()); list usedSurfaces = list(); list curveColors = list(); // go through the list of varieties // produce a list which can be used as input for plotRotatedListFromList() int i; int j; list indList; int ind; ideal itmp; int ncurves; list pd; int k; int surfind; list curSurfColors = list(); list listOfPoints = list(); string str_I = ""; for(i=1; i<=size(varieties); i++) { itmp = varieties[i]; if(plotquality>=3) { itmp = radical(itmp); } itmp = simplify(itmp,1); itmp = simplify(itmp,2); if(size(itmp)==1) { // i.e.: a surface given by one equation surfind = findInList(surfEqn(itmp[1],coords), usedSurfaces); if(surfind==0) { usedSurfaces = usedSurfaces + list(surfEqn(itmp[1],coords)); curSurfColors = list(list("insidecolor:",getInsideColorStr(size(varietiesList[1])+1)), list("outsidecolor:",getOutsideColorStr(size(varietiesList[1])+1))); varietiesList[1] = varietiesList[1] + list(list(list("eqno:",string(size(varietiesList[1])+1)), list("equation:",surfEqn(itmp[1], coords)), curSurfColors[1], curSurfColors[2], list("showcbox:","true"), list("transparency:","0"))); surfind = size(varietiesList[1]); } if(plotquality==1) { varieties = varieties + list(slocus(itmp[1])); } if(plotquality==2 || plotquality==3) { // remove doubled components and // add the 1-dimensional singular components // of the surface to the list of curves: int dsl = dim_slocus(itmp[1]); dsl; if(dsl>=0) { // i.e. there is a singular locus "compute singular locus..."; list eqd; // eqd = equidim(slocus(itmp[1])); ideal tmp_l; tmp_l = std(eqd[size(eqd)]); "dim:",dim(tmp_l); if(dim(tmp_l)==(nvars(basering)-3+2)) { "--- 2-dim."; // we have found a multiple component; // replace it by a simple copy of it itmp = quotient(itmp[1], tmp_l); varieties[i] = itmp[1]; eqd = delete(eqd,size(eqd)); if(size(eqd)>0) { tmp_l = std(eqd[size(eqd)]); } } if(dim(tmp_l)==(nvars(basering)-3+1)) { "--- 1-dim."; // we have found a 1-dimensional singular locus pd = std_primdecGTZ(tmp_l,2); for(k=1; k<=size(pd); k++) { if(pd[k][3]==(nvars(basering)-3+1)) { varieties = varieties + list(pd[k][2]); curveColors[size(varieties)] = curSurfColors; } else { "???"; } } eqd = delete(eqd,size(eqd)); if(size(eqd)>0) { tmp_l = std(eqd[size(eqd)]); } } if(dim(tmp_l)==(nvars(basering)-3+0)) { "--- 0-dim."; // we have found a 0-dimensional singular locus // we compute floating point approximations of the // coordinates of all singular points if(npars(oring)>0) { "str:",parstr(1),rootminpoly(); list all_real_sols = allroots_minpoly(); // "all sols:";all_real_sols; // sprintf("number %s = %s; ", parstr(1), rootminpoly()); int minp; if((npars(basering) == 1) && (minpoly != 0)) { minp = 1; } else { minp = 0; } str_I = ""; if(minp==1) { "minp=1"; string str_para = parstr(1); string str_tmp_l; def cur_ring = basering; if(1) { short=0; str_tmp_l = "ideal eqd_tmp = "+ // string(tmp_l)+","+string(minpoly)+";"; string(tmp_l); "str:",str_tmp_l; string str_num_mp = "number "+parstr(1)+"="+ decstr2ratstr(rootminpoly())+";"; execute("ring Iring = 0,(" // +string(coords)+","+str_para+"),dp;"); +string(coords)+"),dp;"); basering; execute(str_num_mp); execute(str_tmp_l); eqd_tmp; list real_sols = real_solve(eqd_tmp); real_sols; $; setring cur_ring; } } else { // minp==0: we do not know how to handle this "???"; } } else { "no pars"; ideal eqd_tmp = tmp_l; short=0; string str_tmp_l = "ideal eqd_tmp = "+string(tmp_l)+";"; def cur_ring = basering; execute("ring Iring = (real,30),("+string(coords)+"),("+ordstr(oring)+");"); // basering; execute(str_I); execute(str_tmp_l); list real_sols = real_solve(eqd_tmp); setring cur_ring; } "real_sols:";real_sols; for(k=1; k<=size(real_sols); k++) { "search point:"; string(list(real_sols[k][1],real_sols[k][2],real_sols[k][3],string(surfind))); // listOfPoints; if(findInList(string(list(list(real_sols[k][1],real_sols[k][2],real_sols[k][3],string(surfind)))), listOfPoints)==0) { "add pt"; varietiesList[4] = varietiesList[4] + list(list(real_sols[k][1],real_sols[k][2],real_sols[k][3],string(surfind))); listOfPoints = listOfPoints + list(string(list(real_sols[k][1],real_sols[k][2],real_sols[k][3],string(surfind)))); } } } } } } else { // i.e.: more than one equation varietiesList[2] = varietiesList[2] + list(list(list("surfaces:"), list("curveno:", string(size(varietiesList[2])+1)), list("showcbox:","true"))); if(size(curveColors) >= i) { varietiesList[2][size(varietiesList[2])][4] = curveColors[i][1]; varietiesList[2][size(varietiesList[2])][4][1] = "color:"; } ncurves = size(varietiesList[2]); for(j=1; j<=size(itmp); j++) { ind = findInList(surfEqn(itmp[j],coords), usedSurfaces); usedSurfaces = usedSurfaces + list(surfEqn(itmp[1],coords)); // "indList:";indList; if(ind == 0) { // "--------> not in list", surfEqn(itmp[j], coords); if(j==1) { varietiesList[1] = varietiesList[1] + list(list(list("eqno:",string(size(varietiesList[1])+1)), list("equation:",surfEqn(itmp[j], coords)), list("insidecolor:",getInsideColorStr(size(varietiesList[1])+1)), list("outsidecolor:",getOutsideColorStr(size(varietiesList[1])+1)), list("showcbox:","true"), list("transparency:","100"))); } else { varietiesList[1] = varietiesList[1] + list(list(list("eqno:",string(size(varietiesList[1])+1)), list("equation:",surfEqn(itmp[j], coords)), list("insidecolor:",getInsideColorStr(size(varietiesList[1])+1)), list("outsidecolor:",getOutsideColorStr(size(varietiesList[1])+1)), list("showcbox:","false"), list("transparency:","0"))); } ind = size(varietiesList[1]); } else { } varietiesList[2][ncurves][1] = varietiesList[2][ncurves][1] + list(string(ind)); } } } // "------------"; // varietiesList; // "------------"; return(plotRotatedListFromSpecifyList(varietiesList, coords, #)); } example { "Example:"; echo=2; // A cubic surface together with a tritangent plane // (i.e. a plane which cuts out three lines). ring r = 0, (x,y,z), dp; poly cayley_cubic = x^3+y^3+z^3+1^3-1/4*(x+y+z+1)^3; poly plane = 1-x-y-z; plotRotatedList(list(cayley_cubic, plane), list(x,y,z)); // The same cubic and plane. // The plane is not shown but only its intersection with the surface. plotRotatedList(list(cayley_cubic, ideal(cayley_cubic, plane)), list(x,y,z)); } proc plotRotatedListFromSpecifyList(list varietiesList, list #) " USAGE: plotRotatedListFromSpecifyList(list varietiesList, list #); varietiesList has a complicated format (not documented yet); see the example. ASSUME: The basering is of characteristic zero. EXAMPLE: example plotRotatedListFromSpecifyList; " { // make the surfex file string str = getSurfexCodeFromSpecifyList(varietiesList, #); return(plotRotatedFromCode(str, #)); } example { "Example:"; echo=2; // A cubic surface depending on a parameter: ring r = (0,p1), (x,y,z), dp; poly cayley_cubic = x^3+y^3+z^3+1^3-p1*(x+y+z+1)^3; poly plane = 1-x-y-z; plotRotatedListFromSpecifyList(list(list(list(list("eqno:","1"), list("equation:", string(cayley_cubic)) ) ), list(), list(list(1,"0.0","1.0","500","0.25+0.25*sin(PI*p1)")), list() )); } proc plotRotatedListFromStringList(list varieties, list #) " RETURN: the return code of the system command which executes surfex. USAGE: not documented yet. " { // make the surfex file getSurfexCodeFromStringList(varieties, #); string str = getSurfexCodeFromStringList(varieties, #); return(plotRotatedFromCode(str, #)); } proc plotRotatedDirect(list varieties, list #) " USAGE: plotRotatedDirect(list varieties, list #) This opens the external program surfex for drawing the surfaces given by varieties, seen as a surface in the real affine space with coordinates x,y,z. The format for the list varieties is not fully documented yet; please, see the examples below and try to adapt the examples to your needs. ASSUME: Passes the equations directly to surfex, i.e., the variable names should be x,y,z. The advantage is that one can use parameters p1, p2, ...; these will be passed to surfex. " { string str = getSurfexCodeFromListDirect(varieties, #); return(plotRotatedFromCode(str, #)); } example { "Example:"; echo=2; // A cubic surface depending on a parameter: ring r = (0,p1), (x,y,z), dp; poly cayley_cubic = x^3+y^3+z^3+1^3-p1*(x+y+z+1)^3; // The entries of the list of varieties can either be polynomials plotRotatedDirect(list(list(list(cayley_cubic)), list(), list(list(1,"0.0","1.0","500","0.25+0.25*sin(PI*p1)")) )); // or strings which represent surfex-readable polynomials plotRotatedDirect(list(list(list("x^3+y^3+z^3+1^3-p1*(x+y+z+1)^3")), list(), list(list("1","0.0","1.0","500","0.25+0.25*sin(PI*p1)")) )); // More complicated varieties plotRotatedDirect(list(list(list("x^2+y^2-z^2-3^2"), list("x*sin(p1)+y*cos(p1)-3")), list(list(list(1,2))), list(list("1","0.0","1.0","500","2*PI*p1")) )); } proc plotRotatedFromCode(string str, list #) " USAGE: plotRotatedFromCode(string str, list #); This procedure is only for internal usage; it takes the surfex-code as a string and calls surfex. " { // we need a temporary .sux file for surfex string tmpd = "/tmp"; string l="surf"+string(system("pid"))+".sux"; // a temporary file which stores the output of surfex string erg="/tmp/surferg"+string(system("pid")); write(":w "+tmpd+"/"+l, str); string surfex_path=system("Singular"); while(surfex_path[size(surfex_path)]!="/") { surfex_path=surfex_path[1..size(surfex_path)-1]; } surfex_path=surfex_path+"../LIB/surfex"; if (status(surfex_path,"exists")=="no") { // search in SINGULAR_PATH: string surfex_path1=system("SingularLib"); string surfex_path2=surfex_path1; while (find(surfex_path1,":")!=0) { surfex_path2=surfex_path1[1..find(surfex_path1,":")-1]; while(surfex_path2[size(surfex_path2)]==" ") { surfex_path2 = surfex_path2[1..(size(surfex_path2)-1)]; } if (status(surfex_path2+"/surfex","exists")=="yes") break; surfex_path1=surfex_path1[find(surfex_path1,":")+1,size(surfex_path1)]; surfex_path2=surfex_path1[1..(size(surfex_path1)-1)]; while(surfex_path2[size(surfex_path2)]==" ") { surfex_path2 = surfex_path2[1..(size(surfex_path2)-1)]; } } surfex_path=surfex_path2+"/surfex"; } int i=system("sh","surfex \""+surfex_path+"\" -d "+tmpd+" -i " + l +" >"+erg+" 2>/dev/null"); // delete the temporary file i = system("sh","rm " + l +" 2>/dev/null"); return(read(erg)); } /////////////////////////////////////////////////////////// // // procedures used to produce the surf-code: // proc getSurfexCodeFromListDirect(list varieties, list #) " USAGE: getSurfexCodeFromListDirect(list varieties, list #) ASSUME: varieties has four components, - the first is a list of polynomials, say f_1, ..., f_k - the second is a list of lists of numbers in {1, ..., k} describing the curves as intersections of the corresponding f_i - the third is a list of lists describing the parameters used in the polynomials f_i - the fourth is a list of lists of points given by their approximate coordinates (three decimal numbers) RETURN: the surfex code (.sux) " { int i; int j; string str = "this is surfex v0.89.07"+newline; str = str + "TYPE:" + newline; str = str + "specify"+newline; str = str + "EQUATIONS:"+newline; str = str + string(size(varieties[1])) + newline; for(i=1; i<=size(varieties[1]); i++) { str = str + "Equation:"+newline; str = str + "eqno:"+newline; str = str + string(i) + newline; str = str + "equation:"+newline; str = str + surfEqnDir(varieties[1][i][1]) + newline; if(size(varieties[1][i])>=2) { str = str + "showcbox:"+newline; str = str + varieties[1][i][2] + newline; // show it or not if(size(varieties[1][i])>=3) { str = str + "transparency:"+newline; str = str + string(varieties[1][i][3]) + newline; // transparency } } } str = str + "CURVES:"+newline; str = str + string(size(varieties[2])) + newline; for(i=1; i<=size(varieties[2]); i++) { str = str + "Curve:"+newline; str = str + "curveno:"+newline; str = str + string(i) + newline; str = str + "surfaces:"+newline; // "curves:";varieties[2][i]; for(j=1; j<=size(varieties[2][i][1]); j++) { str = str + string(varieties[2][i][1][j]) + newline; } if(size(varieties[2][i])>=2) { str = str + "showcbox:"+newline; str = str + varieties[2][i][2] + newline; // show it or not } } str = str + "PARAMETERS:"+newline; str = str + string(size(varieties[3])) + newline; for(i=1; i<=size(varieties[3]); i++) { str = str + "Parameter:"+newline; str = str + "parno:"+newline; str = str + string(varieties[3][i][1]) + newline; str = str + "fromtoval:"+newline; str = str + varieties[3][i][2] + newline; str = str + varieties[3][i][3] + newline; str = str + string(varieties[3][i][4]) + newline; if(size(varieties[3][i])>=5) { str = str + "function:"+newline; str = str + varieties[3][i][5]+newline; } } // str = str + "////////////////// Parameter: /////////////////////////"+newline; // str = str + "1" + newline; // str = str + "0.0" + newline; // str = str + "1.0" + newline; // str = str + "1000" + newline; // str = str + string(size(varieties[3])) + newline; return(str); } proc getSurfexCodeFromList(list varieties, list coords, list #) " ASSUME: varieties has four components, - the first is a list of polynomials, say f_1, ..., f_k - the second is a list of lists of numbers in {1, ..., k} describing the curves as intersections of the corresponding f_i - the third is a list of lists describing the parameters used in the polynomials f_i - the fourth is a list of lists of points given by their approximate coordinates (three decimal numbers) RETURN: the surfex code (.sux) " { int i; int j; string str = "this is surfex v0.89.07"+newline; str = str + "TYPE:" + newline; str = str + "specify"+newline; str = str + "EQUATIONS:"+newline; str = str + string(size(varieties[1])) + newline; for(i=1; i<=size(varieties[1]); i++) { str = str + "Equation:"+newline; str = str + "eqno:"+newline; str = str + string(i) + newline; str = str + "equation:"+newline; str = str + surfEqn(varieties[1][i][1], coords) + newline; str = str + "showcbox:"+newline; str = str + varieties[1][i][2] + newline; // show it or not str = str + "transparency:"+newline; str = str + string(varieties[1][i][3]) + newline; // transparency } str = str + "CURVES:"+newline; str = str + string(size(varieties[2])) + newline; for(i=1; i<=size(varieties[2]); i++) { str = str + "Curve:"+newline; str = str + "curveno:"+newline; str = str + string(i) + newline; str = str + "surfaces:"+newline; for(j=1; j<=size(varieties[2][i]); j++) { str = str + string(varieties[2][i][1][j]) + newline; } str = str + "showcbox:"+newline; str = str + varieties[2][i][2] + newline; // show it or not } str = str + "PARAMETERS:"+newline; str = str + string(size(varieties[3])) + newline; for(i=1; i<=size(varieties[3]); i++) { str = str + "Parameter:"+newline; str = str + "parno:"+newline; str = str + string(varieties[3][i][1]) + newline; str = str + "fromtoval:"+newline; str = str + surfEqn(varieties[3][i][2], coords) + newline; str = str + surfEqn(varieties[3][i][3], coords) + newline; str = str + string(varieties[3][i][4]) + newline; if(size(varieties[3][i])>=5) { str = str + "function:"+newline; str = str + varieties[3][i][5]+newline; } } // str = str + "////////////////// Parameter: /////////////////////////"+newline; // str = str + "1" + newline; // str = str + "0.0" + newline; // str = str + "1.0" + newline; // str = str + "1000" + newline; // str = str + string(size(varieties[3])) + newline; return(str); } proc getSurfexCodeFromStringList(list varieties, list #) " ASSUME: varieties has three components, - the first is a list of polynomials, say f_1, ..., f_k - the second is a list of lists of numbers in {1, ..., k} describing the curves as intersections of the corresponding f_i - the third is a list of lists describing the parameters used in the polynomials f_i RETURN: the surfex code (.sux) " { int i; int j; string str = "this is surfex v0.89.07"+newline; str = str + "TYPE:" + newline; str = str + "specify"+newline; str = str + "EQUATIONS:"+newline; str = str + string(size(varieties[1])) + newline; for(i=1; i<=size(varieties[1]); i++) { str = str + "Equation:"+newline; str = str + "eqno:"+newline; str = str + string(i) + newline; str = str + "equation:"+newline; str = str + varieties[1][i][1] + newline; str = str + "showcbox:"+newline; str = str + varieties[1][i][2] + newline; // show it or not str = str + "transparency:"+newline; str = str + varieties[1][i][3] + newline; // transparency } str = str + "CURVES:"+newline; str = str + string(size(varieties[2])) + newline; for(i=1; i<=size(varieties[2]); i++) { str = str + "Curve:"+newline; str = str + "curveno:"+newline; str = str + string(i) + newline; str = str + "surfaces:"+newline; for(j=1; j<=size(varieties[2][i][1]); j++) { str = str + string(varieties[2][i][1][j]) + newline; } str = str + "showcbox:"+newline; str = str + varieties[2][i][2] + newline; // show it or not } str = str + "PARAMETERS:"+newline; str = str + string(size(varieties[3])) + newline; for(i=1; i<=size(varieties[3]); i++) { str = str + "Parameter:"+newline; str = str + "parno:"+newline; str = str + string(varieties[3][i][1]) + newline; str = str + "fromtoval:"+newline; str = str + varieties[3][i][2] + newline; str = str + varieties[3][i][3] + newline; str = str + string(varieties[3][i][4]) + newline; if(size(varieties[3][i])>=5) { str = str + "function:"+newline; str = str + varieties[3][i][5]+newline; } } return(str); } proc getSurfexCodeFromSpecifyList(list varieties, list #) " ASSUME: varieties has three components, - the first is a list of polynomials, say f_1, ..., f_k - the second is a list of lists of numbers in {1, ..., k} describing the curves as intersections of the corresponding f_i - the third is a list of lists describing the parameters used in the polynomials f_i - the fourth is a list of lists describing the singular points to be shown as spheres RETURN: the surfex code (.sux) " { int i; int j; int k; string str = "this is surfex v0.89.07"+newline; str = str + "TYPE:" + newline; str = str + "specify"+newline; str = str + "EQUATIONS:"+newline; str = str + string(size(varieties[1])) + newline; for(i=1; i<=size(varieties[1]); i++) { str = str + "Equation:"+newline; for(j=1; j<=size(varieties[1][i]); j++) { str = str + varieties[1][i][j][1] +newline; str = str + varieties[1][i][j][2] +newline; } } str = str + "CURVES:"+newline; str = str + string(size(varieties[2])) + newline; for(i=1; i<=size(varieties[2]); i++) { str = str + "Curve:"+newline; for(j=1; j<=size(varieties[2][i]); j++) { str = str + varieties[2][i][j][1] +newline; if(varieties[2][i][j][1] == "surfaces:") { for(k=2; k<=size(varieties[2][i][j]); k++) { str = str + string(varieties[2][i][j][k]) + newline; } } else { str = str + varieties[2][i][j][2] +newline; } } // str = str + "curveno:"+newline; // str = str + string(i) + newline; // str = str + "surfaces:"+newline; // for(j=1; j<=size(varieties[2][i][1]); j++) { // str = str + string(varieties[2][i][1][j]) + newline; // } // str = str + "showcbox:"+newline; // str = str + varieties[2][i][2] + newline; // show it or not } str = str + "PARAMETERS:"+newline; str = str + string(size(varieties[3])) + newline; for(i=1; i<=size(varieties[3]); i++) { str = str + "Parameter:"+newline; str = str + "parno:"+newline; str = str + string(varieties[3][i][1]) + newline; str = str + "fromtoval:"+newline; str = str + varieties[3][i][2] + newline; str = str + varieties[3][i][3] + newline; str = str + string(varieties[3][i][4]) + newline; if(size(varieties[3][i])>=5) { str = str + "function:"+newline; str = str + varieties[3][i][5]+newline; } } string str_from = "0.0"; string str_to = "5.0"; string str_radius = "50"; str = str + "SOLITARY POINTS:"+newline; str = str + string(size(varieties[4])) + newline; for(i=1; i<=size(varieties[4]); i++) { str = str + "SolitaryPoint:"+newline; str = str + "solPtNo:"+newline; str = str + string(i) + newline; str = str + "surface:"+newline; str = str + varieties[4][i][4] + newline; str = str + "fromtoval:"+newline; str = str + str_from + newline; str = str + str_to + newline; str = str + str_radius + newline; str = str + "coords:" + newline; str = str + varieties[4][i][1] + newline; str = str + varieties[4][i][2] + newline; str = str + varieties[4][i][3] + newline; } return(str); } /////////////////////////////////////////////////////////// // // procedures for standard colors: // proc numBaseColors() " USAGE: numBaseColors() RETURN: the number of predefined surface colors. " { return(6); } proc baseSurfaceColors(int no) " USAGE: baseSurfaceColors(int no) REMARK: There are currently 6=numBaseColors() basic surface colors. You can modify them according to your wishes by just redefining this procedure in your Singular-script. If you want more colors, then you also have to redefine numBaseColors() accordingly. RETURN: a list of three integers describing the RGB values of a color. " { if(no%numBaseColors()==1) { return(list(240,160,0)); } if(no%numBaseColors()==2) { return(list(160,240,0)); } if(no%numBaseColors()==3) { return(list(0,160,240)); } if(no%numBaseColors()==4) { return(list(240,0,160)); } if(no%numBaseColors()==5) { return(list(0,240,160)); } if(no%numBaseColors()==0) { return(list(160,0,240)); } } proc getInsideColorStr(int no) " USAGE: getInsideColorStr(int no) RETURN: a string describing inside color number no where the three integer RGB values are in one line each. " { list bc = baseSurfaceColors(no); string str = string(bc[1])+newline+string(bc[2])+newline+string(bc[3]); return(str); } proc getOutsideColorStr(int no) " USAGE: getOutsideColorStr(int no) RETURN: a string describing outside color number no where the three integer RGB values are in one line each. " { list bc = baseSurfaceColors(no); string str = string(bc[1])+newline+string(bc[2])+newline+string(bc[3]); return(str); } /////////////////////////////////////////////////////////// // // procedures used by the plot procedures: // proc surfEqnDir(list #) " USAGE: surfEqnDir(list #) without any checks etc. RETURN: string(#[1]) where short=0. " { int stmp = short; short = 0; string str = string(#[1]); short = stmp; return(str); } proc surfEqn(poly p, list coords, list #) " USAGE: surfEqn(poly p, list coords) Tries to produce a string for the equation of p which is convenient for surfex. ASSUME: - p defines a plane curve or a surface, - coords is a list of the three coordinates to use, e.g. list(x,y,z), in this way, it is possible to distinguish between x^2+y^2-1 and y^2+z^2-1 RETURN: a string, that one can use with the external program surf EXAMPLE: example surfEqn; shows an example " { int params=0; if(size(#)>0) { params = #[1]; } string err_mes; // string containing error messages def base=basering; int mynvars = nvars(basering); intvec ind=num_of_vars(p); int i,j,n; int minp = 0; n=0; for(i=size(ind);i>0;i--) { if (ind[i]!=0) { n++; } else { if(var(i)==coords[1] || var(i)==coords[2] || var(i)==coords[3]) { ind[i]=1; n++; } } } params = params + npars(basering); n = n + npars(basering); if((npars(basering) == 1) && (minpoly != 0)) { minp = 1; } else { minp = 0; } string str_I = ""; for(i=1; i<=npars(basering); i=i+1) { if(!(parstr(i) == "i")) { if(minp==1) { str_I = str_I + sprintf("number %s = %s; ", parstr(i), rootminpoly()); } else { } } } int bshort = short; short = 0; if(!(minp==1 || npars(basering)==0)) { p=cleardenom(p); err_mes="Cannot plot equations with a parameter without a specified minpoly"; ERROR(err_mes); } str_I = str_I + "poly p = " + string(p) + ";"; short = bshort; if(params==0) { if (n<=2 or n>=4) { err_mes="Cannot plot equations with "+string(n)+" variables"; ERROR(err_mes); // return("0"); } if(n==4) { ring r=(real,30,30),(xx,yy,zz,ww),dp; } else { ring r=(real,30,30),(x,y,z),dp; } } else { if(n-params<=2 || n-params>=4) { err_mes="Cannot plot equations with "+string(n-params)+" variables"; ERROR(err_mes); // return("0"); } else { if(params == 1) { if(n-params==3) { if(minp==1) { // switch to a ring without minimal polynomial: execute("ring rr = (real,30,30),("+varstr(base)+"), dp;"); // rr; // "str_I",str_I; execute(str_I); def base = rr; ring r=(real,30,30),(x,y,z),dp; } else { p=cleardenom(p); ring r=(real,30,30),(x,y,z,p1),dp; } } } if(params == 2) { if(n-params==3) { p=cleardenom(p); ring r=(real,30,30),(x,y,z,p1,p2),dp; } } if(params == 3) { if(n-params==3) { p=cleardenom(p); execute("ring rr = (real,30,30),("+varstr(base)+","+parstr(base)+"), dp;"); rr; "str_I",str_I; execute(str_I); "pnew:",p; def base = rr; ring r=(real,30,30),(x,y,z,p1,p2,p3),dp; } } } } // basering; short=0; map phi=base,0; j=1; for(i=1;i<=mynvars;i++) { if (ind[i]!=0) { phi[i]=var(j); j++; } } poly p=(simplify(phi(p),1)); if (leadcoef(p) <0) { if(size(#)>1) { if(#[2]!=0) { p=-p; } } else { p=-p; } } if(leadcoef(p)!=0) { p = p/leadcoef(p); } string thesurfstr = string(p); if(minp == 1) { // replace k by rootRepl } return (thesurfstr); } // end of surfEqn() example { "EXAMPLE:"; echo =2; ring rr0 = 0,(x(1..3)),dp; poly p = x(1)^3 - x(2)^2; print(surfEqn(p,list(x(1),x(2),x(3)))); ring rr1 = 0,(x,y,z),dp; poly I(1) = 2x2-1/2x3 +1-y+1; print(surfEqn(I(1),list(x,y,z))); // Steiner surface poly J(2) = x^2*y^2+x^2*z^2+y^2*z^2-17*x*y*z; print(surfEqn(J(2),list(x,y,z))); } // end of example surfEqn() proc num_vars_id(ideal I) " USAGE: num_vars_id(ideal I) RETURN: The number of ring-variables occurring in the ideal I. " { intvec v = num_of_vars(I); int num = 0; for(int i=size(v);i>0;i--) { if (v[i]!=0) { num++; } } return(num); } example { "EXAMPLE:"; echo = 2; ring r = 0, (x,y,z),dp; ideal j = x^2-y, x^3-2; num_vars_id(j); } proc findInList(list obj, list l) " USAGE: findInList(list obj, list l) Tries to find the object obj in the list l. ASSUME: the object obj[1] can be compared to the objects in the list l RETURN: if obj[1]=l[i] for some i, then return the first such i, otherwise return 0 " { for(int i=1; i<=size(l); i++) { if(l[i]==obj[1]) { return(i); } } return(0); } example { "EXAMPLE:"; echo = 2; ring r = 0,(x,y,z), dp; list a = list(x^2+y^2+z^2+1, x^2+y^2+z^2-1, x^2+y^2-z^2+1, x^2+y^2-z^2-1); findInList(x^2+y^2+z^2-1, a); findInList(x^2+y^2+z^2, a); } proc std_primdecGTZ(ideal I, list #) " USAGE: std_primdecGTZ(ideal I, list #) Computes a primdary decomposition pd of I using primdecGTZ and then calls std_for_pd(pd). For the output and options, consult the help of std_for_pd. RETURN: see std_for_pd. " { list pd = primdecGTZ(I); return(std_for_pd(pd, #)); } example { "EXAMPLE:"; echo = 2; ring r = 0, (x,y), dp; ideal j = y-x^2,z-x^3; primdecGTZ(j); std_primdecGTZ(j); std_primdecGTZ(j,1); } proc std_for_pd(list pd, list #) " USAGE: std_for_pd(list pd, list #) Call std for each of the prime ideals in the list pd replace the prime ideals by their standard-basis. Compute dim() and mult() of each prime component using these standard bases. If an additional argument is given then do the same for the primary components. ASSUME: pd is in the format produced by primdecGTZ() or primdecSY(). RETURN: A list, say l, of lists, similar to a list returned by primdecSY() or primdecGTZ(). However, each of the entries of l (which is a list l[i]) contains some additional entries: l[1]: the primary ideal l[2]: a standard basis of the associated prime ideal l[3]: dim() of this prime ideal l[4]: mult() of this prime ideal If an additional argument # is given then l[1] changes: l[1]: a standard basis of the primary ideal Morever, there are some more entries: l[5]: dim() of this primary ideal l[6]: mult() of this primary ideal l[7]: l[6] / l[5] " { if(typeof(pd[1])=="ideal") { // this is a Singular bug!? // "bug!";pd;"---"; pd = list(list(pd[1], pd[1])); // pd;$; } list pd_neu; int i; list coords; ideal stdtmp; ideal stdtmp2; for(i=1; i<=size(pd); i++) { stdtmp = std(pd[i][2]); stdtmp2 = pd[i][1]; if(size(#)>0) { stdtmp2 = std(stdtmp2); if(mult(stdtmp)==0) { pd_neu[i] = list(stdtmp2, stdtmp, dim(stdtmp), mult(stdtmp), dim(stdtmp2), mult(stdtmp2), 0); } else { pd_neu[i] = list(stdtmp2, stdtmp, dim(stdtmp), mult(stdtmp), dim(stdtmp2), mult(stdtmp2), mult(stdtmp2)/mult(stdtmp)); } } else { pd_neu[i] = list(stdtmp2, stdtmp, dim(stdtmp), mult(stdtmp)); } } return(pd_neu); } example { "EXAMPLE:"; echo = 2; ring r = 0, (x,y,z), dp; ideal j = y-x^2,z-x^3; list pd = primdecGTZ(j); pd; std_for_pd(pd, 1); } proc real_solve(ideal to_solve) " USAGE: real_solve(ideal to_solve) RETURN: a list of all real solutions (as strings) of the zero-dimensional ideal to_solve (without multiplicities). REMARK: Until now, it may happen that some points appear more than once. " { int k; int i; // def Isolring = solve(to_solve,30,0,60,"nodisplay"); def Isolring = solve(to_solve,9,0,13,"nodisplay"); setring Isolring; // list SOL = solve(to_solve, "oldring", "nodisplay"); list real_sols = list(); list tmpl; for(k=1; k<=size(SOL); k++) { if(find(string(SOL[k]),"I")==0 && find(string(SOL[k]),"i")==0) { tmpl = list(); for(i=1; i<=size(SOL[k]); i++) { tmpl = tmpl + list(string(SOL[k][i])); } real_sols = real_sols + list(tmpl); } } return(real_sols); } example { "EXAMPLE:"; echo = 2; ring r = 0, (x,y), dp; number a = 2; number b = 3; ideal j = (x^2-a),(y^3-b); real_solve(j); } proc rootminpoly(list #) " USAGE: rootminpoly(list #) RETURN: A root of the current minpoly as a string representation of a complex number with the given precision #[1] (default: 30). E.g. ring r=(0,s),x,dp; minpoly = s^2-2; => rootminpoly() 1.41421356237309504880168872421 ASSUME: The current minpoly is non-zero. " { int prec = 30; int k, done; if(size(#)>0) { prec = #[1]; } short = 0; string str_lag = sprintf("list lag = laguerre_solve(%s);", minpoly); string str_ring = sprintf("ring r_sqrt = (complex,prec,I),(%s),lp;", parstr(basering)); execute(str_ring); execute(str_lag); // lag; // choose a real solution, if it exists: done = 0; for(k=1; k<=size(lag) && done==0; k++) { if(find(string(lag[k]),"I")==0) { done = k; } } if(done==0) { // "no real solution."; } if(size(lag)>2) { // return the first real solution return(sprintf("%s",lag[done])); } if(sprintf("%s",lag[1])[1] == "-") { return(sprintf("%s",lag[2])); } else { if(sprintf("%s",lag[1])[1] == "(") { if(sprintf("%s",lag[1])[2] == "-") { return(sprintf("%s",lag[2])); } else { return(sprintf("%s",lag[1])); } } else { return(sprintf("%s",lag[1])); } } short = 1; } example { "EXAMPLE:"; echo =2; ring r=(0,s),x,dp; minpoly = s^2-2; rootminpoly(); ring R=(0,s),x,dp; minpoly = s^2+2; rootminpoly(); } proc allroots_minpoly(list #) " USAGE: allroots_minpoly(list #) RETURN: a list of strings containing all real roots of the minimal polynomial of the active ring. ASSUME: The current minpoly is non-zero. " { int prec = 30; int k, done; if(size(#)>0) { prec = #[1]; } short = 0; string str_lag = sprintf("list lag = laguerre_solve(%s);", minpoly); string str_ring = sprintf("ring r_sqrt = (complex,prec,I),(%s),lp;", parstr(basering)); execute(str_ring); execute(str_lag); // only take the real solutions: done = 0; list real_sols = list(); for(k=1; k<=size(lag) && done==0; k++) { if(find(string(lag[k]),"I")==0) { real_sols = real_sols + list(string(lag[k])); } } return(real_sols); } example { "EXAMPLE:"; echo = 2; ring r=(0,s),x,dp; minpoly = s^3-2; allroots_minpoly(); ring R=(0,s),x,dp; minpoly = s^2-2; allroots_minpoly(); } proc decstr2ratstr(string str) " USAGE: decstr2ratstr(string str) Convert a decimal number of not more than 30 digits to a rational number with 14 digits. REMARK: This procedure still has to be adapted to accept other precisions! " { ring decR = (complex,30,I),(x),lp; execute("number r="+str+";"); execute("r = "+truncdec(r,14)+";"); return(real2ratstr(r)); } proc real2ratstr(number r) " USAGE: real2ratstr(number r) RETURN: A string containing a rational number representing the decimal number r. ASSUME: The current ring has either real or complex base field. " { string ratstr = "number("+string(r*number(10000000000000000))+")/number(10000000000000000)"; return(ratstr); } proc truncdec(number r, int decs) " USAGE: truncdec(number r, int decs) Truncates a decimal number r to the given number (decs) of digits. RETURN: A string representing the truncated number. " { string str = string(r); return(str[1,(decs+2)]); } proc string_of_vars(ideal I) " USAGE: string_of_vars(ideal I) RETURN: A string of all variables contained in the ideal I, separated by commas. " { list listvars = list(); intvec v; int i; poly p; for(i=size(I);i>0;i--) { p=I[i]; while(p!=0) { v=v+leadexp(p); p=p-lead(p); } } for(i=1; i<=nvars(basering); i++) { if(v[i] > 0) { listvars = listvars + list(var(i)); } } string strvars = string(listvars); return(strvars); }