There are no functions for finding the intersection points of two (or
more) arbitrary Paths. This is impossible, so long as 3DLDF
outputs MetaPost code.
3DLDF only “knows” about the Points on a
Path; it doesn't actually generate the curve or other figure
that passes through the Points, and consequently doesn't “know”
how it does this.
In addition, an arbitrary Path can contain connectors.
In 3DLDF, the connectors are
merely strings and are written verbatim to the output file,
however, in MetaPost they influence the form of a Path.
3DLDF can, however, find the intersection points of some
non-arbitrary Paths. So far, it can find the intersection
point of the following combinations of Paths:
Paths, i.e., Paths
for which Path::is_linear() returns true
(see Path Reference; Querying).
In addition, the static Point member function
Point::intersection_points() can be called with four Point
arguments. The first and second arguments are treated as the end points
of one line, and the third and fourth arguments as the end points of the
other.
Polygon. Currently, Reg_Polygon and
Rectangle are the only classes derived from Polygon.
Polygons.
Reg_Cl_Plane_Curve,
see Regular Closed Plane Curve Reference; Intersections). Currently,
Ellipse and Circle are the only classes derived from
Reg_Cl_Plane_Curve.
Ellipses. Since a Circle is also an Ellipse,
one or both of the Ellipses may be a Circle.
See Ellipse Reference; Intersections.
Adding more functions for finding the intersections of various geometric figures is one of my main priorities with respect to extending 3DLDF.
There are currently no special
functions for finding the intersection points
of a line and a Circle or two Circles. Since the
class Circle is derived from class Ellipse,
Circle::intersection_points() resolves to
Ellipse::intersection_points(), which, in turn, calls
Reg_Cl_Plane_Curve::intersection_points().
This does the trick, but it's much easier to find the intersections for
Circles that it is for Ellipses. In particular, the
intersections of two coplanar Circles can be found
algebraically, whereas I've had to implement a numerical solution for
the case of two coplanar Ellipses with different centers and/or
axis orientation. It may also be worthwhile to write
a specialization for
finding the intersection points of a Circle and an
Ellipse.
The theory of intersections is a fascinating and non-trivial branch of
mathematics.1
As I learn more about it, I plan to define more
classes to represent various curves (two-dimensional ones to
start with) and functions for finding their intersection points.
[1] The books on computer graphics and the fairly elementary mathematics books that I own or have referred to don't go into intersections very deeply. One that does is Fischer, Gerd. Ebene Algebraische Kurven, which is a bit over my head.