mirror of
https://github.com/zenorogue/hyperrogue.git
synced 2024-11-28 22:59:54 +00:00
183 lines
7.1 KiB
C++
183 lines
7.1 KiB
C++
/** representation based on the halfplane model; assumes Dim=3 */
|
|
|
|
namespace reps {
|
|
|
|
template<class F> struct sl2 : public array<F, 4> {
|
|
|
|
sl2(F a, F b, F c, F d) { self[0] = a; self[1] = b; self[2] = c; self[3] = d; }
|
|
|
|
sl2 operator * (const sl2& sec) const {
|
|
return sl2(
|
|
self[0] * sec[0] + self[1] * sec[2],
|
|
self[0] * sec[1] + self[1] * sec[3],
|
|
self[2] * sec[0] + self[3] * sec[2],
|
|
self[2] * sec[1] + self[3] * sec[3]
|
|
);
|
|
}
|
|
|
|
std::string print() {
|
|
return hr::lalign(0, "[", self[0], ",", self[1], ";", self[2], ",", self[3], "]");
|
|
}
|
|
|
|
};
|
|
|
|
TD sl2<typename D::Number> split_quaternion_to_sl2(const multivector<D>& h) {
|
|
auto h3 = h[0], h2 = h[1 | 2], h1 = h[1 | 4], h0 = h[2 | 4];
|
|
return sl2(h3 - h1, h2 + h0, -h2 + h0, h3 + h1);
|
|
}
|
|
|
|
TD multivector<D> sl2_to_split_quaternion(const sl2<typename D::Number>& e) {
|
|
auto h0 = (e[1] + e[2]) / 2;
|
|
auto h3 = (e[0] + e[3]) / 2;
|
|
auto h1 = (e[3] - e[0]) / 2;
|
|
auto h2 = (e[1] - e[2]) / 2;
|
|
auto res = zero_vector<multivector_data<D>>();
|
|
res[0] = h3; res[1 | 2] = h2; res[1 | 4] = h1; res[2 | 4] = h0;
|
|
return res;
|
|
}
|
|
|
|
template<class N> using sl2c = sl2<std::complex<N>>;
|
|
|
|
TD sl2c<typename D::Number> split_biquaternion_to_sl2c(const multivector<D>& h) {
|
|
using cn = std::complex<typename D::Number>;
|
|
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]));
|
|
}
|
|
|
|
TD multivector<D> sl2c_to_split_biquaternion(const sl2c<typename D::Number>& e) {
|
|
auto res = zero_vector<multivector_data<D>>();
|
|
res[0] = +(real(e[0]) + real(e[3])) / 2;
|
|
res[3] = +(real(e[1]) - real(e[2])) / 2;
|
|
res[5] = -(imag(e[1]) + imag(e[2])) / 2;
|
|
res[6] = +(imag(e[3]) - imag(e[0])) / 2;
|
|
res[9] = +(real(e[3]) - real(e[0])) / 2;
|
|
res[10] = +(real(e[1]) + real(e[2])) / 2;
|
|
res[12] = +(imag(e[2]) - imag(e[1])) / 2;
|
|
res[15] = +(imag(e[0]) + imag(e[3])) / 2;
|
|
return res;
|
|
}
|
|
|
|
TD struct rep_halfplane {
|
|
|
|
using data = D;
|
|
using N = typename D::Number;
|
|
using point = std::complex<N>;
|
|
using isometry = sl2<N>;
|
|
|
|
static isometry cspin(int i, int j, N alpha) {
|
|
// return split_quaternion_to_sl2( rep_clifford<D>::cspin(i, j, alpha) );
|
|
if(i>j) std::swap(i, j), alpha = -alpha; alpha /= 2;
|
|
auto ca = cos(alpha), sa = sin(alpha);
|
|
return isometry(ca, -sa, sa, ca);
|
|
}
|
|
static isometry cspin90(int i, int j, N alpha) {
|
|
// return split_quaternion_to_sl2( rep_clifford<D>::cspin(i, j, alpha) );
|
|
auto ca = sqrt(N(2)), sa = sqrt(N(2));
|
|
if(i>j) std::swap(i, j), sa = -sa;
|
|
return isometry(ca, -sa, sa, ca);
|
|
}
|
|
static isometry lorentz(int i, int j, N alpha) {
|
|
// return split_quaternion_to_sl2( rep_clifford<D>::lorentz(i, j, alpha) );
|
|
if(i>j) std::swap(i, j); alpha /= 2;
|
|
if(i == 0) return isometry(exp(-alpha), N(0), N(0), exp(alpha));
|
|
if(i == 1) {
|
|
auto ca = cosh(alpha), sa = sinh(alpha);
|
|
return isometry(ca, sa, sa, ca);
|
|
}
|
|
throw hr::hr_exception("bad lorentz");
|
|
}
|
|
static isometry id() { return isometry(N(1),N(0),N(0),N(1)); };
|
|
static point center() { return point(N(0), N(1)); };
|
|
static point apply(const isometry& T, const point& x) {
|
|
return (T[0] * x + T[1] * 1) / (T[2] * x + T[3] * 1);
|
|
};
|
|
static isometry apply(const isometry& T, const isometry& U) { return T * U; };
|
|
|
|
static typename rep_clifford<D>::point to_poincare(const point& x) {
|
|
auto a = real(x), b = imag(x);
|
|
|
|
auto tmp = isometry(sqrt(b), a/sqrt(b), N(0), N(1)/sqrt(b));
|
|
auto sq = sl2_to_split_quaternion<D>(tmp);
|
|
|
|
// 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;
|
|
|
|
sq = despin(sq);
|
|
return typename rep_clifford<D>::point({{sq}});
|
|
}
|
|
|
|
static isometry inverse(isometry T) { return isometry(T[3], -T[1], -T[2], T[0]); }
|
|
static isometry push(const point& p) { return split_quaternion_to_sl2<D>(to_poincare(p)[0]); }
|
|
|
|
static N dist0(const point& x) { return rep_clifford<D>::dist0(to_poincare(x)); }
|
|
static N angle(const point& x) { return rep_clifford<D>::angle(to_poincare(x)); }
|
|
static N get_coord(const point& x, int i) { return rep_clifford<D>::get_coord(to_poincare(x), i); }
|
|
|
|
// imag may be very small and still important, so do not use the default complex print
|
|
static std::string print(const point& x) { return hr::lalign(0, "{real:", real(x), " imag:", imag(x), "}"); }
|
|
static std::string print(const isometry& x) { return x.print(); }
|
|
};
|
|
|
|
TD struct rep_halfspace {
|
|
|
|
using data = D;
|
|
using N = typename D::Number;
|
|
struct point { std::complex<N> xy; N z; };
|
|
using isometry = sl2c<N>;
|
|
|
|
static isometry cspin(int i, int j, N alpha) {
|
|
return split_biquaternion_to_sl2c( rep_clifford<D>::cspin(i, j, alpha) );
|
|
}
|
|
static isometry cspin90(int i, int j) {
|
|
return split_biquaternion_to_sl2c( rep_clifford<D>::cspin90(i, j) );
|
|
}
|
|
static isometry lorentz(int i, int j, N alpha) {
|
|
return split_biquaternion_to_sl2c( rep_clifford<D>::lorentz(i, j, alpha) );
|
|
}
|
|
static isometry id() { return isometry(N(1),N(0),N(0),N(1)); }
|
|
static point center() { return point{ .xy = N(0), .z = N(1) }; }
|
|
static point apply(const isometry& T, const point& x) {
|
|
auto nom = T[0] * x.xy + T[1] * N(1);
|
|
auto nomz= T[0] * x.z;
|
|
auto den = T[2] * x.xy + T[3] * N(1);
|
|
auto denz= T[2] * x.z;
|
|
// D = den + denz * j
|
|
auto dnorm = std::norm(den) + std::norm(denz);
|
|
using std::conj;
|
|
// conj(D) = conj(den) - denz * j
|
|
// N / D = (nom + nomz * j) / (den + denz * j) =
|
|
// = (nom + nomz * j) * (conj(den) - denz * j) / dnorm
|
|
|
|
// auto rxy = (nom * conj(den) - nomz * j * denz * j);
|
|
// auto rz*j = (-nom * denz * j + nomz * j * conj(den))
|
|
|
|
// apply the formula: j * a = conj(a) * j
|
|
|
|
auto rxy = (nom * conj(den) + nomz * conj(denz));
|
|
auto rz = (nomz * den - nom * denz); // todo only real part
|
|
// println(hlog, "imag of rz = ", imag(rz));
|
|
return point { .xy = rxy / dnorm, .z = real(rz) / dnorm };
|
|
};
|
|
static isometry apply(const isometry& T, const isometry& U) { return T * U; };
|
|
|
|
static typename rep_clifford<D>::point to_poincare(const point& x) {
|
|
auto tmp = isometry(sqrt(x.z), x.xy/sqrt(x.z), N(0), N(1)/sqrt(x.z));
|
|
auto sq = sl2c_to_split_biquaternion<D>(tmp);
|
|
sq = despin(sq);
|
|
return typename rep_clifford<D>::point({{sq}});
|
|
}
|
|
|
|
static isometry inverse(isometry T) { return isometry(T[3], -T[1], -T[2], T[0]); }
|
|
static isometry push(const point& p) { return split_biquaternion_to_sl2c<D>(to_poincare(p)[0]); }
|
|
|
|
static N dist0(const point& x) { return rep_clifford<D>::dist0(to_poincare(x)); }
|
|
static N angle(const point& x) { return rep_clifford<D>::angle(to_poincare(x)); }
|
|
static N get_coord(const point& x, int i) { return rep_clifford<D>::get_coord(to_poincare(x), i); }
|
|
|
|
// imag may be very small and still important, so do not use the default complex print
|
|
static std::string print(const point& x) { return hr::lalign(0, "{x:", real(x.xy), " y:", imag(x.xy), " z:", x.z, "}"); }
|
|
static std::string print(const isometry& x) { return x.print(); }
|
|
};
|
|
|
|
template<class D> using rep_half = typename std::conditional<D::Dim==3, rep_halfplane<D>, rep_halfspace<D>>::type;
|
|
|
|
}
|