gns.geom module

gns.geom.cartCirc2Point(x, y, l, u)

Convert from Cartesian coordinates back to 1-d angle parameterising circle. Then converts back to physical point value based on upper and lower limits Note y = 0 gives p = 0 for all x

gns.geom.cartSphere2Point(x, y, z, l1, l2, u1, u2)

Convert from Cartesian coordinates back to 2-d angle parameterising surface of sphere. Then converts back to physical point value based on upper and lower limits Again assumes r = 1

gns.geom.cartTorus2Point(x, y, z, l1, l2, u1, u2)

Convert from Cartesian coordinates on torus back to 2-d angle parameterising torus. Then converts back to physical point value based on upper and lower limits Again assumes r = R = 1

gns.geom.cartesianFromSpherical(phi, theta)

get cartesians from sphericals

gns.geom.getPoleOrthogs(v1)

get vector orthogonal to v1, which points towards positive z-pole, then finds vector orthogonal to v1 and v2 by using cross product

gns.geom.getPoleVec(v)

takes vector v, which is normal to plane tangential to surface of unit sphere (i.e. unit vector from origin), and finds vector in perpendicular plane which points to (positive) z axis. Since v is also a point on the perpendicular plane, only need single vector to get pole vector from n_hat dot (x - a) = 0 i.e. n_hat = a = v. ASSUMES v is normalised

gns.geom.getRandOrthogs(v1)

get two vectors (mutually) orthogonal to v1. ASSUMES v1 is normalised

gns.geom.physPeriod2Pi(p, l, u)

gns.geom.physPeriod2TwoPi(p, l, u)

gns.geom.piPeriod2Phys(phi, l, u)

gns.geom.point2CartCirc(p, l, u)

Takes a periodic 1-d physical point along with its bounds, projects it onto the unit circle (r = 1) and returns its 2-d Cartesian coordinates on this circle

gns.geom.point2CartSphere(p1, p2, l1, l2, u1, u2)

Takes a periodic 2-d physical point along with its bounds, projects it onto surface of ‘unit sphere’ which is parameterised by these periodic values and returns its 3-d Cartesian coordinates on the sphere. Sphere is defined to have radius 1

gns.geom.point2CartTorus(p1, p2, l1, l2, u1, u2)

Takes a periodic 2-d physical point along with its bounds, Converts it into a point on a torus which is parameterised by these periodic values and returns its 3-d Cartesian coordinates on the torus. Torus is defined to have greater radius = 2 and lesser radius = 1

gns.geom.projectCart2Circ(x, y)

Takes arbitrary point in x-y plane and projects onto unit circle. Does same as first part of cartCirc2Point() Returns angle measured from positive x-axis

gns.geom.projectCart2Sphere(x, y, z)

Takes arbitrary point in 3-d cartesian coordinates and projects it onto sphere centred on origin. Essentially does same as first part of cartSphere2Point() but for arbitrary radius r. Returns phi and theta

gns.geom.projectCart2Torus(x, y, z, R)

Takes arbitrary point in 3-d cartesian coordinates and projects it onto torus centred on origin with greater torus radius R. First calculates azimuthal angle by considering projection of point onto circle in x-y plane with radius equal to R. Then calculates angle around tube by taking arcsin of ratio of z component of original point to distance from original point to centre of tube at azimuthal angle. Returns phi and theta

gns.geom.rodriguezRot(k, v, alpha)

Rotate vector v about angle alpha (defined b y right hand rule) in plane defined by unit vector k (plane perpendicular to k).

gns.geom.sphericalTranslation(phi0, theta0, dPhi, dTheta)

phi0 and theta0 correspond to x0, y0, z0 dphi, dtheta correspond to some dx, dy, dz returns coordinates xt, yt, zt

gns.geom.switchPolarSys(phi)

gns.geom.switchTorusSys(theta, R, rho)

gns.geom.testCircleProposalSymmetry(p, l, u, cov)

takes a single point on circle (parameterised by angle) and evaluates 2-d gaussian in cartesian coords at a random point which is sampled from same 2-d Gaussian, centred on same input point. Then projects random point onto circle, and evaluates pdf of a gaussian centred on the proposed point at input point. Checks if these two pdf values are the same Essentially checks if (phi’ | phi) = p(phi | phi’) where p is proposal distribution of Metropolis algo

gns.geom.testOrthog(vs)

gns.geom.testSphereProposalSymmetry(p1, p2, l1, l2, u1, u2, cov)

takes a single point on surface of sphere (parameterised by 2 angles) and evaluates 3-d gaussian in cartesian coords at a random point which is sampled from same 3-d Gaussian, centred on same input point. Then projects random point onto sphere, and evaluates pdf of a gaussian centred on the proposed point at input point. Checks if these two pdf values are the same Essentially checks if p(phi’, theta’ | phi, theta) = p(phi, theta | phi’, theta’) where p is proposal distribution of Metropolis algo

gns.geom.testTorusProposalSymmetry(p1, p2, l1, l2, u1, u2, cov)

takes a single point on torus (parameterised by 2 angles) and evaluates 3-d gaussian in cartesian coords at a random point which is sampled from same 3-d Gaussian, centred on same input point. Then projects random point onto torus, and evaluates pdf of a gaussian centred on the proposed point at input point. Checks if these two pdf values are the same Essentially checks if p(phi’, theta’ | phi, theta) = p(phi, theta | phi’, theta’) where p is proposal distribution of Metropolis algo

gns.geom.translateAxes(axes, dPhi, dTheta)

calls translateAxis on each vector in axes, and shifts each by dPhi, dTheta

gns.geom.translateAxis(v, dPhi, dTheta)

Translates input (Cartesian) vector by dphi, dtheta, returns translated (Cartesian) vector

gns.geom.twoPiPeriod2Phys(phi, l, u)