// IntMat.java (part of FunDomain java program) /* -------------------------------------------------------------- FunDomain: Program for drawing fundamental domains of subgroups of SL_2(Z) Copyright (C) 2001 Helena A. Verrill This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. Helena A. Verrill UK address: 23 Roper Close Rugby CV21 4PF England email: verrill@math.ku.dk see web page: http://hverrill.net for most recent address and email address. more information about the GPL can be found at: http://www.gnu.org/copyleft/gpl.html ------------------------------------------------------------------ */ import java.awt.*; // This class defines integer 2 by 2 matrices, and various operations on them // For future: might be better to define an extension class of // fractional linear transformations (ie, different definition of equality) public class IntMat{ public int a,b,c,d; //constructor public IntMat(){ this.a=1; this.b=0; this.c=0; this.d=1; } public IntMat(int a, int b, int c, int d){ this.a=a; this.b=b; this.c=c; this.d=d; } // some 'constant' matrices static public final IntMat T = new IntMat(1,1,0,1); static public final IntMat Tm = new IntMat(1,-1,0,1); static public final IntMat S = new IntMat(0,-1,1,0); static public final IntMat R = new IntMat(0,-1,1,1); static public final IntMat Id = new IntMat(1,0,0,1); //methods: multiply public IntMat mult(IntMat B){ int a1,b1,c1,d1; a1 = a * B.a + b * B.c; b1 = a * B.b + b * B.d; c1 = c * B.a + d * B.c; d1 = c * B.b + d * B.d; IntMat M = new IntMat(a1,b1,c1,d1); return M; } public static IntMat T(int n){ IntMat N = new IntMat(1,n,0,1); return N; } public static IntMat mult(IntMat A, IntMat B){ int a1,b1,c1,d1; a1 = A.a * B.a + A.b * B.c; b1 = A.a * B.b + A.b * B.d; c1 = A.c * B.a + A.d * B.c; d1 = A.c * B.b + A.d * B.d; IntMat M = new IntMat(a1,b1,c1,d1); return M; } //methods: determinant public int det(){ return a*d - b*c; } //methods: invert public IntMat inv(){ //??? need to learn about 'exceptions' perhaps. if (det()!=1) ???; IntMat M = new IntMat(d,-b,-c,a); return M; } //methods: divide public IntMat div(IntMat B){ //??? how to cope with exceptions like det B = 0?? int a1,b1,c1,d1; a1 = a * B.d - b * B.c; b1 = -a * B.b + b * B.a; c1 = c * B.d - d * B.c; d1 = -c * B.b + d * B.a; IntMat M = new IntMat(a1,b1,c1,d1); return M; } //??? There seem to be some errors coming up associated to the // following function. What's wrong??? public static IntMat div(IntMat A, IntMat B){ //??? how to cope with exceptions like det B = 0?? int a1,b1,c1,d1; a1 = A.a * B.d - A.b * B.c; b1 = -A.a * B.b + A.b * B.a; c1 = A.c * B.d - A.d * B.c; d1 = -A.c * B.b + A.d * B.a; IntMat M = new IntMat(a1,b1,c1,d1); return M; } // method: check equality public boolean equal(IntMat A){ if (this.a == A.a && this.b == A.b && this.c == A.c && this.d == A.d) return true; else return false; } // method: check equality as fractional linear transformation public boolean fracEqual(IntMat A){ if ((this.a == A.a && this.b == A.b && this.c == A.c && this.d == A.d)|| (this.a == -A.a && this.b == -A.b && this.c == -A.c && this.d == -A.d)) return true; else return false; } // method: negate public IntMat negate(){ IntMat M = new IntMat(-a,-b,-c,-d); return M; } // method: conjugate // note, it's assumed that after conjugating we still end up // with an integer matrix. This might cause problems if the // program is extended // also it's assumed det(B) not = 0 // this will cause errors if this is not true. public void conjugate(IntMat B){ int det = B.det(); int newa = (-this.b*B.c + this.a*B.d)*B.a + (-this.d*B.c + this.c*B.d)*B.b; int newb = this.b*B.a*B.a + (-this.a + this.d)*B.b*B.a - this.c*B.b*B.b; int newc = -this.b*B.c*B.c + (this.a - this.d)*B.d*B.c + this.c*B.d*B.d ; int newd = (this.b*B.c + this.d*B.d)*B.a + (-this.a*B.c - this.c*B.d)*B.b; this.a = newa/det; this.b = newb/det; this.c = newc/det; this.d = newd/det; } // image of x + iy from fractional transformation defined by M public double XIm(double x, double y){ double A = (double) this.a; double B = (double) this.b; double C = (double) this.c; double D = (double) this.d; double denom; denom= (C*x+D)*(C*x+D)+C*C*y*y; if (denom!=0) return ((A*x+B)*(C*x+D) + A*C*y*y)/denom; else return (-A*x+B)*(-C*x+D)/((-C*x+D)*(-C*x+D)); // this value of x if y is infinity is choosen so that // maxY can be defined properly in HypTriangle class } public double YIm(double x, double y){ double A = (double) this.a; double B = (double) this.b; double C = (double) this.c; double D = (double) this.d; double denom; denom= (C*x+D)*(C*x+D)+C*C*y*y; if (denom!=0) return (-C*y*(A*x+B)+A*y*(C*x+D))/denom; else return YIm(0.5,HypTriangle.sqrt3/2); // the above choice is so that HypTri draws the triangle correctly } public double XIm(String infinity, int which){ double A = (double) this.a; if (this.c!=0) { double C = (double) this.c; return A/C; } else { double B = (double) this.b; double D = (double) this.d; return (2*B+which*A)/D/2; } } public double YIm(String infinity, double max){ if (this.c!=0) return 0; else return max; } public IntMat move_to_origin(){ if (c==0) { return IntMat.Id; } else { int translation = a/c; IntMat MMM = new IntMat(1,-translation,0,1).mult(this); return MMM; } } }