# Make mathematica create equations and put them into NSolve

I want to be able to define n shapes and find the intersection between them and a line. I'm treating the shapes as "one" shape, i.e. I do not want any internal intersections (hence the volume equations).

I can do this by typing out each shape and equation as follows:

line = InfiniteLine[{{0.5, 0.5, 0.5}, {0.5, 0.5, 1}}];

(*shapes - I can define these*)
shape1 = Ball[];
shape2 = Cone[{{0, 0, 0}, {1, 1, 1}}, 1/2];

(*------------ I want from here  automated ---------*)
(*surface equations *)
surfaceequations1 = RegionMember[RegionBoundary[shape1], {x, y, z}];
surfaceequations2 = RegionMember[RegionBoundary[shape2], {x, y, z}];

volumeequation1 = RegionMember[shape1, {x, y, z}];
volumeequation2 = RegionMember[shape2, {x, y, z}];

intersection =
NSolve[{x, y, z} \[Element]
line && (surfaceequations1 ||
surfaceequations2) && ! (volumeequation1 && volumeequation2), {x,
y, z}];

(* ---------- automation can stop here ---------- *)

points = Point[{x, y, z}] /. intersection

Graphics3D[{{Opacity[0.5], shape1}, {Opacity[0.7], shape2},
line, {Red, PointSize[0.015], points}}]


But this method means I have to type out each equation for each shape. If I change the number of shapes, I have to go through code and delete some.

## Edit:

The correct logic to find the intersection with the shape as whole (rather than the intersections with the individual n shapes) is actually:

 intersection =
NSolve[{x, y, z} \[Element]
line && (surfaceequations1 || surfaceequations2 ||
surfaceequations3) && ! (volumeequation1 &&
volumeequation2) && ! (volumeequation2 &&
volumeequation3) && ! (volumeequation1 && volumeequation3), {x,
y, z}]


i.e. for the intersection with the line to not to be within the shape, it must not be within any two shapes. This means its properly treated as one shape.

So, now shape1 = Ball[]; shape2 = Cone[]; shape3 = Cuboid[]; works as expected.

## The Problem

I want to be able to define n shapes, then mathematica defines the n surface equations, and the n volume equations and puts them into NSolve.

Also; apologies for the bad question title - I'm not sure what to call this problem.

ClearAll[constraints, intersections]
constraints[shapes__]:= And[##& @@ (Not /@
Through[(RegionMember[RegionIntersection @ ##] & @@@ Subsets[{shapes}, {2}])@#]),
RegionMember[RegionUnion @@ (RegionBoundary /@ {shapes})]@#] &

intersections[l_, s__]:=NSolve[#\[Element] l && constraints[s][#], #]& @
({x, y, z}[[;; RegionEmbeddingDimension[l]]])


Examples:

line = InfiniteLine[{{0.5, 0.5, 0.5}, {0.5, 0.5, 1}}];
shape1 = Ball[];
shape2 = Cone[{{0, 0, 0}, {1, 1, 1}}, 1/2];
shape3 = Cuboid[];

intersections[line, shape1, shape2]


{{x -> 0.5 ,y -> 0.5, z -> -0.7071067811865475},
{x -> 0.5, y -> 0.5, z -> 0.7542812707978059}}

 Graphics3D @ {line, Opacity[.5, Green], shape1, Opacity[.5, Blue], shape2,
PointSize[.05], Opacity[1, Red],
Point[{x, y,z} /. intersections[line, shape1,shape2]]}


intersections[line, shape1, shape2, shape3]


{{x -> 0.5, y -> 0.5, z -> 1.}, {x -> 0.5, y -> 0.5, z -> 1.}, {x -> 0.5, y -> 0.5, z -> -1.}}

Graphics3D @ {line, Opacity[.5, Green], shape1, Opacity[.5, Blue], shape2,
Opacity[.5, Yellow], shape3, PointSize[.05], Opacity[1, Red],
Point[{x, y,z} /. intersections[line, shape1,shape2, shape3]]}


Replace shape3 with Cuboid[{0, 0, -3/2}] to get

2D examples

 DeleteDuplicates @ intersections[InfiniteLine[{-1, -1}, {1, 1}],
Disk[{0, 0}, .5], Triangle[]]


{{x -> -0.353553, y -> -0.353553}, {x -> 0.5, y -> 0.5}}

Graphics @ {#, Opacity[.5, Green], #2, Opacity[.5, Blue], #3,
PointSize[.05], Opacity[1, Red], Point[{x, y} /. intersections[##]]}& @@
{InfiniteLine[{-1, -1}, {1, 1}],Disk[{0, 0}, .5], Triangle[]}


 DeleteDuplicates @ intersections[InfiniteLine[{-1, -1}, {1, 1}],
Disk[{0, 0}, .5], Triangle[], Disk[{1, 1}, .5]]


{{x -> 1.35355, y -> 1.35355}, {x -> 0.646447, y -> 0.646447}, {x -> -0.353553, y -> -0.353553}, {x -> 0.5, y -> 0.5}}

Graphics@{#, Opacity[.5, Green], #2, Opacity[.5, Blue], #3,
Opacity[.5, Magenta], #4, PointSize[.05], Opacity[1, Red],
Point[{x, y} /. intersections[##]]} & @@ {InfiniteLine[{-1, -1}, {1, 1}],
Disk[{0, 0}, .5], Triangle[], Disk[{1,1}, .5]}


DeleteDuplicates @ intersections[InfiniteLine[{-1, -1}, {1, 1}],
Disk[{0, 0}, .5], Triangle[], Disk[{1, 1}, .75]]


{{x -> 1.53033, y -> 1.53033}, {x -> -0.353553, y -> -0.353553}}

Graphics@{#, Opacity[.5, Green], #2, Opacity[.5, Blue], #3,
Opacity[.5, Magenta], #4,PointSize[.05], Opacity[1, Red],
Point[{x, y} /. intersections[##]]} & @@ {InfiniteLine[{-1, -1}, {1, 1}],
Disk[{0, 0}, .5], Triangle[], Disk[{1,1}, .75]}


• Is there anyway I can have a list of shapes e.g shapes = {shape1, shape2}. Then use go something like: intersections[line,shapes] ?
– Tomi
Commented Jan 24, 2018 at 17:37
• The best I can do is: intersections[line, ##] & @@ shapes but this only gives the first intersection.
– Tomi
Commented Jan 24, 2018 at 17:49
• @Tomi, intersections[line, ##] & @@ shapes gives two solutions on Wolfram Cloud. It will be more convenient to have intersections[l_, {s__}] := ...  in the line defining intersections to make the second argument a list so that intersections[line, shapes] works.
– kglr
Commented Jan 24, 2018 at 18:48
• This solution won't work for more than 2 shapes. It will consider the points give intersections in between the shapes. I have added an edit which gives the correct logic to find the intersection with the line. So, the current solution doesn't actually answer the original question - but that was because of my bad logic. .
– Tomi
Commented Jan 27, 2018 at 14:48
• @Tomi, please see the updated version.
– kglr
Commented Jan 30, 2018 at 21:59