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Tomographer
v1.0a
Tomographer C++ Framework Documentation
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Tomographer
v1.0a
Tomographer C++ Framework Documentation
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Functions | |
| template<typename Der1 , typename Der2 > | |
| void | cart_to_sph (Eigen::MatrixBase< Der2 > &rtheta, const Eigen::MatrixBase< Der1 > &cart) |
| Convert Cartesian coordinates to spherical coordinates in N dimensions. More... | |
| template<typename Der1 , typename Der2 > | |
| void | sphsurf_to_cart (Eigen::MatrixBase< Der2 > &cart, const Eigen::MatrixBase< Der1 > &theta, const typename Eigen::MatrixBase< Der1 >::Scalar R=1.0) |
| Convert spherical angles to Cartesian coordinates in N dimensions. More... | |
| template<typename Der1 , typename Der2 > | |
| void | sph_to_cart (Eigen::MatrixBase< Der2 > &cart, const Eigen::MatrixBase< Der1 > &rtheta) |
| Convert spherical coordinates to Cartesian coordinates in N dimensions. More... | |
| template<typename Der1 > | |
| Eigen::MatrixBase< Der1 >::Scalar | cart_to_sph_jacobian (const Eigen::MatrixBase< Der1 > &rtheta) |
| Volume element of the hypersphere. More... | |
| template<typename Der1 > | |
| Eigen::MatrixBase< Der1 >::Scalar | surf_sph_jacobian (const Eigen::MatrixBase< Der1 > &theta) |
| Surface element of the hypersphere. More... | |
| template<typename Der1 , typename Der2 > | |
| void | sphsurf_diffjac (Eigen::ArrayBase< Der1 > &dxdtheta, const Eigen::MatrixBase< Der2 > &theta) |
| The differential of passing from spherical to cartesian coordinates on the sphere of unit radius. More... | |
| template<typename Der1 , typename Der2 > | |
| void | sphsurf_diffjac2 (Eigen::ArrayBase< Der1 > &ddxddtheta, const Eigen::MatrixBase< Der2 > &theta) |
| The second order differential of passing from spherical to cartesian coordinates on the sphere of unit radius. More... | |
Utilities for hyperspherical coordinates. See also the corresponding theory page Hyperspherical Coordinates.
| void Tomographer::SphCoords::cart_to_sph | ( | Eigen::MatrixBase< Der2 > & | rtheta, |
| const Eigen::MatrixBase< Der1 > & | cart | ||
| ) |
Convert Cartesian coordinates to spherical coordinates in N dimensions.
See Hyperspherical Coordinates for information about the conventions and ranges of the coordinates.
| rtheta | (output), vector of N entries, will store the \( r\) and \(\theta_i\) values corresponding to the spherical coordinates representation of the point represented by cart. The value rtheta(0) is the \(r\) coordinate and rtheta(1),..,rtheta(N-1) are the angle coordinates, corresponding respectively to \(\theta_1\ldots\theta_{N-1}\). All \( \theta_i \) 's but the last range from \(0..\pi\), while \(\theta_{N-1}\) ranges from \(-\pi\) to \(+\pi\). |
| cart | is a vector of N entries, corresponding to the cartesian coordinates of the point to transform. |
Definition at line 64 of file sphcoords.h.
| Eigen::MatrixBase<Der1>::Scalar Tomographer::SphCoords::cart_to_sph_jacobian | ( | const Eigen::MatrixBase< Der1 > & | rtheta | ) |
Volume element of the hypersphere.
Calculates the volume element, or Jacobian, of the conversion from cartesian coordinates to spherical coordinates. More precisely, this function computes
\[ J = \left\vert\det \frac{\partial(x_i)}{\partial(r,\theta_j)}\right\vert = r^{N-1} \sin^{N-2}\left(\theta_1\right)\sin^{N-3}\left(\theta_2\right) \ldots\sin\left(\theta_{N-2}\right)\ , \]
where N is the dimension of the Euclidean space in which the N-1-sphere is embedded.
See Hyperspherical Coordinates for more info and conventions.
Definition at line 208 of file sphcoords.h.
| void Tomographer::SphCoords::sph_to_cart | ( | Eigen::MatrixBase< Der2 > & | cart, |
| const Eigen::MatrixBase< Der1 > & | rtheta | ||
| ) |
Convert spherical coordinates to Cartesian coordinates in N dimensions.
See Hyperspherical Coordinates for information about the conventions and ranges of the coordinates.
| cart | (output) is a vector of N entries, which will be set to the cartesian coordinates of the point represented by rtheta. |
| rtheta | vector of N entries which specifies the \(r\) and \(\theta_i\) spherical coordinates to transform. rtheta(0) is the \(r\) coordinate and rtheta(1),..,rtheta(N-1) are the angle coordinates, corresponding respectively to \(\theta_1\ldots\theta_{N-1}\). |
Definition at line 174 of file sphcoords.h.
| void Tomographer::SphCoords::sphsurf_diffjac | ( | Eigen::ArrayBase< Der1 > & | dxdtheta, |
| const Eigen::MatrixBase< Der2 > & | theta | ||
| ) |
The differential of passing from spherical to cartesian coordinates on the sphere of unit radius.
The input parameter theta is a list of spherical angles, with theta(i-1) (C/C++ offset) corresponding to \( \theta_i \). The coordinate \( r\) is set equal to one. The dimension N of the Euclidian space is given by the length of the theta vector plus one.
After this function returns, dxdtheta(k-1,i-1) (C/C++ offsets) is set to the value of
\[ \frac{\partial x_k}{\partial \theta_i} , \]
with k=1,...,N, and i=1,..,N-1.
Definition at line 274 of file sphcoords.h.
| void Tomographer::SphCoords::sphsurf_diffjac2 | ( | Eigen::ArrayBase< Der1 > & | ddxddtheta, |
| const Eigen::MatrixBase< Der2 > & | theta | ||
| ) |
The second order differential of passing from spherical to cartesian coordinates on the sphere of unit radius.
The input parameter theta is a list of spherical angles, with theta(i-1) (C/C++ offset) corresponding to \( \theta_i \). The coordinate \( r\) is set equal to one. The dimension N of the Euclidian space is given by the length of the theta vector plus one.
After this function returns, ddxddtheta(k-1,(i-1)+(N-1)*(j-1)) (C/C++ offsets) is set to the value of
\[ \frac{\partial^2 x_k}{\partial \theta_i \partial \theta_j}\ , \]
with k = 1,...,N, and i,j = 1,..,N-1.
Definition at line 358 of file sphcoords.h.
|
inline |
Convert spherical angles to Cartesian coordinates in N dimensions.
See Hyperspherical Coordinates for information about the conventions and ranges of the coordinates.
This function behaves exactly like sph_to_cart(), but takes the theta arguments separately from the R argument. This is useful if you only have angle coordinates that parameterize a fixed-radius hypersphere.
| cart | (output) is a vector of N entries, which will be set to the cartesian coordinates of the point represented by the spherical coordinates (R, theta). |
| theta | vector of N-1 entries which specifies the \(\theta_i\) spherical coordinates to transform. The value theta(0) is \(\theta_1\), and so on. |
| R | the radial coordinate part of the hyperspherical coordinates of the point to transform. |
Definition at line 134 of file sphcoords.h.
| Eigen::MatrixBase<Der1>::Scalar Tomographer::SphCoords::surf_sph_jacobian | ( | const Eigen::MatrixBase< Der1 > & | theta | ) |
Surface element of the hypersphere.
Calculates the volume element, or Jacobian, of the conversion from cartesian coordinates to spherical coordinates on the surface of the hypersphere of fixed radius R=1. More precisely, this function computes
\[ \left\vert J\right\vert_{r=1} = \left\vert\det \frac{\partial(x_i)}{\partial(r,\theta_j)}\right\vert_{r=1} = \sin^{N-2}\left(\theta_1\right)\sin^{N-3}\left(\theta_2\right) \ldots\sin\left(\theta_{N-2}\right)\ , \]
where N is the dimension of the Euclidean space in which the N-1-sphere is embedded.
See Hyperspherical Coordinates for more info and conventions.
| theta | is the vector of the N-1 spherical angles, \(\theta_1\ldots\theta_{N-1}\). |
Definition at line 242 of file sphcoords.h.
