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[Efi Fogel <>]

Hi KHartmann,

I've looked a bit more carefully into the second issue. Apparently, there is a good reason to use Arr_conic_traits_2 (or at least not to use Arr_circle_segment_traits_2).

The Arr_circle_segment_traits_2 supports segments and circular arcs. It uses an efficient, yet limited number type called CGAL::Sqrt_extension. If you reach the limitation, you fall back into using a more general traits, e.g., Arr_conic_traits_2, and in this case you do.

The segments supported by Arr_circle_segment_traits_2 can have endpoints with coordinates that are not necessarily rational. However, the underlying line must have rational coefficient. When constructing the (exact) offset we need to construct segments that do not necessarily lie on rational lines. More precisely, say you have a segment S = (s,t) in your original polygon. Let L be the underlying line of the offset segment. L : a * x + b * y + c = 0. Then, a and b are rational, but c in not necessarily so. 

Efi

 

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On Sun, Jan 8, 2017 at 2:35 AM, Efi Fogel <> wrote:

Your observation(s) are correct and points to some deficiencies.

1. Conic_point_2 (or _Conic_x_monotone_arc_2 for that matter) are the types of the curves handled by Arr_conic_traits_2. They do not support convenient member functions that return the type of the underlying conic curve (being either a circle, an ellipse, a parabola, or a hyperbola), and if the underlying conic curve is a circle, for example, they do not support a member function that returns the square of the radius for example. However, it is not hard to extract this information from the coefficients of the conic, r, s, t, u, v, and w. Recall that the conic curve is the zero set of the following polynomial:

p(x,y) = r*x^2 + s*y^2 + t*xy + u*x + v*y + w

and a conic that represents a circle is the zero set of the following polynomial:

p(x,y) = (x-a)^2 + (y-b)^2 - R^2

in other words

x^2 + y^2 - 2*a*x - 2*b*y + (a^2 + b^2 - R^2) = 0

which implies that r == s, t == 0, etc.

2. I do not see a reason why CGAL::offset_polygon_2() cannot use Arr_circle_segment_traits_2 instead of Arr_conic_traits_2. (I might be wrong though.) I may try to provide an overload of CGAL::offset_polygon_2() that uses Arr_circle_segment_traits_2 (but it may take some time).

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On Thu, Jan 5, 2017 at 11:12 PM, KHartmann <> wrote:

If I am using CGAL::Gps_circle_segment_traits_2<Kernel>,
I can get the arc center points and
the arc orientations for any X_monotone_curve_2 with the code
shown below:

  Gen_Polygon_2::Curve_const_iterator curveIter;
  for (curveIter = poly.curves_begin();
       curveIter != poly.curves_end();
       curveIter++)
  {
    const X_monotone_curve_2& curve = (*curveIter);

    if (curve.is_circular())
    {
      double arc_center_x =
         CGAL::to_double( curve.supporting_circle().center().x() );
      double arc_center_y =
         CGAL::to_double( curve.supporting_circle().center().y() );
      bool clockWise =
         curve.orientation() == CGAL::CLOCKWISE
    }
  }

But how would I do the same, if I am using
CGAL::Arr_conic_traits_2<Rat_kernel, Alg_kernel, Nt_traits>?

I don't have "is_circular" or "supporting_circle" available to me
when I am using conic_traits.

I am stuck using conic_traits, since that is what CGAL::offset_polygon_2
uses.
Even though CGAL::offset_polygon_2 always returns line segments and circular
arcs.





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