sdf_shape_3D.h

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/*
  This file is part of MADNESS.
  
  Copyright (C) 2007,2010 Oak Ridge National Laboratory
  
  This program is free software; you can redistribute it and/or modify
  it under the terms of the GNU General Public License as published by
  the Free Software Foundation; either version 2 of the License, or
  (at your option) any later version.
  
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  along with this program; if not, write to the Free Software
  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
  
  For more information please contact:
  
  Robert J. Harrison
  Oak Ridge National Laboratory
  One Bethel Valley Road
  P.O. Box 2008, MS-6367
  
  email: harrisonrj@ornl.gov
  tel:   865-241-3937
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  $Id$
*/
 
/**
  \file mra/sdf_shape_3D.h
  \brief Implements the SignedDFInterface for common 3-D geometric objects.
  \ingroup mrabcint
 
  This file provides signed distance functions for common 3-D geometric objects:
  - Plane
  - Sphere
  - Cone
  - Paraboloid
  - Box
  - Cube
  - Ellipsoid
  - Cylinder
  
  \note The signed distance functions should be the shortest distance between
  a point and \b any point on the surface.  This is hard to calculate in many
  cases, so we use contours here.  The surface layer may not be equally thick
  around all points on the surface.  Some surfaces (plane, sphere, paraboloid)
  use the exact signed distance functions.  All others use the contours, which
  may be extremely problematic and cause excessive refinement.  The sdf
  function of the sphere class outlines how to calculate the exact signed
  distance functions, if needed.
*/  
  
#ifndef MADNESS_MRA_SDF_SHAPE_3D_H__INCLUDED
#define MADNESS_MRA_SDF_SHAPE_3D_H__INCLUDED
 
#include <madness/mra/sdf_domainmask.h>
 
namespace madness {
 
    /// \brief A plane surface (3 dimensions)
    class SDFPlane : public SignedDFInterface<3> {
    protected:
        const coord_3d normal; ///< The normal vector pointing OUTSIDE the surface
        const coord_3d point; ///< A point in the plane
 
    public:
        
        /** \brief SDF for a plane transecting the entire simulation volume
 
            \param normal The outward normal definining the plane
            \param point A point in the plane */
        SDFPlane(const coord_3d& normal, const coord_3d& point)
            : normal(normal*(1.0/sqrt(normal[0]*normal[0] + normal[1]*normal[1] + normal[2]*normal[2])))
            , point(point) 
        {}
        SDFPlane(const coord_3d& normal, const coord_3d& point) {…}
 
        /** \brief Computes the normal distance
 
            This SDF is exact, and easy to show.
 
            \param pt Point at which to compute the distance from the surface
            \return The signed distance from the surface */
        double sdf(const coord_3d& pt) const {
            return (pt[0]-point[0])*normal[0] + (pt[1]-point[1])*normal[1] + (pt[2]-point[2])*normal[2];
        }
        double sdf(const coord_3d& pt) const  {…}
 
        /** \brief Computes the gradient of the SDF.
 
            \param pt Point at which to compute the gradient
            \return the gradient */
        coord_3d grad_sdf(const coord_3d& pt) const {
            MADNESS_EXCEPTION("gradient method is not yet implemented for this shape",0);
        }
        coord_3d grad_sdf(const coord_3d& pt) const  {…}
    };
    class SDFPlane : public SignedDFInterface<3> {…};
 
    /// \brief A spherical surface (3 dimensions)
    class SDFSphere : public SignedDFInterface<3> {
    protected:
        const double radius; ///< Radius of sphere
        const coord_3d center; ///< Center of sphere
 
    public:
        /** \brief SDF for a sphere
 
            \param radius The radius of the sphere
            \param center The center of the sphere */
        SDFSphere(const double radius, const coord_3d &center) 
            : radius(radius)
            , center(center) 
        {}
        SDFSphere(const double radius, const coord_3d &center)  {…}
 
        /** \brief Computes the normal distance
 
            This SDF is exact, and easy to show.
 
            \param pt Point at which to compute the distance from the surface
            \return The signed distance from the surface */
        double sdf(const coord_3d& pt) const {
            double temp, r;
            int i;
            
            r = 0.0;
            for(i = 0; i < 3; ++i) {
                temp = pt[i] - center[i];
                r += temp * temp;
            }
            
            return sqrt(r) - radius;
        }
        double sdf(const coord_3d& pt) const  {…}
 
        /** \brief Computes the gradient of the SDF.
 
            \param pt Point at which to compute the gradient
            \return the gradient */
        coord_3d grad_sdf(const coord_3d& pt) const {
            double x = pt[0] - center[0];
            double y = pt[1] - center[1];
            double z = pt[2] - center[2];
            double r = sqrt(x*x + y*y + z*z);
            coord_3d g;
            g[0] = x/r;
            g[1] = y/r;
            g[2] = z/r;
            return g;
        }
        coord_3d grad_sdf(const coord_3d& pt) const  {…}
    };
    class SDFSphere : public SignedDFInterface<3> {…};
 
    /** \brief A cone (3 dimensions)
 
        The cone is defined by 
 
        \f[ \sqrt{x^2 + y^2} - c * z = 0 \f]
 
        where \f$ z \f$ is along the cone's axis. */
    class SDFCone : public SignedDFInterface<3> {
    protected:
        const coord_3d apex; ///< The apex
        const double c; ///< The radius
        const coord_3d dir; ///< The direction of the axis, from the apex INSIDE
 
    public:
        /** \brief Constructor for cone
 
            \param c Parameter \f$ c \f$ in the definition of the cone
            \param apex Apex of cone
            \param direc Oriented axis of the cone */
        SDFCone(const double c, const coord_3d &apex, const coord_3d &direc) 
            : apex(apex)
            , c(c)
            , dir(direc*(1.0/sqrt(direc[0]*direc[0] + direc[1]*direc[1] + direc[2]*direc[2])))
        {}
        SDFCone(const double c, const coord_3d &apex, const coord_3d &direc)  {…}
 
        /** \brief Computes the normal distance
 
            This SDF naively uses contours, and should be improved before
            serious usage.
 
            \param pt Point at which to compute the distance from the surface
            \return The signed distance from the surface */
        double sdf(const coord_3d& pt) const {
            coord_3d diff;
            unsigned int i;
            double dotp;
            
            for(i = 0; i < 3; ++i)
                diff[i] = pt[i] - apex[i];
            
            dotp = diff[0]*dir[0] + diff[1]*dir[1] + diff[2]*dir[2];
            
            for(i = 0; i < 3; ++i)
                diff[i] -= dotp * dir[i];
            
            return sqrt(diff[0]*diff[0] + diff[1]*diff[1] + diff[2]*diff[2])
                - c * dotp;
        }
        double sdf(const coord_3d& pt) const  {…}
 
        /** \brief Computes the gradient of the SDF.
 
            \param pt Point at which to compute the gradient
            \return the gradient */
        coord_3d grad_sdf(const coord_3d& pt) const {
            MADNESS_EXCEPTION("gradient method is not yet implemented for this shape",0);
        }
        coord_3d grad_sdf(const coord_3d& pt) const  {…}
    };
    class SDFCone : public SignedDFInterface<3> {…};
 
    /** \brief A paraboloid (3 dimensions)
 
        The surface is defined by 
 
        \f[ x^2 + y^2 - c * z == 0 \f]
 
        where \f$ z \f$ is along the paraboloid's axis. */
    class SDFParaboloid : public SignedDFInterface<3> {
    protected:
        const coord_3d apex; ///< The apex
        const double c; ///< Curvature/radius of the surface
        const coord_3d dir;///< The direction of the axis, from the apex INSIDE
        const long double zero; ///< Numerical zero for root-finding in sdf
        const long double rootzero; ///< Numerical zero for the roots
 
    public:
        /** \brief Constructor for paraboloid
 
            \param c Parameter \f$ c \f$ in the definition of the paraboloid
            \param apex Apex of paraboloid
            \param direc Oriented axis of the paraboloid */
        SDFParaboloid(const double c, const coord_3d &apex, const coord_3d &direc) 
            : apex(apex)
            , c(c)
            , dir(direc*(1.0/sqrt(direc[0]*direc[0] + direc[1]*direc[1] + direc[2]*direc[2])))
            , zero(1.0e-10L)
            , rootzero(1.0e-14L)
        {}
        SDFParaboloid(const double c, const coord_3d &apex, const coord_3d &direc)  {…}
 
        /** \brief Computes the normal distance.
 
            This SDF is exact.
 
            Given a point, pt=(x, y, z), the goal is to find another point,
            pt0=(x0, y0, z0), on the surface that minimizes |pt - pt0|^2.  The
            root of this minimized square distance (and a sign) is the sdf.
 
            For simplicity (here), I will assume that the paraboloid's axis
            is along the positive z-axis and that the origin is the apex.
            Note that the code does NOT make these assumptions.
 
            Thus, we want to minimize
               (x-x0)^2 + (y-y0)^2 + (z-z0)^2
            subject to
               x0^2 + y0^2 - c z0 == 0.
 
            Using Lagrange multipliers, the system of equations is
            -2(x-x0) == L 2 x0
            -2(y-y0) == L 2 y0
            -2(z-z0) == L (-c)
            x0^2 + y0^2 - c z0 == 0.
 
            After some algebra, we get a cubic equation for L,
            (x^2 + y^2 - c z) + (2c z + c^2/2) L - c (z + c) L^2
               + c^2/2 L^3 == 0
 
            This can be solved analytically in Mathematica, producing a long,
            messy equation.  This equation has some stability issues (large
            cancellations in some areas) and is not implemented below.
            Instead we use an iterative root finder to locate the roots of the
            cubic.  The simple bisection method is used, perhaps someone will
            improve the code someday).
 
            Once we have the correct Lagrange multiplier,
            |pt - pt0|^2 = c L^2 (z - L c/2 + c/4).
            The square root of this quantity (with the appropriate sign for
            inside/outside) gives the sdf.
            
            \param pt Point at which to compute the distance from the surface
            \return The signed distance from the surface */
        double sdf(const coord_3d& pt) const {
            // work in extended precision since this calculation can have
            // large cancellations
            Vector<long double, 3> diff;
            unsigned int i, n;
            long double dotp, d;
            long double dist, temp[2], upper;
        //double lambda;
            std::vector<long double> roots;
 
            // silence warnings ... doesn't work
            //lambda = 0.0;
            dist = 0.0;
 
            // avoid lots of casting
            long double c = (long double)this->c;
 
            for(i = 0; i < 3; ++i)
                diff[i] = pt[i] - apex[i];
 
            dotp = diff[0]*dir[0] + diff[1]*dir[1] + diff[2]*dir[2];
            
            for(i = 0; i < 3; ++i)
                diff[i] -= dotp * dir[i];
 
            d = diff[0]*diff[0] + diff[1]*diff[1] + diff[2]*diff[2]
                - c * dotp;
            //printf("   %.16Le %.16Le %.16Le\n", c, d, dotp);
 
            // get the real roots (Lagrange multipliers) and find the one
            // that minimizes the distance.
            // note that real roots must be no bigger than (4dotp+c)/(2c)
            // from the analytical form
            roots = get_roots(c, d, dotp);
            upper = (4.0L*dotp + c) * 0.5L / c;
            n = 0;
            for(std::vector<long double>::iterator iter = roots.begin();
                iter != roots.end(); ++iter) {
 
                temp[0] = *iter;
 
                // are we in range?
                if(temp[0] <= upper) {
 
                    // calculate the distance
                    temp[1] = c*(dotp - temp[0]*c*0.5L + c*0.25L);
                    if(temp[1] < 0.0L) {
                        // temp[1] is analytically non-negative
                        // if we're here, it's noise
                        temp[1] = 0.0L;
                    }
                    else {
                        temp[1] = std::abs(temp[0])*sqrt(temp[1]);
                    }
 
                    if(n == 0) {
              //lambda = temp[0];
                        dist = temp[1];
                    }
                    else {
                        if(temp[1] < dist) {
              //lambda = temp[0];
                            dist = temp[1];
                        }
                    }
                    ++n;
                }
            }
            //printf("   L = %.16Le D = %.16Le\n", lambda, dist);
 
            MADNESS_ASSERT(n > 0); // we should have at least one real root...
 
            if(d > 0.0L)
                return (double)dist;
            else
                return (double)(-dist);
        }
        double sdf(const coord_3d& pt) const  {…}
 
        /** \brief Computes the gradient of the SDF.
 
            \param pt Point at which to compute the gradient
            \return the gradient */
        coord_3d grad_sdf(const coord_3d& pt) const {
            MADNESS_EXCEPTION("gradient method is not yet implemented for this shape",0);
        }
        coord_3d grad_sdf(const coord_3d& pt) const  {…}
 
        protected:
        /** \brief Finds real root(s) of the cubic polynomial in the sdf
                   function.
 
            The cubic equation can successfully solve the cubic analytically,
            but evaluating the expression can be numerically irksome.
            Analytical results show that at most one root appears in each of
            the domains
            - (-Infinity, (2z+2c-|c-2z|)/(3c))
            - ((2z+2c-|c-2z|)/(3c), (2z+2c+|c-2z|)/(3c))
            - ((2z+2c-|c-2z|)/(3c), Infinity).
 
            Furthermore, the cubic is always increasing in the first and
            third domains, and always decreasing in the second.
 
            Since the cubic is easy to evaluate, we use simple bisections
            to find the roots...  this can probably be improved.
 
            Cubic equation of interest:
            d + c(2z+c/2)*L - c(c+z)L^2 + c^2/2L^3 == 0
 
            \param[in] c Parameter c for the paraboloid.
            \param[in] d The d parameter: x^2+y^2-cz = d
            \param[in] z The distance from the apex along the paraboloid's
                         axis (called dotp in sdf)
            \return The root(s)
        */
        std::vector<long double> get_roots(const long double c,
                             const long double d, const long double z) const {
 
            long double lower, upper, bound;
            std::vector<long double> roots(0);
            long double derivroot1, derivroot2, temp;
            bool hasregion2, keepgoing, foundroot;
 
            // calculate the region boundaries
            lower = 2.0L*(c+z);
            upper = std::abs(c - 2.0L*z);
            hasregion2 = (upper >= 1.0e-10L);
 
            derivroot1 = (lower - upper) / (3.0L*c);
            derivroot2 = (lower + upper) / (3.0L*c);
 
            // first region -------------------------------------------------
            foundroot = false;
            upper = derivroot1;
 
            // if cubic(upper) < 0.0, there's no root in this region
            bound = eval_cubic(upper, c, d, z);
            if(std::abs(bound) < zero) {
                // this is the root!
                roots.push_back(upper);
                hasregion2 = false;
            }
            if(bound > 0.0L) {
                // there's a root in region 1: find it!
 
                // need to find a lower bound
                temp = -1.0;
                do {
                    lower = upper + temp;
                    bound = eval_cubic(lower, c, d, z);
                    if(std::abs(bound) < zero) {
                        // found the root!
                        roots.push_back(lower);
                        foundroot = true;
                        break;
                    }
 
                    keepgoing = (bound > 0.0L);
                    // if we didn't cross the root, we have a better upperbound
                    if(keepgoing)
                        upper = lower;
                } while(keepgoing);
 
                // with a lower and upper bound, find the root!
                if(!foundroot) {
                    temp = find_root(lower, upper, c, d, z, true);
                    roots.push_back(temp);
                }
            }
            else {
                // no root in region 1; hence no root in region 2
                hasregion2 = false;
            }
 
            // second region ------------------------------------------------
            if(hasregion2) {
                lower = derivroot1;
                upper = derivroot2;
                // the lower range has to be positive the upper range has to be
                // negative
                if(lower > 0.0L && upper < 0.0L) {
                    temp = find_root(lower, upper, c, d, z, false);
                    roots.push_back(temp);
                }
            }
 
            // third region -------------------------------------------------
            foundroot = false;
            lower = derivroot2;
 
            // if cubic(upper) > 0.0, there's no root in this region
            bound = eval_cubic(lower, c, d, z);
            if(std::abs(bound) < zero) {
                // this is the root!
                roots.push_back(lower);
            }
            if(bound < 0.0L) {
                // there's a root in region 3: find it!
 
                // need to find an upper bound
                temp = 1.0;
                do {
                    upper = lower + temp;
                    bound = eval_cubic(upper, c, d, z);
                    if(std::abs(bound) < zero) {
                        // found the root!
                        roots.push_back(upper);
                        foundroot = true;
                        break;
                    }
 
                    keepgoing = (bound < 0.0L);
                    // if we didn't cross the root, we have a better lowerbound
                    if(keepgoing)
                        lower = upper;
                } while(keepgoing);
 
                // with a lower and upper bound, find the root!
                if(!foundroot) {
                    temp = find_root(lower, upper, c, d, z, true);
                    roots.push_back(temp);
                }
            }
 
            return roots;
        }
        std::vector<long double> get_roots(const long double c, {…}
 
        /** Finds a root in the specified range for the cubic equation.
 
            This uses a simple bisection method... perhaps someone will
            improve it one day...
 
            \param[in] lower The bottom of the range
            \param[in] upper The top of the range
            \param[in] c Parameter c for the paraboloid.
            \param[in] d The d parameter: x^2+y^2-cz = d
            \param[in] z The distance from the apex along the paraboloid's
                         axis (called dotp in sdf)
            \param[in] dir True if the function is increasing in the domain,
                           false for decreasing
            \return the root.
        */
        long double find_root(long double lower, long double upper,
                              const long double c, const long double d,
                              const long double z, bool dir) const {
 
            long double value, middle;
            do {
                middle = 0.5L*(upper + lower);
                value = eval_cubic(middle, c, d, z);
 
                if(std::abs(value) < zero)
                    return middle;
 
                if(value < 0.0L) {
                    if(dir)
                        lower = middle;
                    else
                        upper = middle;
                }
                else {
                    if(dir)
                        upper = middle;
                    else
                        lower = middle;
                }
            } while(std::abs(upper-lower) > rootzero);
 
            return middle;
        }
        long double find_root(long double lower, long double upper, {…}
 
        /** \brief Evaluates the cubic equation for the Lagrangian multipliers
 
            Cubic equation of interest:
            d + c(2z+c/2)*L - c(c+z)L^2 + c^2/2L^3 == 0
 
            \param[in] x Value at which to evaluate.
            \param[in] c Parameter c for the paraboloid.
            \param[in] d The d parameter: x^2+y^2-cz = d
            \param[in] z The distance from the apex along the paraboloid's
                         axis (called dotp in sdf)
            \return the value
        */
        long double eval_cubic(const long double x, const long double c,
                               const long double d, const long double z)
                    const {
            return ((c*0.5L*x - (c + z))*x + (2.0L*z + c*0.5L))*c*x + d;
        }
        long double eval_cubic(const long double x, const long double c, {…}
 
        /** \brief Evaluates the derivative of the cubic equation for the
                   Lagrangian multipliers
 
            Cubic equation of interest:
            d + c(2z+c/2)*L - c(c+z)L^2 + c^2/2L^3 == 0
 
            Derivative:
            c(2z+c/2) - 2c(c+z)L + 3c^2/2 L^2
 
            \param[in] x Value at which to evaluate.
            \param[in] c Parameter c for the paraboloid.
            \param[in] d The d parameter: x^2+y^2-cz = d
            \param[in] z The distance from the apex along the paraboloid's
                         axis (called dotp in sdf)
            \return the value
        */
        long double eval_cubic_deriv(const long double x, const long double c,
                               const long double d, const long double z)
                    const {
            return c*((1.5L*c*x - 2.0L*(c+z))*x + 2.0L*z + 0.5L*c);
        }
        long double eval_cubic_deriv(const long double x, const long double c, {…}
    };
    class SDFParaboloid : public SignedDFInterface<3> {…};
 
    /** \brief A box (3 dimensions)
 
        \note LIMIT -- the 3 primary axes must be x, y, and z */
    class SDFBox : public SignedDFInterface<3> {
    protected:
        const coord_3d lengths;  ///< Half the length of each side of the box
        const coord_3d center; ///< the center of the box
 
    public:
        /** \brief Constructor for box
 
            \param length The lengths of the box
            \param center The center of the box */
        SDFBox(const coord_3d& length, const coord_3d& center) 
            : lengths(length*0.5), center(center) 
        {}
        SDFBox(const coord_3d& length, const coord_3d& center)  {…}
 
        /** \brief Computes the normal distance
 
            This SDF naively uses contours, and should be improved before
            serious usage.  If far from the corners, the SDF is easy (similar
            to a plane), and is essentially what's implemented.
 
            \param pt Point at which to compute the distance from the surface
            \return The signed distance from the surface */
        double sdf(const coord_3d& pt) const {
            double diff, max;
            int i;
            
            max = fabs(pt[0] - center[0]) - lengths[0];
            for(i = 1; i < 3; ++i) {
                diff = fabs(pt[i] - center[i]) - lengths[i];
                if(diff > max)
                    max = diff;
            }
            
            return max;
        }
        double sdf(const coord_3d& pt) const  {…}
 
        /** Computes the gradient of the SDF.
 
            \param pt Point at which to compute the gradient
            \return the gradient */
        coord_3d grad_sdf(const coord_3d& pt) const {
            MADNESS_EXCEPTION("gradient method is not yet implemented for this shape",0);
        }
        coord_3d grad_sdf(const coord_3d& pt) const  {…}
    };
    class SDFBox : public SignedDFInterface<3> {…};
 
    /** \brief A cube (3 dimensions)
 
        \note LIMIT -- the 3 primary axes must be x, y, and z */
    class SDFCube : public SDFBox {
    public:
        /** \brief Constructor for box
 
            \param length The length of each side of the cube
            \param center The center of the cube */
        SDFCube(const double length, const coord_3d& center) 
            : SDFBox(coord_3d(length), center)
        {}
        SDFCube(const double length, const coord_3d& center)  {…}
    };
    class SDFCube : public SDFBox {…};
 
    /** \brief An ellipsoid (3 dimensions)
    
        \note LIMIT -- the 3 primary axes must be x, y, and z */
    class SDFEllipsoid : public SignedDFInterface<3> {
    protected:
        coord_3d radii; ///< the directional radii
        coord_3d center; ///< the center
        
    public:
        /** \brief Constructor for ellipsoid
 
            \param radii The directional radii of the ellipsoid
            \param center The center of the ellipsoid */
        SDFEllipsoid(const coord_3d& radii, const coord_3d& center) 
            : radii(radii)
            , center(center) 
        {}
        SDFEllipsoid(const coord_3d& radii, const coord_3d& center)  {…}
        
        /** \brief Computes the normal distance
 
            This SDF naively uses contours, and should be improved before
            serious usage.
 
            \param pt Point at which to compute the distance from the surface
            \return The signed distance from the surface */
        double sdf(const coord_3d& pt) const {
            double quot, sum;
            int i;
            
            sum = 0.0;
            for(i = 0; i < 3; ++i) {
                quot = (pt[i] - center[i]) / radii[i];
                sum += quot * quot;
            }
            
            return sum - 1.0;
        }
        double sdf(const coord_3d& pt) const  {…}
 
        /** Computes the gradient of the SDF.
 
            \param pt Point at which to compute the gradient
            \return the gradient */
        coord_3d grad_sdf(const coord_3d& pt) const {
            MADNESS_EXCEPTION("gradient method is not yet implemented for this shape",0);
        }
        coord_3d grad_sdf(const coord_3d& pt) const  {…}
    };
    class SDFEllipsoid : public SignedDFInterface<3> {…};
    
    /// \brief A cylinder (3 dimensions)
    class SDFCylinder : public SignedDFInterface<3> {
    protected:
        double radius; ///< the radius of the cylinder
        double a; ///< half the length of the cylinder
        coord_3d center; ///< the central axial point of the cylinder (distance a from both ends)
        coord_3d axis; ///< the axial direction of the cylinder
 
    public:
        /** \brief Constructor for cylinder
 
            \param radius The radius of the cylinder
            \param length The length of the cylinder
            \param axpt The central axial point of the cylinder (equidistant
                        from both ends)
            \param axis The axial direction of the cylinder */
        SDFCylinder(const double radius, const double length, const coord_3d& axpt, const coord_3d& axis) 
            : radius(radius)
            , a(length / 2.0)
            , center(axpt)
            , axis(axis*(1.0/sqrt(axis[0]*axis[0] + axis[1]*axis[1] + axis[2]*axis[2]))) 
        {}
        SDFCylinder(const double radius, const double length, const coord_3d& axpt, const coord_3d& axis)  {…}
        
        /** \brief Computes the normal distance
 
            This SDF naively uses contours, and should be improved before
            serious usage.
 
            \param pt Point at which to compute the distance from the surface
            \return The signed distance from the surface */
        double sdf(const coord_3d& pt) const {
            double dist;
            coord_3d rel, radial;
            int i;
            
            // axial distance
            dist = 0.0;
            for(i = 0; i < 3; ++i) {
                rel[i] = pt[i] - center[i];
                dist += rel[i] * axis[i];
            }
            
            // get the radial component
            for(i = 0; i < 3; ++i)
                radial[i] = rel[i] - dist * axis[i];
            
            return std::max(fabs(dist) - a, sqrt(radial[0]*radial[0] + radial[1]*radial[1]
                                                 + radial[2]*radial[2]) - radius);
        }
        double sdf(const coord_3d& pt) const  {…}
 
        /** Computes the gradient of the SDF.
 
            \param pt Point at which to compute the gradient
            \return the gradient */
        coord_3d grad_sdf(const coord_3d& pt) const {
            MADNESS_EXCEPTION("gradient method is not yet implemented for this shape",0);
        }
        coord_3d grad_sdf(const coord_3d& pt) const  {…}
    };
    class SDFCylinder : public SignedDFInterface<3> {…};
 
} // end of madness namespace
 
#endif // MADNESS_MRA_SDF_SHAPE_3D_H__INCLUDED