MADNESS
0.10.1
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A paraboloid (3 dimensions) More...
#include <sdf_shape_3D.h>
Public Member Functions | |
SDFParaboloid (const double c, const coord_3d &apex, const coord_3d &direc) | |
Constructor for paraboloid. More... | |
coord_3d | grad_sdf (const coord_3d &pt) const |
Computes the gradient of the SDF. More... | |
double | sdf (const coord_3d &pt) const |
Computes the normal distance. More... | |
Public Member Functions inherited from madness::SignedDFInterface< 3 > | |
virtual | ~SignedDFInterface () |
virtual Vector< double, NDIM > | grad_sdf (const Vector< double, NDIM > &x) const=0 |
Returns the gradient of the signed distance from the surface (i.e., dsdf(x)/dx [i] ) More... | |
virtual double | sdf (const Vector< double, NDIM > &x) const=0 |
Returns the signed distance from the surface,. More... | |
Protected Member Functions | |
long double | eval_cubic (const long double x, const long double c, const long double d, const long double z) const |
Evaluates the cubic equation for the Lagrangian multipliers. More... | |
long double | eval_cubic_deriv (const long double x, const long double c, const long double d, const long double z) const |
Evaluates the derivative of the cubic equation for the Lagrangian multipliers. More... | |
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 |
std::vector< long double > | get_roots (const long double c, const long double d, const long double z) const |
Finds real root(s) of the cubic polynomial in the sdf function. More... | |
Protected Attributes | |
const coord_3d | apex |
The apex. More... | |
const double | c |
Curvature/radius of the surface. More... | |
const coord_3d | dir |
The direction of the axis, from the apex INSIDE. More... | |
const long double | rootzero |
Numerical zero for the roots. More... | |
const long double | zero |
Numerical zero for root-finding in sdf. More... | |
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Constructor for paraboloid.
c | Parameter in the definition of the paraboloid |
apex | Apex of paraboloid |
direc | Oriented axis of the paraboloid |
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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
[in] | x | Value at which to evaluate. |
[in] | c | Parameter c for the paraboloid. |
[in] | d | The d parameter: x^2+y^2-cz = d |
[in] | z | The distance from the apex along the paraboloid's axis (called dotp in sdf) |
Referenced by find_root(), and get_roots().
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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
[in] | x | Value at which to evaluate. |
[in] | c | Parameter c for the paraboloid. |
[in] | d | The d parameter: x^2+y^2-cz = d |
[in] | z | The distance from the apex along the paraboloid's axis (called dotp in sdf) |
References c.
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Finds a root in the specified range for the cubic equation.
This uses a simple bisection method... perhaps someone will improve it one day...
[in] | lower | The bottom of the range |
[in] | upper | The top of the range |
[in] | c | Parameter c for the paraboloid. |
[in] | d | The d parameter: x^2+y^2-cz = d |
[in] | z | The distance from the apex along the paraboloid's axis (called dotp in sdf) |
[in] | dir | True if the function is increasing in the domain, false for decreasing |
References std::abs(), c, d(), dir, eval_cubic(), lower, rootzero, upper, and zero.
Referenced by get_roots().
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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
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
[in] | c | Parameter c for the paraboloid. |
[in] | d | The d parameter: x^2+y^2-cz = d |
[in] | z | The distance from the apex along the paraboloid's axis (called dotp in sdf) |
References std::abs(), c, d(), eval_cubic(), find_root(), lower, upper, and zero.
Referenced by sdf().
Computes the gradient of the SDF.
pt | Point at which to compute the gradient |
References MADNESS_EXCEPTION.
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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
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.
pt | Point at which to compute the distance from the surface |
References std::abs(), apex, c, d(), diff(), dir, dist(), get_roots(), MADNESS_ASSERT, and upper.
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Curvature/radius of the surface.
Referenced by eval_cubic(), eval_cubic_deriv(), find_root(), get_roots(), and sdf().
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The direction of the axis, from the apex INSIDE.
Referenced by find_root(), and sdf().
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Numerical zero for the roots.
Referenced by find_root().
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Numerical zero for root-finding in sdf.
Referenced by find_root(), and get_roots().