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`

:

- Two linear
`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. - A line and a
`Polygon`

. Currently,`Reg_Polygon`

and`Rectangle`

are the only classes derived from`Polygon`

. - Two
`Polygons`

. - A line and a Regular Closed Plane Curve (
`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`

. - Two
`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.