#ifndef frb_geo_H
#define frb_geo_H

/* Miscellaneous geometry functions. */
/* Last edited on 2005-01-16 14:45:33 by stolfi */
/* Copyright © 2005 Universidade Estadual de Campinas. See note at end of file. */

#include <r3.h>
#include <r4x4.h>
#include <frb_types.h>

r3_t frb_linear_interpolate
  ( double t, 
    double a, 
    r3_t *pa, 
    double b, 
    r3_t *pb 
  );
  /*
    Interpolates linearly the position at time {t} between 
    points {pa} and {b}, assuming they have times {a} and {b}. */

void frb_hermite_interpolate 
  ( double t, 
    double a, 
    r3_t *pa, 
    r3_t *va, 
    double b, 
    r3_t *pb, 
    r3_t *vb,  
    r3_t *p,
    r3_t *v 
  );
  /*
    Performs cubic Hermite interpolation for argument {t} given
    positions {pa} and {pb} and velocities {va} and {vb} at times {a}
    and {b}. Best used for {t} between {a} and {b}. */

r3_t frb_estimate_velocity_Q 
  ( double a, 
    r3_t *pa, 
    double b, 
    r3_t *pb, 
    double c, 
    r3_t *pc 
  );

r3_t frb_estimate_velocity_L
  ( double a, 
    r3_t *pa, 
    double b, 
    r3_t *pb, 
    double c, 
    r3_t *pc 
  );

r3_t frb_estimate_velocity_C
  ( double a, 
    r3_t *pa, 
    double b, 
    r3_t *pb, 
    double c, 
    r3_t *pc 
  );
  /* 
    These procedures return estimates of a curve's velocity at time {b}, 
    given its positions {pa}, {pb}, {pc} at three successive times {a}, {b}, {c}.
    
    frb_estimate_velocity_Q uses quadratic interpolation. It is precise for
    quadratic functions but is unstable when {b} is near {a} or {c}.
    
    frb_estimate_velocity_L uses linear interpolation between {(a,pa)} and {(c,pc)}.
    It ignores {b} and {pb}.  It is therefore robust for uneven intervals,
    but is not precise for quadratics. 
    
    frb_estimate_velocity_C uses a formula that degenerates to linear
    interpolation between {(a,pa)} and {(b,pb)} when {b} is close to
    {c}, and to linear interpolation between {(b,pb)} and {(c,pc)}
    when {b} is close to {a}.  It thus gives an {unconstrained
    derivative} effect at double nodes, which is useful for
    modeling. It is more robust than frb_estimate_velocity_Q, but 
    is not precise for quadratics. */

double frb_hermite_curve_length
  ( double a, 
    r3_t *pa, 
    r3_t *va, 
    double b, 
    r3_t *pb, 
    r3_t *vb 
  );
  /*
    Computes the approximate length between {pa} and {pb}
    of the cubic curve that interpolates {(a,pa)} and {(b,pb)}, 
    with velocities {va}, {vb} at those points. */

r3_t frb_B_spline_approximation
  ( double t,
    double a, 
    double b, 
    r3_t *Pabc,
    double c, 
    r3_t *Pbcd,
    double d, 
    r3_t *Pcde,
    double e, 
    r3_t *Pdef,
    double f 
  );
  /*
    Computes the position at time {t} of a cubic B-Spline curve
    with control points {Pabc}, {Pbcd}, {Pcde}, {Pdef},
    and knot sequence {a},... {f}.  Assumes {t} lies between {c}
    and {d}. */

double frb_avg_seg_dist
  ( r3_t *a1,
    r3_t *a2,
    r3_t *b1,
    r3_t *b2 
  );
  /* 
    A segment distance function.
    
    The result is the root mean square distance between {a(t)}
    and {b(t)}, which travel along the segments {a1--a2} and {b1--b2}
    at constant speed. */

double frb_avg_seg_dist_sqr
  ( r3_t *a1,
    r3_t *a2,
    r3_t *b1,
    r3_t *b2 
  );
  /* 
    The square of {frb_avg_seg_dist}, only faster. */

r4x4_t frb_translation_matrix ( r3_t *p );
  /*
    Builds a translation matrix that takes the origin of R^3 to 
    point {p} (in Cartesian coordinates). */

r4x4_t frb_rotation_matrix ( r3_t *u, r3_t *v );
  /*
    Builds a rotation matrix that takes the vector {u} to the
    vector {v} by the shortest route. */

r3_t frb_seg_dir ( r3_t *p, r3_t *q );
  /*
    Direction from {p} to {q}. */

r3_t frb_seg_mid ( r3_t *p, r3_t *q );
  /*
    Midpoint of segment from {p} to {q}. */


#endif
/* COPYRIGHT AND AUTHORSHIP NOTICE

  Copyright © 2005 Universidade Estadual de Campinas (UNICAMP).
  Created by Helena C. G. Leitão and Jorge Stolfi in 1995--2005.
  
  This source file can be freely distributed, used, and modified,
  provided that this copyright and authorship notice is preserved in
  all copies, and that any modified versions of this file are clearly
  marked as such.
  
  This software has NO WARRANTY of correctness or applicability for
  any purpose. Neither the authors nor their employers shall be held
  responsible for any losses or damages that may result from its use.

  END OF NOTICE */