triangle.scad

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//! Triangle shapes, conversions, properties, and tests functions.
/***************************************************************************//**
  \file
  \author Roy Allen Sutton
  \date   2015-2019
 
  \copyright
 
    This file is part of [omdl] (https://github.com/royasutton/omdl),
    an OpenSCAD mechanical design library.
 
    The \em omdl is free software; you can redistribute it and/or modify
    it under the terms of the [GNU Lesser General Public License]
    (http://www.gnu.org/licenses/lgpl.html) as published by the Free
    Software Foundation; either version 2.1 of the License, or (at
    your option) any later version.
 
    The \em omdl 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
    Lesser General Public License for more details.
 
    You should have received a copy of the GNU Lesser General Public
    License along with the \em omdl; if not, write to the Free Software
    Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
    02110-1301, USA; or see <http://www.gnu.org/licenses/>.
 
  \details
 
    \amu_define group_name  (Triangles)
    \amu_define group_brief (Triangle mathematical functions.)
 
  \amu_include (include/amu/pgid_path_pstem_pg.amu)
*******************************************************************************/
 
//----------------------------------------------------------------------------//
// group and macros.
//----------------------------------------------------------------------------//
 
/***************************************************************************//**
  \amu_include (include/amu/group_in_parent_start.amu)
  \amu_include (include/amu/includes_required.amu)
 
  \amu_define html_image_w  (512)
  \amu_define latex_image_w (2.25in)
  \amu_include (include/amu/scope_diagram_2d_object.amu)
 
  \details
 
    See [Wikipedia](https://en.wikipedia.org/wiki/Triangle)
    for more information.
*******************************************************************************/
 
//----------------------------------------------------------------------------//
// shape generation
//----------------------------------------------------------------------------//
 
//! \name Shapes
//! @{
 
//! Compute the side lengths of a triangle given its vertex coordinates.
/***************************************************************************//**
  \param    c <coords-3d | coords-2d>  A list, [v1, v2, v3], the 3d or 2d
            vertex coordinates.
 
  \returns  <decimal-list-3> A list of side lengths [s1, s2, s3].
 
  \details
 
    Each side length is opposite the corresponding vertex.
 
    \amu_eval ( object=triangle_ppp2sss ${object_diagram_2d} )
 
    No verification is performed to ensure that the given sides specify
    a valid triangle. See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Solution_of_triangles
*******************************************************************************/
function triangle_ppp2sss
(
  c
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    s1 = distance_pp(v2, v3),
    s2 = distance_pp(v3, v1),
    s3 = distance_pp(v1, v2)
  )
  [s1, s2, s3];
 
//! Compute the side lengths of a triangle given two sides and the included angle.
/***************************************************************************//**
  \param    v <decimal-list-3> A list, [s1, a3, s2], the side lengths
            and the included angle.
 
  \returns  <decimal-list-3> A list of side lengths [s1, s2, s3].
 
  \details
 
    Each side length is opposite the corresponding vertex.
 
    \amu_eval ( object=triangle_sas2sss ${object_diagram_2d} )
 
    No verification is performed to ensure that the given sides specify
    a valid triangle. See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Solution_of_triangles
*******************************************************************************/
function triangle_sas2sss
(
  v
) =
  let
  (
    s1 = v[0],
    a3 = v[1],
    s2 = v[2],
 
    s3 = sqrt( s1*s1 + s2*s2 - 2*s1*s2*cos(a3) )
  )
  [s1, s2, s3];
 
//! Compute the side lengths of a triangle given a side and two adjacent angles.
/***************************************************************************//**
  \param    v <decimal-list-3> A list, [a1, s3, a2], the side length
            and two adjacent angles.
 
  \returns  <decimal-list-3> A list of side lengths [s1, s2, s3].
 
  \details
 
    Each side length is opposite the corresponding vertex.
 
    \amu_eval ( object=triangle_asa2sss ${object_diagram_2d} )
 
    No verification is performed to ensure that the given sides specify
    a valid triangle. See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Solution_of_triangles
*******************************************************************************/
function triangle_asa2sss
(
  v
) =
  let
  (
    a1 = v[0],
    s3 = v[1],
    a2 = v[2],
 
    a3 = 180 - a1 - a2,
 
    s1 = s3 * sin( a1 ) / sin( a3 ),
    s2 = s3 * sin( a2 ) / sin( a3 )
  )
  [s1, s2, s3];
 
//! Compute the side lengths of a triangle given a side, one adjacent and the opposite angle.
/***************************************************************************//**
  \param    v <decimal-list-3> A list, [a1, a2, s1], a side length,
            one adjacent and the opposite angle.
 
  \returns  <decimal-list-3> A list of side lengths [s1, s2, s3].
 
  \details
 
    Each side length is opposite the corresponding vertex.
 
    \amu_eval ( object=triangle_aas2sss ${object_diagram_2d} )
 
    No verification is performed to ensure that the given sides specify
    a valid triangle. See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Solution_of_triangles
*******************************************************************************/
function triangle_aas2sss
(
  v
) =
  let
  (
    a1 = v[0],
    a2 = v[1],
    s1 = v[2],
 
    a3 = 180 - a1 - a2,
 
    s2 = s1 * sin( a2 ) / sin( a1 ),
    s3 = s1 * sin( a3 ) / sin( a1 )
  )
  [s1, s2, s3];
 
//! Compute a set of vertex coordinates for a triangle given its side lengths in 2D.
/***************************************************************************//**
  \param    v <decimal-list-3> A list, [s1, s2, s3], the side lengths.
  \param    a <integer> The axis alignment index
            < \b x_axis_ci | \b y_axis_ci >.
  \param    cw <boolean> Order vertices clockwise.
 
  \returns  <coords-2d> A list of vertex coordinates [v1, v2, v3],
            when \p cw = \b true, else [v1, v3, v2].
 
  \details
 
    Each side length is opposite the corresponding vertex. The triangle
    will be constructed with \em v1 at the origin. Side \em s2 will be
    on the 'x' axis or side \em s3 will be on the 'y' axis as
    determined by parameter \p a.
 
    \amu_eval ( object=triangle2d_sss2ppp ${object_diagram_2d} )
 
    No verification is performed to ensure that the given sides specify
    a valid triangle. See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Solution_of_triangles
*******************************************************************************/
function triangle2d_sss2ppp
(
  v,
  a  = x_axis_ci,
  cw = true
) =
  let
  (
    s1  = v[0],
    s2  = v[1],
    s3  = v[2],
 
    v1  = origin2d,
 
    v2  = (a == x_axis_ci) ?
          // law of cosines
          let(x = (-s1*s1 + s2*s2 + s3*s3) / (2*s2))
          [x, sqrt(s3*s3 - x*x)]
        : [0, s3],
 
    v3  = (a == x_axis_ci) ?
          [s2, 0]
          // law of cosines
        : let(y = (-s1*s1 + s2*s2 + s3*s3) / (2*s3))
          [sqrt(s2*s2 - y*y), y]
  )
    (cw == true) ? [v1, v2, v3] : [v1, v3, v2];
 
//! @}
 
//----------------------------------------------------------------------------//
// shape properties
//----------------------------------------------------------------------------//
 
//! \name Properties
//! @{
 
//! Compute the area of a triangle given its vertex coordinates in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
  \param    s <boolean> Return the vertex ordering sign.
 
  \returns  <decimal> The area of the given triangle.
*******************************************************************************/
function triangle2d_area
(
  c,
  s = false
) =
  let
  (
    sa = is_left_ppp(p1=c[0], p2=c[1], p3=c[2]) / 2
  )
    (s == false) ? abs(sa) : sa;
 
//! Compute the centroid of a triangle.
/***************************************************************************//**
  \param    c <coords-3d | coords-2d>  A list of 3d or 2d vertex
            coordinates [v1, v2, v3].
  \param    d <integer> The number of dimensions [2:3].
 
  \returns  <point-3d | point-2d> The centroid coordinate.
 
  \details
 
    When \p d is assigned \b 0, \p d is set to the minimun number of
    coordinates of the vertices in \p c.
 
    See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Centroid
*******************************************************************************/
function triangle_centroid
(
  c,
  d = 2
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    e  = (d == 0) ? min(len(v1), len(v2), len(v3)) : d
  )
    [ for (i=[0:e-1]) (v1[i] + v2[i] + v3[i])/3 ];
 
//! Compute the center coordinate of a triangle's incircle in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
 
  \returns  <point-2d> The incircle center coordinate [x, y].
 
  \details
 
    The interior point for which distances to the sides of the triangle
    are equal. See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle
*******************************************************************************/
function triangle2d_incenter
(
  c
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    d1 = distance_pp(v2, v3),
    d2 = distance_pp(v3, v1),
    d3 = distance_pp(v1, v2)
  )
    [
      ( (v1[0] * d1 + v2[0] * d2 + v3[0] * d3) / (d3 + d1 + d2) ),
      ( (v1[1] * d1 + v2[1] * d2 + v3[1] * d3) / (d3 + d1 + d2) )
    ];
 
//! Compute the inradius of a triangle's incircle in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
 
  \returns  <decimal> The incircle radius.
 
  \details
 
    See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle
*******************************************************************************/
function triangle2d_inradius
(
  c
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    d1 = distance_pp(v2, v3),
    d2 = distance_pp(v3, v1),
    d3 = distance_pp(v1, v2)
  )
    sqrt( ((-d3+d1+d2) * (+d3-d1+d2) * (+d3+d1-d2)) / (d3+d1+d2) ) / 2;
 
//! Compute the center coordinate of a triangle's excircle in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
  \param    v <integer> Return coordinate opposite vertex \p v.
 
  \returns  <point-2d> The excircle center coordinate [x, y].
 
  \details
 
    A circle outside of the triangle specified by \p v1, \p v2, and \p
    v3, tangent to the side opposite \p v and tangent to the
    extensions of the other two sides away from \p v. See [Wikipedia]
    for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle
*******************************************************************************/
function triangle2d_excenter
(
  c,
  v = 1
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    d1 = distance_pp(v2, v3),
    d2 = distance_pp(v3, v1),
    d3 = distance_pp(v1, v2)
  )
    (v == 1) ? [ ((-d1*v1[0]+d2*v2[0]+d3*v3[0])/(-d1+d2+d3)),
                  ((-d1*v1[1]+d2*v2[1]+d3*v3[1])/(-d1+d2+d3)) ]
  : (v == 2) ? [ ((+d1*v1[0]-d2*v2[0]+d3*v3[0])/(+d1-d2+d3)),
                  ((+d1*v1[1]-d2*v2[1]+d3*v3[1])/(+d1-d2+d3)) ]
  : (v == 3) ? [ ((+d1*v1[0]+d2*v2[0]-d3*v3[0])/(+d1+d2-d3)),
                  ((+d1*v1[1]+d2*v2[1]-d3*v3[1])/(+d1+d2-d3)) ]
  : origin2d;
 
//! Compute the exradius of a triangle's excircle in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
  \param    v <integer> Return coordinate opposite vertex \p v.
 
  \returns  <decimal> The excircle radius of the excircle opposite \p v.
 
  \details
 
    See [Wikipedia] for more information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle
*******************************************************************************/
function triangle2d_exradius
(
  c,
  v = 1
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    d1 = distance_pp(v2, v3),
    d2 = distance_pp(v3, v1),
    d3 = distance_pp(v1, v2),
    s  = (+d1+d2+d3)/2
  )
    (v == 1) ? sqrt(s * (s-d2) * (s-d3) / (s-d1))
  : (v == 2) ? sqrt(s * (s-d1) * (s-d3) / (s-d2))
  : (v == 3) ? sqrt(s * (s-d1) * (s-d2) / (s-d3))
  : 0;
 
//! Compute the coordinate of a triangle's circumcenter.
/***************************************************************************//**
  \param    c <coords-3d | coords-2d>  A list of 3d or 2d vertex
            coordinates [v1, v2, v3].
  \param    d <integer> The number of dimensions [2:3].
 
  \returns  <point-3d | point-2d> The circumcenter coordinate.
 
  \details
 
    A circle that passes through all of the vertices of the triangle.
    The radius is the distance from the circumcenter to any of vertex.
    When \p d is assigned \b 0, \p d is set to the minimun number of
    coordinates of the vertices in \p c. See [Wikipedia] for more
    information.
 
  [Wikipedia]: https://en.wikipedia.org/wiki/Circumscribed_circle
*******************************************************************************/
function triangle_circumcenter
(
  c,
  d = 2
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    s2a = sin( 2 * angle_ll([v1, v3], [v1, v2]) ),
    s2b = sin( 2 * angle_ll([v2, v1], [v2, v3]) ),
    s2c = sin( 2 * angle_ll([v3, v2], [v3, v1]) ),
 
    e  = (d == 0) ? min(len(v1), len(v2), len(v3)) : d
  )
    [ for (i=[0:e-1]) (v1[i]*s2a + v2[i]*s2b + v3[i]*s2c) / (s2a+s2b+s2c) ];
 
//! @}
 
//----------------------------------------------------------------------------//
// shape property tests
//----------------------------------------------------------------------------//
 
//! \name Tests
//! @{
 
//! Test the vertex ordering, or orientation, of a triangle in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
 
  \returns  <boolean> \b true if the vertices are ordered clockwise,
            \b false if the vertices are ordered counterclockwise, and
            \b undef if the ordering can not be determined.
*******************************************************************************/
function triangle2d_is_cw
(
  c
) =
  let
  (
    il = is_left_ppp(p1=c[0], p2=c[1], p3=c[2])
  )
    (il < 0) ? true
  : (il > 0) ? false
  : undef;
 
//! Test if a point is inside a triangle in 2d.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
  \param    p <point-2d> A test point coordinate [x, y].
 
  \returns  <boolean> \b true when the point is inside the polygon and
            \b false otherwise.
 
  \details
 
    See [Wikipedia] for more information.
 
    [Wikipedia]: https://en.wikipedia.org/wiki/Barycentric_coordinate_system
*******************************************************************************/
function triangle2d_is_pit
(
  c,
  p
) =
  let
  (
    v1 = c[0],
    v2 = c[1],
    v3 = c[2],
 
    d = ((v2[1]-v3[1]) * (v1[0]-v3[0]) + (v3[0]-v2[0]) * (v1[1]-v3[1]))
  )
    (d == 0) ? true
  : let
  (
    a = ((v2[1]-v3[1]) * ( p[0]-v3[0]) + (v3[0]-v2[0]) * ( p[1]-v3[1])) / d
  )
    (a < 0) ? false
  : let
  (
    b = ((v3[1]-v1[1]) * ( p[0]-v3[0]) + (v1[0]-v3[0]) * ( p[1]-v3[1])) / d
  )
    (b < 0) ? false
  : ((a + b) < 1);
 
//! @}
 
//----------------------------------------------------------------------------//
// shape rounding
//----------------------------------------------------------------------------//
 
//! \name Rounding
//! @{
 
//! Compute the rounding center coordinate for a given radius of a triangle vertex in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
  \param    r <decimal> The vertex rounding radius.
 
  \returns  <decimal> The rounding center coordinate.
*******************************************************************************/
function triangle2d_vround3_center
(
  c,
  r
) = let( ir = triangle2d_inradius(c) )
    (c[1]-r/(r-ir) * triangle2d_incenter(c)) * (ir-r)/ir;
 
//! Compute the rounding tangent coordinates for a given radius of a triangle vertex in 2D.
/***************************************************************************//**
  \param    c <coords-2d> A list of vertex coordinates [v1, v2, v3].
  \param    r <decimal> The vertex rounding radius.
 
  \returns  <decimal> The rounding tangent coordinates [t1, t2].
*******************************************************************************/
function triangle2d_vround3_tangents
(
  c,
  r
) = let
    (
      rc = triangle2d_vround3_center(c, r),
      im = sqrt( pow(distance_pp(c[1], rc),2) - pow(r,2) ),
      t1 = c[1] + im * unit_l([c[1], c[0]]),
      t2 = c[1] + im * unit_l([c[1], c[2]])
    )
    [t1, t2];
 
//! @}
 
//! @}
//! @}
 
//----------------------------------------------------------------------------//
// openscad-amu auxiliary scripts
//----------------------------------------------------------------------------//
 
/*
BEGIN_SCOPE diagram;
  BEGIN_OPENSCAD;
    include <omdl-base.scad>;
    include <tools/operation_cs.scad>;
    include <tools/drafting/draft-base.scad>;
 
    module dt (vl = empty_lst, al = empty_lst, sl = empty_lst)
    {
      s = 10;
      t = [6, 8, 7];
      r = min(t)/len(t)*s;
 
      c = triangle2d_sss2ppp(t)*s;
 
      draft_polygon(c, w=s*2/3);
      draft_axes([[0,12], [0,7]]*s, ts=s/2);
 
      for (i = [0 : len(c)-1])
      {
        cv = c[i];
        os = shift(shift(v=c, n=i, r=false), 1, r=false, c=false);
 
        if ( !is_empty( find( mv=i, v=vl )) )
        draft_dim_leader(cv, v1=[mean(os), cv], l1=5, t=str("v", i+1), bs=0, cmh=s*1, cmv=s);
 
        if ( !is_empty( find( mv=i, v=al )) )
        draft_dim_angle(c=cv, r=r, v1=[cv, second(os)], v2=[cv, first(os)], t=str("a", i+1), a=0, cmh=s, cmv=s);
 
        if ( !is_empty( find( mv=i, v=sl )) )
        draft_dim_line(p1=first(os), p2=second(os), t=str("s", i+1), cmh=s, cmv=s);
      }
    }
 
    object = "triangle_sas2sss";
 
    if (object == "triangle_ppp2sss") dt( vl = [0,1,2]);
    if (object == "triangle_sas2sss") dt( vl = [0,1,2], al = [2], sl = [0,1]);
    if (object == "triangle_asa2sss") dt( vl = [0,1,2], al = [0,1], sl = [2]);
    if (object == "triangle_aas2sss") dt( vl = [0,1,2], al = [0,1], sl = [0]);
    if (object == "triangle2d_sss2ppp") dt( sl = [0,1,2]);
  END_OPENSCAD;
 
  BEGIN_MFSCRIPT;
    include --path "${INCLUDE_PATH}" {var_init,var_gen_svg}.mfs;
 
    defines   name "objects" define "object"
              strings "
                triangle_ppp2sss
                triangle_sas2sss
                triangle_asa2sss
                triangle_aas2sss
                triangle2d_sss2ppp
              ";
    variables add_opts_combine "objects";
    variables add_opts "--viewall --autocenter";
 
    include --path "${INCLUDE_PATH}" scr_make_mf.mfs;
  END_MFSCRIPT;
END_SCOPE;
*/
 
//----------------------------------------------------------------------------//
// end of file
//----------------------------------------------------------------------------//