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183 lines
7.1 KiB
C++
183 lines
7.1 KiB
C++
/** representation based on the halfplane model; assumes Dim=3 */
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namespace reps {
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template<class F> struct sl2 : public array<F, 4> {
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sl2(F a, F b, F c, F d) { self[0] = a; self[1] = b; self[2] = c; self[3] = d; }
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sl2 operator * (const sl2& sec) const {
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return sl2(
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self[0] * sec[0] + self[1] * sec[2],
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self[0] * sec[1] + self[1] * sec[3],
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self[2] * sec[0] + self[3] * sec[2],
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self[2] * sec[1] + self[3] * sec[3]
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);
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}
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std::string print() {
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return hr::lalign(0, "[", self[0], ",", self[1], ";", self[2], ",", self[3], "]");
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}
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};
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TD sl2<typename D::Number> split_quaternion_to_sl2(const multivector<D>& h) {
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auto h3 = h[0], h2 = h[1 | 2], h1 = h[1 | 4], h0 = h[2 | 4];
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return sl2(h3 - h1, h2 + h0, -h2 + h0, h3 + h1);
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}
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TD multivector<D> sl2_to_split_quaternion(const sl2<typename D::Number>& e) {
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auto h0 = (e[1] + e[2]) / 2;
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auto h3 = (e[0] + e[3]) / 2;
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auto h1 = (e[3] - e[0]) / 2;
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auto h2 = (e[1] - e[2]) / 2;
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auto res = zero_vector<multivector_data<D>>();
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res[0] = h3; res[1 | 2] = h2; res[1 | 4] = h1; res[2 | 4] = h0;
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return res;
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}
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template<class N> using sl2c = sl2<std::complex<N>>;
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TD sl2c<typename D::Number> split_biquaternion_to_sl2c(const multivector<D>& h) {
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using cn = std::complex<typename D::Number>;
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return sl2(cn(h[0]-h[9], h[15]-h[6]), cn(h[3]+h[10], -h[5]-h[12]), cn(h[10]-h[3], h[12]-h[5]), cn(h[0]+h[9], h[6]+h[15]));
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}
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TD multivector<D> sl2c_to_split_biquaternion(const sl2c<typename D::Number>& e) {
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auto res = zero_vector<multivector_data<D>>();
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res[0] = +(real(e[0]) + real(e[3])) / 2;
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res[3] = +(real(e[1]) - real(e[2])) / 2;
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res[5] = -(imag(e[1]) + imag(e[2])) / 2;
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res[6] = +(imag(e[3]) - imag(e[0])) / 2;
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res[9] = +(real(e[3]) - real(e[0])) / 2;
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res[10] = +(real(e[1]) + real(e[2])) / 2;
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res[12] = +(imag(e[2]) - imag(e[1])) / 2;
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res[15] = +(imag(e[0]) + imag(e[3])) / 2;
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return res;
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}
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TD struct rep_halfplane {
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using data = D;
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using N = typename D::Number;
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using point = std::complex<N>;
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using isometry = sl2<N>;
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static isometry cspin(int i, int j, N alpha) {
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// return split_quaternion_to_sl2( rep_clifford<D>::cspin(i, j, alpha) );
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if(i>j) std::swap(i, j), alpha = -alpha; alpha /= 2;
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auto ca = cos(alpha), sa = sin(alpha);
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return isometry(ca, -sa, sa, ca);
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}
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static isometry cspin90(int i, int j, N alpha) {
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// return split_quaternion_to_sl2( rep_clifford<D>::cspin(i, j, alpha) );
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auto ca = sqrt(N(2)), sa = sqrt(N(2));
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if(i>j) std::swap(i, j), sa = -sa;
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return isometry(ca, -sa, sa, ca);
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}
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static isometry lorentz(int i, int j, N alpha) {
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// return split_quaternion_to_sl2( rep_clifford<D>::lorentz(i, j, alpha) );
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if(i>j) std::swap(i, j); alpha /= 2;
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if(i == 0) return isometry(exp(-alpha), N(0), N(0), exp(alpha));
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if(i == 1) {
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auto ca = cosh(alpha), sa = sinh(alpha);
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return isometry(ca, sa, sa, ca);
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}
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throw hr::hr_exception("bad lorentz");
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}
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static isometry id() { return isometry(N(1),N(0),N(0),N(1)); };
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static point center() { return point(N(0), N(1)); };
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static point apply(const isometry& T, const point& x) {
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return (T[0] * x + T[1] * 1) / (T[2] * x + T[3] * 1);
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};
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static isometry apply(const isometry& T, const isometry& U) { return T * U; };
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static typename rep_clifford<D>::point to_poincare(const point& x) {
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auto a = real(x), b = imag(x);
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auto tmp = isometry(sqrt(b), a/sqrt(b), N(0), N(1)/sqrt(b));
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auto sq = sl2_to_split_quaternion<D>(tmp);
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// sq[0] = (sqrt(b) + 1/sqrt(b)) / 2;; sq[1 | 2] = a/sqrt(b)/2; sq[1 | 4] = (1/sqrt(b) - sqrt(b)) / 2; sq[2 | 4] = a/sqrt(b)/2;
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sq = despin(sq);
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return typename rep_clifford<D>::point({{sq}});
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}
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static isometry inverse(isometry T) { return isometry(T[3], -T[1], -T[2], T[0]); }
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static isometry push(const point& p) { return split_quaternion_to_sl2<D>(to_poincare(p)[0]); }
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static N dist0(const point& x) { return rep_clifford<D>::dist0(to_poincare(x)); }
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static N angle(const point& x) { return rep_clifford<D>::angle(to_poincare(x)); }
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static N get_coord(const point& x, int i) { return rep_clifford<D>::get_coord(to_poincare(x), i); }
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// imag may be very small and still important, so do not use the default complex print
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static std::string print(const point& x) { return hr::lalign(0, "{real:", real(x), " imag:", imag(x), "}"); }
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static std::string print(const isometry& x) { return x.print(); }
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};
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TD struct rep_halfspace {
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using data = D;
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using N = typename D::Number;
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struct point { std::complex<N> xy; N z; };
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using isometry = sl2c<N>;
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static isometry cspin(int i, int j, N alpha) {
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return split_biquaternion_to_sl2c( rep_clifford<D>::cspin(i, j, alpha) );
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}
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static isometry cspin90(int i, int j) {
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return split_biquaternion_to_sl2c( rep_clifford<D>::cspin90(i, j) );
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}
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static isometry lorentz(int i, int j, N alpha) {
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return split_biquaternion_to_sl2c( rep_clifford<D>::lorentz(i, j, alpha) );
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}
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static isometry id() { return isometry(N(1),N(0),N(0),N(1)); }
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static point center() { return point{ .xy = N(0), .z = N(1) }; }
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static point apply(const isometry& T, const point& x) {
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auto nom = T[0] * x.xy + T[1] * N(1);
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auto nomz= T[0] * x.z;
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auto den = T[2] * x.xy + T[3] * N(1);
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auto denz= T[2] * x.z;
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// D = den + denz * j
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auto dnorm = std::norm(den) + std::norm(denz);
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using std::conj;
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// conj(D) = conj(den) - denz * j
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// N / D = (nom + nomz * j) / (den + denz * j) =
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// = (nom + nomz * j) * (conj(den) - denz * j) / dnorm
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// auto rxy = (nom * conj(den) - nomz * j * denz * j);
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// auto rz*j = (-nom * denz * j + nomz * j * conj(den))
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// apply the formula: j * a = conj(a) * j
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auto rxy = (nom * conj(den) + nomz * conj(denz));
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auto rz = (nomz * den - nom * denz); // todo only real part
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// println(hlog, "imag of rz = ", imag(rz));
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return point { .xy = rxy / dnorm, .z = real(rz) / dnorm };
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};
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static isometry apply(const isometry& T, const isometry& U) { return T * U; };
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static typename rep_clifford<D>::point to_poincare(const point& x) {
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auto tmp = isometry(sqrt(x.z), x.xy/sqrt(x.z), N(0), N(1)/sqrt(x.z));
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auto sq = sl2c_to_split_biquaternion<D>(tmp);
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sq = despin(sq);
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return typename rep_clifford<D>::point({{sq}});
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}
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static isometry inverse(isometry T) { return isometry(T[3], -T[1], -T[2], T[0]); }
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static isometry push(const point& p) { return split_biquaternion_to_sl2c<D>(to_poincare(p)[0]); }
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static N dist0(const point& x) { return rep_clifford<D>::dist0(to_poincare(x)); }
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static N angle(const point& x) { return rep_clifford<D>::angle(to_poincare(x)); }
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static N get_coord(const point& x, int i) { return rep_clifford<D>::get_coord(to_poincare(x), i); }
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// imag may be very small and still important, so do not use the default complex print
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static std::string print(const point& x) { return hr::lalign(0, "{x:", real(x.xy), " y:", imag(x.xy), " z:", x.z, "}"); }
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static std::string print(const isometry& x) { return x.print(); }
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};
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template<class D> using rep_half = typename std::conditional<D::Dim==3, rep_halfplane<D>, rep_halfspace<D>>::type;
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}
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