File ControlPointDerivatives.h¶

Copyright

This code is licensed under the 3-clause BSD license.

Copyright ETH Zurich, Laboratory for Physical Chemistry, Reiher Group.

See LICENSE.txt for details.

namespace Scine

This header file contains functions that allow for common notation for common things that can be done at a different degree of derivatives.

This header contains alias definitions defining which classes to use for the different degrees of derivatives.

namespace Utils

namespace BSplines

namespace ControlPointDerivatives¶

Calculation of the derivatives of BSpline curves C (or of dC/du, d2C/du2, …) with respect to the control points.

Not only can the derivatives of the curve itself with respect to the control points be calculated, but also of dC/du, d2C/du2, etc. The degree of the curve to derive is given by “curveDerivative” in some functions below.

The functions exist in two variants: one to calculate the derivatives of the curve with respect to one control point, or with respect to all the control points.

Note that the derivative of some control point coordinate i is non-zero only for the coordinate i of the curve.

Functions

Eigen::VectorXd oneDerivative(const BSpline &spline, double u, int controlPointIndex, int curveDerivative)¶

Eigen::MatrixXd allDerivatives(const BSpline &spline, double u, int curveDerivative)¶

Eigen::VectorXd curveDerivatives(const BSpline &spline, double u, int controlPointIndex)¶: Get the derivatives of the curve at a given u with respect to some control point coordinates. returns a vector of dCk / dPk for k in [0, nDimensions). NB: all other dCi / dPj are zero.

Eigen::MatrixXd curveDerivatives(const BSpline &spline, double u)¶: Get the derivatives of the curve at a given u with respect to all control point coordinates. returns a matrix of dCk / dP^A_k for k in [0, nDimensions), and A being a given control point. NB: all other dCi / dPj are zero.

Eigen::VectorXd firstOrderCurveDerivatives(const BSpline &spline, double u, int controlPointIndex)¶: Get the derivatives of the curve tangent at a given u with respect to some control point coordinates. returns a vector of dC’k / dPk for k in [0, nDimensions). NB: all other dC’i / dPj are zero.

Eigen::MatrixXd firstOrderCurveDerivatives(const BSpline &spline, double u)¶: Get the derivatives of the curve tangent at a given u with respect to all control point coordinates. returns a matrix of dC’k / dP^A_k for k in [0, nDimensions), and A being a given control point. NB: all other dC’i / dPj are zero.

Eigen::VectorXd secondOrderCurveDerivatives(const BSpline &spline, double u, int controlPointIndex)¶: Get the derivatives of the curve tangent at a given u with respect to some control point coordinates. returns a vector of dC’k / dPk for k in [0, nDimensions). NB: all other dC’i / dPj are zero.

Eigen::MatrixXd secondOrderCurveDerivatives(const BSpline &spline, double u)¶: Get the derivatives of the curve tangent at a given u with respect to all control point coordinates. returns a matrix of dC’k / dP^A_k for k in [0, nDimensions), and A being a given control point. NB: all other dC’i / dPj are zero.