Skip to main content

graphics3d - Finding points in a plane


The end of the Star Wars saga is near, so I feel compelled to design a Mouse Droid. Based upon the Paul Murphy's schematics I've reconstructed the outline of the top shell:


enter image description here


Since my goal is to create a 3D-printable object, I'd like to carve out the inside of the shell, and this objective has tested the limits of my trigonometry knowledge.



The goal is to create an identical shape (excluding the brim) that is scaled to provide an arbitrary thickness in the x,y,z directions (in practice, the x and y thicknesses will be identical and typically thinner than the z thickness). Because the object will be 3D printed, thickness must be defined in the [x,y,z] dimensions, so the approach I'm using is to assign the z thickness, find points in a new plan that intersect with the shell outline, and translate as appropriate in the x and y directions to get the coordinates for the cutout. I'm stuck here:


pts = {{1.53685, 1, 0.6}, {2.77444, 2.81657, 7.6187}, {15.5486, 2.81657, 
7.6187}, {20.4632, 1, 0.6}, {1.53685, 11.25, 0.6}, {2.77444,
9.43343, 7.6187}, {15.5486, 9.43343, 7.6187}, {20.4632, 11.25,
0.6}, {0, 0, 0}, {0, 12.25, 0}, {22, 12.25, 0}, {22, 0, 0}, {0, 0,
0.6}, {0, 12.25, 0.6}, {22, 12.25, 0.6}, {22, 0, 0.6}};
pl1 = pts[[{0, 4, 7, 3} + 1]];
pl2 = pts[[{1, 5, 6, 2} + 1]];
Graphics3D[{
Red, Thick, MapThread[Line[{#1, #2}] &, {pl1, pl2}],

Red, Opacity[0.1], Polygon[pl1],
Blue, Opacity[0.1], Polygon[pl2],
Black, Polygon[# + {0, 0, 5.6187} & /@ pl1]
}, Boxed -> False,
PlotLabel ->
"What is the point on red line that intersects black plane?"]

enter image description here


Given the 8 points on two parallel planes and a third parallel plane of some known distance (2 units in this case) from the top plane, how to I find the four points where the red lines intersect the black plane? The internal angles of the Mouse Droid are 80 and 55 degrees around x and 75 degrees around y.



Answer




lines = MapThread[Line[{#1, #2}] &, {pl1, pl2}];
plane = Polygon[# + {0, 0, 5.6187} & /@ pl1];
intersections = (RegionIntersection[plane, #] & /@ lines)[[All, 1, 1]];

Graphics3D[{Red, Thick, lines, Opacity[0.1], Polygon[pl1],
Blue, Polygon[pl2], Black, plane, Green, Polygon@intersections,
Opacity[1], Sphere[#, .3] & /@ intersections}, Boxed -> False]

enter image description here


Update: An alternative approach to find the intersections:



scale = Rescale[.6 + 5.6187, MinMax[{pl1[[1, -1]], pl2[[1, -1]]}], {0, 1}];
intersections2 = pl1 + scale (pl2 - pl1) ;

intersections2 == intersections


True



Graphics3D[{Red, Thick, lines, Opacity[.1],  Blue, Hexahedron[pts], 
Black, plane, Green, Polygon@intersections2, Opacity[1],

Sphere[#, .2] & /@ intersections2}, Boxed -> False]

enter image description here


Update 2: A purely graphical approach using ParametricPlot3D (as in Cesareo's answer) with MeshFunctions and Mesh options to find the desired intersections:


Show[ParametricPlot3D[pl1 + λ (pl2 - pl1), {λ, 0, 1}, 
PlotStyle -> Directive[Red, Thick],
MeshFunctions -> {#3 &},
Mesh -> {{.06 + 5.6187}},
MeshStyle -> ({Green, Sphere[#, .2] & @@ #} &)],
Graphics3D[{Opacity[0.1], Red, Polygon[pl1], Blue, Polygon[pl2], Black, plane}],

Boxed -> False, Axes -> False]

enter image description here


Comments

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...