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/*
* Copyright (C) 1998, 2000-2007, 2010, 2011, 2012, 2013 SINTEF ICT,
* Applied Mathematics, Norway.
*
* Contact information: E-mail: tor.dokken@sintef.no
* SINTEF ICT, Department of Applied Mathematics,
* P.O. Box 124 Blindern,
* 0314 Oslo, Norway.
*
* This file is part of SISL.
*
* SISL is free software: you can redistribute it and/or modify
* it under the terms of the GNU Affero General Public License as
* published by the Free Software Foundation, either version 3 of the
* License, or (at your option) any later version.
*
* SISL is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Affero General Public License for more details.
*
* You should have received a copy of the GNU Affero General Public
* License along with SISL. If not, see
* <http://www.gnu.org/licenses/>.
*
* In accordance with Section 7(b) of the GNU Affero General Public
* License, a covered work must retain the producer line in every data
* file that is created or manipulated using SISL.
*
* Other Usage
* You can be released from the requirements of the license by purchasing
* a commercial license. Buying such a license is mandatory as soon as you
* develop commercial activities involving the SISL library without
* disclosing the source code of your own applications.
*
* This file may be used in accordance with the terms contained in a
* written agreement between you and SINTEF ICT.
*/
#include "sisl-copyright.h"
/*
*
* $Id: s2511.c,v 1.3 2001-06-12 11:07:34 jbt Exp $
*
*/
#define S2511
#include "sislP.h"
#if defined(SISLNEEDPROTOTYPES)
void
s2511(SISLSurf *surf, int ider, double derive[], double normal[],
double *mehlum, int *jstat)
#else
void s2511(surf, ider, derive, normal, mehlum, jstat)
SISLSurf *surf;
int ider;
double derive[];
double normal[];
double *mehlum;
int *jstat;
#endif
/*
***************************************************************************
*
***************************************************************************
* PURPOSE : To compute the third order Mehlum curvature M(u,v) of a
* surface for given values (u,v). This is a lower level
* routine, used for evaluation of many M(u,v)'s.
* INPUT :
* surf - Pointer to the surface to evaluate.
* ider - Only implemented for ider=0 (derivative order).
* derive - Array containing derivatives from routine s1421().
* Size = idim*6.
* normal - Array containing the normal from routine s1421().
* Size = 3.
*
* OUTPUT :
* mehlum - Third order Mehlum curvature of the surface.
* jstat - Status messages
*
* = 0 : Ok.
* < 0 : Error.
*
* METHOD : The third order Mehlum curvature is given by
*
* M(u,v) = (5G^3P^2
* + (EG + 4F^2)
* *(9GQ^2 + 9ES^2 + 6GPS + 6EQT)
* + 5E^3T^2
* - 2F(3EG + 2F^2)(PT + 9QS)
* - 30F(G^2PQ + E^2ST))
* /(16(EG - F^2)^3).
*
* The variables E,F,G,P,Q,S and T
* are the coefficients of the first and third fundamental form.
* They are given by:
* E = <Xu,Xu>, F = <Xu,Xv> and G = <Xv,Xv>.
* P = <N,Xuuu> + 3(a*alpha + b*beta),
* Q = <N,Xuuv> + c*alpha + d*beta + 2a*gamma + 2b*delta,
* S = <N,Xuvv> + 2c*gamma + 2d*delta + a*epsilon + b*mu,
* T = <N,Xvvv> + 3(c*epsilon + d*mu),
* where N is normalized, and
* a = Ff - Ge,
* b = Fe - Ef,
* c = Fg - Gf,
* d = Ff - Eg,
* e, f and g being the second fundamental form coefficients
* (e = <N,Xuu>, f = <N,Xuv> and g = <N,Xvv>), and
* alpha = <Xuu,Xu>/||N||^2,
* beta = <Xuu,Xv>/||N||^2,
* gamma = <Xuv,Xu>/||N||^2,
* delta = <Xuv,Xv>/||N||^2,
* epsilon = <Xvv,Xu>/||N||^2,
* mu = <Xvv,Xv>/||N||^2.
*
* REFERENCES : Differential Geometry of Curves and Surfaces,
* (Manfredo P. Do Carmo, Prentice Hall,
* ISBN: 0-13-212589-7).
*-
* CALLS : s2513()
*
* LIMITATIONS :
* (i) If the surface is degenerated (not regular) at the point
* (u,v), it makes no sense to speak about the Mehlum
* M(u,v).
* (ii) If the surface is closed to degenerate, the Mehlum
* M(u,v) can be numerical unstable.
* (iii) The dimension of the space in which the surface lies must
* be 1,2 or 3, if not, jstat = -105 is returned.
*
*
* WRITTEN BY : Johannes Kaasa, SINTEF, Oslo, Norway. Date: 1995-9
*****************************************************************************
*/
{
double fundform[10]; /* The coefficients of the fundamental forms.
The sequence is: E, F, G, e, f, g, P, Q, S, T. */
double length; /* Square of normal length. */
double a, b, c, d; /* Utility coefficients. */
double alpha; /* Utility coefficient. */
double beta; /* Utility coefficient. */
double gamma; /* Utility coefficient. */
double delta; /* Utility coefficient. */
double epsilon; /* Utility coefficient. */
double mu; /* Utility coefficient. */
double P, Q, S, T; /* Third order coefficients. */
double numerator; /* Value of a numerator. */
double denominator; /* Value of a denominator. */
if (ider != 0) goto err178;
if (surf->idim == 1 || surf->idim == 3) /* 1D and 3D surface */
{
s2513(surf, ider, 3, 1, derive, normal, fundform, jstat);
if (*jstat < 0) goto error;
if (surf->idim == 3)
length = normal[0]*normal[0] + normal[1]*normal[1] +
normal[2]*normal[2];
else if (surf->idim == 1)
length = 1. + derive[1]*derive[1] + derive[2]*derive[2];
alpha = s6scpr(&derive[3*(surf->idim)], &derive[surf->idim],
surf->idim)/length;
beta = s6scpr(&derive[3*(surf->idim)], &derive[2*(surf->idim)],
surf->idim)/length;
gamma = s6scpr(&derive[4*(surf->idim)], &derive[surf->idim],
surf->idim)/length;
delta = s6scpr(&derive[4*(surf->idim)], &derive[2*(surf->idim)],
surf->idim)/length;
epsilon = s6scpr(&derive[5*(surf->idim)], &derive[surf->idim],
surf->idim)/length;
mu = s6scpr(&derive[5*(surf->idim)], &derive[2*(surf->idim)],
surf->idim)/length;
a = fundform[1]*fundform[4] - fundform[2]*fundform[3];
b = fundform[1]*fundform[3] - fundform[0]*fundform[4];
c = fundform[1]*fundform[5] - fundform[2]*fundform[4];
d = fundform[1]*fundform[4] - fundform[0]*fundform[5];
P = fundform[6] + 3*(a*alpha + b*beta);
Q = fundform[7] + c*alpha + d*beta + 2*a*gamma + 2*b*delta;
S = fundform[8] + 2*c*gamma + 2*d*delta + a*epsilon + b*mu;
T = fundform[9] + 3*(c*epsilon + d*mu);
numerator = 5*fundform[2]*fundform[2]*fundform[2]*P*P
+ (fundform[0]*fundform[2] + 4*fundform[1]*fundform[1])
*(9*fundform[2]*Q*Q
+ 9*fundform[0]*S*S
+ 6*fundform[2]*P*S
+ 6*fundform[0]*Q*T)
+ 5*fundform[0]*fundform[0]*fundform[0]*T*T
- 2*fundform[1]*(3*fundform[0]*fundform[2] + 2*fundform[1]*fundform[1])
*(P*T + 9*Q*S)
- 30*fundform[1]*(fundform[2]*fundform[2]*P*Q
+ fundform[0]*fundform[0]*S*T);
denominator = fundform[0]*fundform[2] - fundform[1]*fundform[1];
*mehlum = numerator/(16*denominator*denominator*denominator);
}
else if (surf->idim == 2) /* 2D surface */
{
/* The surface lies in a plane => K(u,v) = 0 */
*mehlum = 0.0;
}
else /* When surf->idim != 1,2 or 3 */
{
goto err105;
}
/* Successful computations */
*jstat = 0;
goto out;
/* Error in input, surf->idim != 1,2 or 3 */
err105:
*jstat = -105;
s6err("s2511",*jstat,0);
goto out;
/* Illegal derivative requested. */
err178:
*jstat = -178;
s6err("s2511",*jstat,0);
goto out;
error:
s6err("s2511",*jstat,0);
goto out;
out:
return;
}
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