/* This is a simple but rather interesting program that graphs equations in 3 dimensions. You enter equations in terms of x, y, and z something like one of the following: y = sin[x] + cos[z] x^2 + y^2 + z^2 = 81 (This is a sphere) y cos[x] = x sin[z] This version of the program can also graph INEQUALITIES, which have less-than or greater-than symbols instead of just equals. Inequalities are important for graphing infinitely-thin objects and making them printable and sliceable. For example, you might need to convert: x^2 + y^2 - z^2 = 81 into abs[x^2 + y^2 - z^2 - 81] <= 2 to give the walls some thickness and give your slicer a chance to print it successfully. Also increasing the value of "res" below will make the voxels in the .obj file larger. Here is an egg. Modify the 1.5 and 1.1 to change its aspect ratio: (x^2 + y^2 + z^2)^2 = 6 (1.5 z^3 + (1.5 - 1.1) z (y^2 + x^2)) Here is a Klein bottle: (x^2 + y^2 + z^2 + 2y - 1)((x^2 + y^2 + z^2 - 2y - 1)^2 - 8 z^2) + 16 x z (x^2 + y^2 + z^2 - 2y - 1) = 0 Here is a heart: 320 (-x^2 z^3 - 9 y^2 z^3 / 80 + (x^2 + 9 y^2 / 4 + z^2 - 1)^3) <= 0 This uses a recursive method to subdivide and test cuboids. You can also use logical relations like AND and OR to combine multiple shapes. */ lasteq = "" xmin = -10 xmax = 10 ymin = -10 ymax = 10 zmin = -10 zmax = 10 // Change the doublings to vary the number of voxels. This is the number // of doublings, so if the number is 10 we have 2^10=1024 doublings for // a resolution of 1024x1024x1024. // Be warned that increasing the doublings by 1 makes 8 times as many voxels! doublings = 9 r = 2^doublings // Number of voxels on each axis res = 254/in // Resolution at which to render the .obj file // If there are arguments to the program, graph them, otherwise prompt. while func = (length[ARGS] > 0 ? ARGS@0 : input["Enter equation: ", lasteq]) { hasInequality = false certEq = undef lasteq = certFunc = func // If there's an inequality, let's make a test equation to see if we can // fill in an entire cuboid using the "CERTAINLY" comparators. if func =~ %r/([<>]|!=)/ { hasInequality = true certFunc =~ %s/<=/ CLE /g // Replace <= with certainly less than or equals certFunc =~ %s/>=/ CGE /g // Replace >= with certainly greater than or equals certFunc =~ %s// CGT /g // Replace > with certainly greater than certFunc =~ %s/!=/ CNE /g // Replace = with certainly not equals certFunc =~ %s/=/ CEQ /g // Replace = with certainly equals certEq = parseToExpression[certFunc] } // These replacements turn normal comparator and equality tests into // "POSSIBLY EQUALS" tests. func =~ %s/<=/ PLE /g // Replace <= with possibly less than or equals func =~ %s/>=/ PGE /g // Replace >= with possibly greater than or equals func =~ %s// PGT /g // Replace > with possibly greater than func =~ %s/!=/ PNE /g // Replace = with possibly not equals func =~ %s/=/ PEQ /g // Replace = with possibly equals eq = parseToExpression[func] println[inputForm[eq]] if certEq != undef println[inputForm[certEq]] // Scale factors on each axis sx = r / (xmax-xmin) sy = r / (ymax-ymin) sz = r / (zmax-zmin) v = callJava["frink.graphics.VoxelArray", "construct", [xmin sx, xmax sx, ymin sy, ymax sy, zmin sz, zmax sz, false]] // Perform the recursive testing of the volume testCube[xmin, xmax, ymin, ymax, zmin, zmax, v, eq, certEq, doublings, sx, sy, sz] // To convert from voxel coordinate v to original coordinate x, // x = minx + v (xmax - xmin) / (vmax - vmin) // inverse: // v = (x - xmin) (vmax - vmin) / (xmax - xmin) // println["Center of mass (in voxel coords): " + v.centerOfMass[].toString[]] v.projectX[undef].show["X"] v.projectY[undef].show["Y"] v.projectZ[undef].show["Z"] filename = "graph3D.obj" print["Writing $filename..."] w = new Writer[filename] w.println[v.toObjFormat["graph3D", 1 / (res mm)]] w.close[] println["done."] if length[ARGS] > 0 exit[] } // Recursive function to test an interval containing the specified bounds. // If no possible solution exists, the recursion halts. If the entire cube // is filled with a "certainly" equation, it is filled and recursion halts. // If only a possible solution exists, this breaks it down into 8 sub-cuboids // and tests each of them recursively. // level is the maximum number of levels to split, so the total // resolution of the final graph will be 2^level. testCube[x1, x2, y1, y2, z1, z2, v, eq, certEq, level, sx, sy, sz] := { nextLevel = level - 1 x = new interval[x1, x2] y = new interval[y1, y2] z = new interval[z1, z2] // Test the cuboid. If it possibly contains solutions, recursively // subdivide. res = eval[eq] if res or res==undef { if (nextLevel >= 0) { // Do we have inequalities and a CERTAINLY test? if (certEq != undef) AND (eval[certEq] == true) { // If the entire cuboid is a solution, then fill the rectangle // and stop further recursion on this cuboid. v.fillCube[x1 sx, x2 sx, y1 sy, y2 sy, z1 sz, z2 sz, true] return } // Further subdivide the cuboid into 8 octants and recursively // test them all cx = (x1 + x2)/2 cy = (y1 + y2)/2 cz = (z1 + z2)/2 testCube[x1, cx, y1, cy, z1, cz, v, eq, certEq, nextLevel, sx, sy, sz] testCube[cx, x2, y1, cy, z1, cz, v, eq, certEq, nextLevel, sx, sy, sz] testCube[x1, cx, cy, y2, z1, cz, v, eq, certEq, nextLevel, sx, sy, sz] testCube[cx, x2, cy, y2, z1, cz, v, eq, certEq, nextLevel, sx, sy, sz] testCube[x1, cx, y1, cy, cz, z2, v, eq, certEq, nextLevel, sx, sy, sz] testCube[cx, x2, y1, cy, cz, z2, v, eq, certEq, nextLevel, sx, sy, sz] testCube[x1, cx, cy, y2, cz, z2, v, eq, certEq, nextLevel, sx, sy, sz] testCube[cx, x2, cy, y2, cz, z2, v, eq, certEq, nextLevel, sx, sy, sz] } else if (res) // Valid point at lowest level; fill it v.fillCube[x1 sx, x2 sx, y1 sy, y2 sy, z1 sz, z2 sz,true] } }