> script("files/hands_on_tropical_geometry/3d_printing_helper_functions.pl");
> application "tropical";
===== I. Modelling Tropical Surfaces =====
==== I.1 Constructing a tropical surface in polymake ====
An easy way to construct a tropical surface in $\mathbb{R}^3$ of type ''%%fan::PolyhedralComplex%%'' in ''%%polymake%%'' is by constructing and converting a ''%%tropical::Hypersurface%%''. This can be done by specifying a tropical polynomial as a string.
> $f = toTropicalPolynomial("min(1+2*w,1+2*x,1+2*y,1+2*z,w+x,w+y,w+z,x+y,x+z,y+z)");
> $tropicalSurfaceTmp = new tropical::Hypersurface(POLYNOMIAL=>$f);
> Important: Since ''%%polymake%%'' uses homogeneous coordinates, i.e., it identifies polyhedra in $\mathbb{R}^3$ with cones in $\mathbb{R}^4$, tropical polynomials need to be tetravariate and homogeneous instead of trivariate and inhomogeneous.
Alternatively, one can specifying an exponent matrix and a coefficient vector:
> $exponentVectors = [[2,0,0,0], [1,1,0,0], [1,0,1,0], [1,0,0,1], [0,1,1,0], [0,1,0,1], [0,0,1,1], [0,2,0,0], [0,0,2,0], [0,0,0,2]];
> $coefficients = [1,0,0,0,0,0,0,1,1,1];
> $tropicalSurfaceTmp = new Hypersurface(MONOMIALS=>$exponentVectors, COEFFICIENTS=>$coefficients);
To convert the ''%%tropical::Hypersurface%%'' to ''%%fan::PolyhedralComplex%%'':
> $tropicalSurface = new fan::PolyhedralComplex(
> VERTICES=>$tropicalSurfaceTmp->VERTICES->minor(All,~[1]),
> MAXIMAL_POLYTOPES=>$tropicalSurfaceTmp->MAXIMAL_POLYTOPES,
> INPUT_LINEALITY=>$tropicalSurfaceTmp->LINEALITY_SPACE->minor(All,~[1]));
> $tropicalSurface->VISUAL;
> $boundingBox = generateBoundingBox($tropicalSurface);
> Note: ''%%generateBoundingBox($tropicalSurface)%%'' draws a box around all vertices of ''%%$tropicalSurface%%'' with a margin of 1, ''%%generateBoundingBox($tropicalSurface,3,4,5)%%'' draw a box around all vertices of ''%%$tropicalSurface%%'' with margins 3, 4, 5 in x-, y-, z-direction respectively.
or by specifying a custom ''%%polytope%%'', which need not be a cuboid:
> $boundingBox = polytope::scale(polytope::cube(3),2);
To construct a bounded polyhedral complex use the command ''%%intersectWithBoundingBox%%'', which intersects any ''%%fan::PolyhedralComplex%%'' with a ''%%polytope%%''.
> $tropicalSurfaceBounded = intersectWithBoundingBox($tropicalSurface,$boundingBox);
> $tropicalSurfaceBounded->VISUAL;
> $filename = "foo.scad";
> generateSCADFileForSurface($tropicalSurfaceBounded,$filename);
==== I.3 Solidifying a polyhedral complex in OpenSCAD and exporting it for 3D-printing ====
To solidify the surface into a three-dimensional model, open the exported file in ''%%OpenSCAD%%''. Below is a preview of a solidified tropical quadratic surface. {{:tutorials:release:4.11:hands_on_tropical_geometry:OpenSCADSurface.png|OpenSCADSurface.png}}
The top of the ''%%OpenSCAD%%'' file contains parameters which control the thickness of the model amongst other things:
colorSurface = "SlateGray"; // color of surface
scalingFactor = 1; // global scaling factor
thicknessSurface = 0.05; // thickness of surface
- ''%%colorSurface%%'' is the color of the surface in the render. It can be specified either by an HTML color name or by RGB value.
- ''%%scalingFactor%%'' is a factor by which the polyhedral complex is scaled. It can be used to scale the model to a desired size.
- ''%%thicknessSurface%%'' controls the thickness of the surface. ''%%OpenSCAD%%'' solidifies the surface by replacing every vertex with a ball with specified thickness and taking convex hulls.
> Important: In order to 3D-print the tropical surface, every face needs to have a minimal thickness. As a rule of thumb for printing with PLA, we recommend a minimal thickness of 2mm for a print of size 10cm3.
===== II. Modelling Tropical Curves =====
==== II.1 Constructing a tropical curve in polymake ====
An easy way to construct a tropical curve in $\mathbb{R}^3$ of type ''%%fan::PolyhedralComplex%%'' in ''%%polymake%%'' is by intersecting two tropical surfaces. Note however that not all tropical curves can be constructed this way.
Below is an example of how to define a tropical sextic curve as an intersection of a tropical quadratic and a cubic surface.
> # constructing tropical quadric surface
> $mQuadric = [[2,0,0,0], [1,1,0,0], [1,0,1,0], [1,0,0,1],
> [0,1,1,0], [0,1,0,1], [0,0,1,1], [0,2,0,0],
> [0,0,2,0], [0,0,0,2]];
> $cQuadric = [1,-1/4,-2/4,-3/4,-3/4,-4/4,-5/4,2/4,0,-2/4];
> $TQuadric = new tropical::Hypersurface(MONOMIALS=>$mQuadric,
> COEFFICIENTS=>$cQuadric);
>
> # constructing tropical cubic surface
> $mCubic = [[3,0,0,0], [0,3,0,0], [0,0,3,0], [0,0,0,3],
> [1,1,1,0], [1,1,0,1], [1,0,1,1], [0,1,1,1],
> [2,1,0,0], [2,0,1,0], [2,0,0,1], [1,2,0,0],
> [1,0,2,0], [1,0,0,2], [0,2,1,0], [0,2,0,1],
> [0,1,2,0], [0,1,0,2], [0,0,2,1], [0,0,1,2]];
> $cCubic = [3,3,3,3,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1];
> $TCubic = new tropical::Hypersurface(MONOMIALS=>$mCubic,
> COEFFICIENTS=>$cCubic);
>
> # constructing a tropical sextic curve as the stable intersection
> # of a quadratic and a cubic surface
> $tropicalCurveTmp = tropical::intersect($TQuadric,$TCubic);
> Important: ''%%intersect%%'' is the intersection of tropical cycles, i.e., it tcomputes the so-called stable intersection, not the set-theoretic intersection.
As before, we convert the tropical curve to type ''%%fan::PolyhedralComplex%%'':
> $tropicalCurve = new fan::PolyhedralComplex(
> VERTICES=>$tropicalCurveTmp->VERTICES->minor(All,~[1]),
> MAXIMAL_POLYTOPES=>$tropicalCurveTmp->MAXIMAL_POLYTOPES,
> INPUT_LINEALITY=>$tropicalCurveTmp->LINEALITY_SPACE->minor(All,~[1]));
> $tropicalCurve->VISUAL;
> $boundingBox = generateBoundingBox($tropicalCurve);
> $tropicalCurveBounded = intersectWithBoundingBox($tropicalCurve,$boundingBox);
For the sake of stability, the tropical curve requires a frame. This is best done using a tropical surface containing it:
> $tropicalSurface = new fan::PolyhedralComplex(
> VERTICES=>$TQuadric->VERTICES->minor(All,~[1]),
> MAXIMAL_POLYTOPES=>$TQuadric->MAXIMAL_POLYTOPES);
> $tropicalSurfaceFrame = intersectWithBoundingBoxForFraming($tropicalSurface,$boundingBox);
> compose($tropicalCurveBounded->VISUAL,$tropicalSurfaceFrame->VISUAL);
> #todo: adjust edge thicknesses to distinguish frame from curve
> $filename = "foo.scad";
> generateSCADFileForCurve($tropicalSurfaceFrame,$tropicalCurve,$filename);
==== II.3 Solidifying a tropical curve in OpenSCAD and exporting it for 3D-printing ====
To solidify the curve into a three-dimensional model, open the exported file in ''%%OpenSCAD%%''. Here is the preview of the solidified tropical sextic curve (with frame) constructed above:
{{:tutorials:release:4.11:hands_on_tropical_geometry:OpenSCADCurve.png|OpenSCADCurve.png}}
As before, the top of the ''%%OpenSCAD%%'' file contains parameters which control the thickness of the model amongst other things:
colorFrame = "SlateGray"; // color of frame
colorCurve = [0.83,.15,0.27]; // color of curve
scalingFactor = 1; // global scaling factor
thicknessFrame = 0.05; // thickness of frame
thicknessCurve = 0.05; // thickness of curve
- ''%%colorFrame%%'' and ''%%colorCurve%%'' are the colors of the surface in the render. They can be specified either by an HTML color name or by RGB value.
- ''%%scalingFactor%%'' is a factor by which the polyhedral complex is scaled. It can be used to scale the model to the desired size.
- ''%%thicknessFrame%%'' and ''%%thicknessCurve%%'' control the thickness of the frame and curve. ''%%OpenSCAD%%'' solidifies both by replacing every vertex with a ball with diameter equal to the specified radius and taking convex hulls.
===== III. Modelling Tropical Curves on Tropical Surfaces =====
==== III.1 Constructing a curve and surface in polymake ====
The previous section showed an easy way to construct a tropical curve lying on a tropical surface in ''%%polymake%%'': start with the surface and construct the curve by intersecting it with another surface. However, as mentioned before, this may not always possible or easy to do. One option that always works is to construct the curve manually by specifying its vertices and edges:
> $linearEquation = toTropicalPolynomial("max(y,z,w)",qw(w x y z));
> $tropicalSurfaceTmp = new tropical::Hypersurface(POLYNOMIAL=>$linearEquation);
> $tropicalSurface = new fan::PolyhedralComplex(
> POINTS=>$tropicalSurfaceTmp->VERTICES->minor(All,~[1]),
> INPUT_POLYTOPES=>$tropicalSurfaceTmp->MAXIMAL_POLYTOPES,
> INPUT_LINEALITY=>$tropicalSurfaceTmp->LINEALITY_SPACE->minor(All,~[1]));
>
> $tropicalCurveVertices =
> [[1,0,0,0],[1,-8,0,0],[1,-8,-6,0],[1,12,0,0],[0,0,-1,0],[1,-6,-6,0],[1,-11,-3,0],
> [1,-11,-6,0],[0,-1,0,0],[1,-12,-7,0],[1,4,0,-2],[1,10,0,-2],[1,8,0,-6],[0,0,0,-1],
> [0,1,1,1],[1,-4,6,6],[1,-4,5,5],[1,-3,5,5],[1,-5,4,4],[1,-3,3,3],[1,-5,3,3]];
> $tropicalCurveEdges =
> [[3,4],[0,5],[2,5],[4,5],[1,6],[6,7],[6,8],[2,7],[7,9],[8,9],[4,9],[10,11],[3,11],[11,12],
> [0,10],[10,12],[12,13],[1,13],[14,15],[8,15],[15,16],[14,17],[16,17],[16,18],[17,19],[8,18],
> [18,20],[0,19],[19,20],[1,20],[3,14]];
> $tropicalCurve = new fan::PolyhedralComplex(
> POINTS=>$tropicalCurveVertices,
> INPUT_POLYTOPES=>$tropicalCurveEdges);
(for a preview of the constructed tropical plane with a tropical quartic curve see below)
==== III.2 Bounding a tropical surface in polymake and exporting it to OpenSCAD ====
Constructing a bounding box is easiest done using the command ''%%generateBoundingBox%%'' as before:
> $boundingBox = generateBoundingBox($tropicalCurve);
> $tropicalSurfaceBounded = intersectWithBoundingBox($tropicalSurface,$boundingBox);
> $tropicalCurveBounded = intersectWithBoundingBox($tropicalCurve,$boundingBox);
> compose($tropicalSurfaceBounded->VISUAL,$tropicalCurveBounded->VISUAL);
> $filename = "foo.scad";
> generateSCADFileForSurfaceAndCurve($tropicalSurfaceBounded,$tropicalCurveBounded,$filename);
==== III.3 Solidifying a tropical surface with curve in OpenSCAD and exporting it for 3D-printing ====
To solidify the surface and curve into a three-dimensional model, open the exported file in ''%%OpenSCAD%%''. Here is a preview of the solidified tropical plane with a tropical quartic curve constructed above:
{{:tutorials:release:4.11:hands_on_tropical_geometry:OpenSCADSurfaceAndCurve.png|OpenSCADSurfaceAndCurve.png}}
The top of the ''%%OpenSCAD%%'' file contains parameters which control the thickness of the model amongst other things:
colorSurface = "SlateGray"; // color of surface
colorCurve = [0.83,.15,0.27]; // color of curve
scalingFactor = 1; // global scaling factor
thicknessSurface = 0.01; // thickness of surface
thicknessCurve = 0.1; // thickness of curve
- ''%%colorSurface%%'' and ''%%colorCurve%%'' are the colors of the surface and curve in the render. It can be specified either by an HTML color name or by RGB value.
- ''%%scalingFactor%%'' is a factor by which the polyhedral complex is scaled. It can be used to scale the model to the desired size.
- ''%%thicknessSurface%%'' and ''%%thicknessCurve%%'' control the thickness of the surface and curve. ''%%OpenSCAD%%'' solidifies the surface by replacing every vertex with a ball with diameter equal to the specified radius and taking convex hulls.
===== IV. Printing tropical models using Prusa =====
To 3D-print our models using ''%%Prusa%%'', we suggest exporting them to ''%%3mf%%''. The advantage of the ''%%3mf%%'' file format is that it may contain multiple objects. The relative positions of the object is stored, so that - once loaded into ''%%PrusaSlicer%%'' - they are aligned as intended and we could assign them differently colored materials. The export of a single ''%%3mf%%'' file containing multiple objects can be done with [[https://github.com/jschobben/colorscad|''%%Colorscad%%'']].