Reduce the burden of mathematics when playing OpenSCAD

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Reduce the burden of mathematics/algorithm when playing OpenSCAD.

## Introduction

Some of my 3D models require complex mathematics/algorithm. I extract them into dotSCAD. Hope it helps when you're playing OpenSCAD.

## Getting started

OpenSCAD uses three library locations, the installation library, built-in library, and user defined libraries. Check Setting OPENSCADPATH in OpenSCAD User Manual/Libraries for details.

I set `OPENSCADPATH` to the `src` folder of dotSCAD so all examples here start searching modules/functions from `src`.

Every public module/function has the same name as the .scad file. Here's an example using the `line2d` module:

``````use <line2d.scad>

line2d(p1 = [0, 0], p2 = [5, 0], width = 1);
``````

The library uses directories to categorize some modules/functions. For example, vx_circle.scad exists in `voxel` directory. Prefix the directory name when using `vx_circle`.

``````use <voxel/vx_circle.scad>

for(pt = points) {
translate(pt) square(1);
}
``````

# API Reference

## 2D Module

Signature Description
arc(radius, angle[, width, width_mode"]) create an arc.
hexagons(radius, spacing, levels) create hexagons in a hexagon.
line2d(p1, p2[, width, p1Style, p2Style]) create a line from two points.
multi_line_text(lines[, line_spacing, size, font, ...]) create multi-line text from a list of strings.
pie(radius, angle) create polyline2de a pie (circular sector).
polyline2d(points[, width, startingStyle, endingStyle, ...]) create a polyline from a list of `[x, y]` coordinates.
polygon_hull(points) create a convex polygon by hulling a list of points. It avoids using hull and small 2D primitives to create the polygon.
rounded_square(size, corner_r[, center]) create a rounded square in the first quadrant.

## 3D Module

Signature Description
crystal_ball(radius[, theta, phi, thickness]) create a crystal ball based on spherical coordinates (r, θ, φ) used in mathematics.
line3d(p1, p2[, diameter, p1Style, p2Style]) create a 3D line from two points.
loft(sections[, slices]) develop a smooth skin between crosssections with different geometries.
polyhedron_hull(points) create a convex polyhedron by hulling a list of points. It avoids using `hull` and small 3D primitives to create the polyhedron.
polyline3d(points, diameter[, startingStyle, endingStyle]) create a polyline from a list of `[x, y, z]`.
rounded_cube(size, corner_r[, center]) create a cube in the first octant.
rounded_cylinder(radius, h, round_r[, convexity, center]) create a rounded cylinder.
sweep(sections[, triangles]) develop a smooth skin from crosssections with the same number of sides.

## Transformation

Signature Description
along_with(points, angles[, twist, scale, method]) put children along the given path. If there's only one child, put the child for each point.
bend(size, angle[, frags]) bend a 3D object.
hollow_out(shell_thickness) hollow out a 2D object.
shear([sx, sy, sz]) shear all child elements along the X-axis, Y-axis, or Z-axis.
select(i) select module objects.
polyline_join(points) place a join on each point. Hull each pair of joins and union all convex hulls.

## 2D Function

Signature Description
bijection_offset(pts, d[, epsilon]) move 2D outlines outward or inward by a given amount. Each point of the offsetted shape is paired with exactly one point of the original shape.
contours(points, threshold) compute contour polygons by applying marching squares to a rectangular list of numeric values.
in_shape(shapt_pts, pt[, include_edge, epsilon]) check whether a point is inside a shape.
trim_shape(shape_pts, from, to[, epsilon]) trim a tangled-edge shape to a non-tangled shape.

## 2D/3D Function

Signature Description
angle_between(vt1, vt2) return the angle between two vectors.
bezier_smooth(path_pts, round_d[, t_step, closed, angle_threshold]) use bezier curves to smooth a path.
cross_sections(shape_pts, path_pts, angles[, twist, scale]) given a 2D shape, points and angles along the path, this function returns all cross-sections.
in_polyline(line_pts, pt[, epsilon]) check whether a point is on a line.
lines_intersection(line1, line2[, ext, epsilon]) find the intersection of two line segments. Return `[]` if lines don't intersect.
path_scaling_sections(shape_pts, edge_path) given an edge path with the first point at the outline of a shape, this function uses the path to calculate scaling factors and returns all scaled sections in the reversed order of the edge path.
midpt_smooth(points, n[, closed]) given a 2D path, this function constructs a mid-point smoothed version by joining the mid-points of the lines of the path.

## Path

Signature Description
arc_path(radius, angle) create an arc path.
archimedean_spiral(arm_distance, init_angle, point_distance, num_of_points[, rt_dir]) get all points and angles on the path of an archimedean spiral.
bauer_spiral(n, radius = 1[, rt_dir]) create visually even spacing of n points on the surface of the sphere. Successive points will all be approximately the same distance apart.
bezier_curve(t_step, points) given a set of control points, this function returns points of the Bézier path.
bspline_curve(t_step, degree, points, knots, weights) B-spline interpolation using de Boor's algorithm.
curve(t_step, points[, tightness]) create a curved path. An implementation of Centripetal Catmull-Rom spline.
fibonacci_lattice(n, radius = 1[, dir]) create visually even spacing of n points on the surface of the sphere. Nearest-neighbor points will all be approximately the same distance apart.
golden_spiral(from, to, point_distance[, rt_dir)] get all points and angles on the path of a golden spiral based on Fibonacci numbers. The distance between two points is almost constant.
helix(radius, levels, level_dist[, vt_dir, rt_dir]) get all points on the path of a spiral around a cylinder.
sphere_spiral(radius, za_step[, z_circles, begin_angle, end_angle, ...]) create all points and angles on the path of a spiral around a sphere. It returns a vector of `[[x, y, z], [ax, ay, az]]`.
torus_knot(p, q, phi_step) generate a path of The (p,q)-torus knot.

## Extrusion

Signature Description
bend_extrude(size, thickness, angle[, frags]) extrude and bend a 2D shape.
box_extrude(height, shell_thickness, bottom_thickness[, offset_mode, chamfer, ...]) create a box (container) from a 2D object.
ellipse_extrude(semi_minor_axis, height[, center, convexity, twist, slices]) extrude a 2D object along the path of an ellipse from 0 to 180 degrees.
rounded_extrude(size, round_r[, angle, twist, convexity]) extrude a 2D object roundly from 0 to 180 degrees.
stereographic_extrude(shadow_side_leng) take a 2D polygon as input and extend it onto a sphere.

## 2D Shape

Signature Description
shape_arc(radius, angle, width[, width_mode]) return points on the path of an arc shape.
shape_circle(radius, n) return points on the path of a circle.
shape_cyclicpolygon(sides, circle_r, corner_r) return points on the path of a regular cyclic polygon.
shape_ellipse(axes) return points on the path of an ellipse.
shape_liquid_splitting(radius, centre_dist[, tangent_angle, t_step]) return shape points of two splitting liquid shapes, kind of how cells divide.
shape_path_extend(stroke_pts, path_pts[, scale, closed]) extend a 2D stroke along a path to create a 2D shape.
shape_pentagram(r) return shape points of a pentagram.
shape_pie(radius, angle) return shape points of a pie (circular sector) shape.
shape_square(size[, corner_r]) return shape points of a rounded square or rectangle.
shape_superformula(phi_step, m1, m2, n1, [n2, n3, a, b]) return shape points of Superformula.
shape_taiwan(h[, distance]) return shape points of Taiwan.
shape_trapezium(length, h[, corner_r]) return shape points of an isosceles trapezoid.

## 2D Shape Extrusion

Signature Description
archimedean_spiral_extrude(shape_pts, arm_distance, init_angle, point_distance, num_of_points, ...) extrude a 2D shape along the path of an archimedean spiral.
golden_spiral_extrude(shape_pts, from, to, point_distance, ...) extrude a 2D shape along the path of a golden spiral.
helix_extrude(shape_pts, radius, levels, level_dist, ...) extrude a 2D shape along a helix path.
path_extrude(shape_pts, path_pts, ...) extrude a 2D shape along a path.
ring_extrude(shape_pts, radius[, angle = 360]) rotational extrusion spins a 2D shape around the Z-axis.
sphere_spiral_extrude(shape_pts, radius, za_step, ...) extrude a 2D shape along the path of a sphere spiral.

## Util

### util/list

Signature Description
util/binary_search(sorted, target[, lo, hi]) search a value in a sorted list.
util/contains(lt, elem) return `true` if `lt` contains `elem`.
util/find_index(lt, test) return the index of the first element that satisfies the testing function.
util/dedup(lt, ...) eliminate duplicate vectors.
util/flat(lt[, depth]) return a new list with all sub-list elements concatenated into it recursively up to the specified depth.
util/reverse(lt) reverse a list.
util/slice(lt, begin, end) return a list selected from `begin` to `end`, or to the `end` of the list (`end` not included).
util/sorted(lt[, cmp, key, reverse]) sort a list.
util/sum(lt) use `+` to sum up all elements in a list.
util/swap(lt, i, j) swap two elements in a list.
util/zip(lts, combine) make a list that aggregates elements from each of the lists.
util/every(lt, test) test whether all elements in the list pass the test implemented by the provided function.
util/some(lt, test) test whether at least one element in the list passes the test implemented by the provided function.
util/count(lt, test) return the number of times `test` return `true` in the list.

### util/random

Signature Description
util/choose(choices, seed) choose an element from the given list.
util/rand([min_value, max_value, seed_value]) generate a pseudo random number.
util/shuffle(lt[, seed]) randomizes the order of the elements.

### util/string

Signature Description
util/parse_number(t) parse the string argument as an number.
util/split_str(t, delimiter) split the given string around matches of the given delimiting character.
util/sub_str(t, begin, end) return the part of the string from `begin` to `end`, or to the `end` of the string (`end` not included).

### util/math

Signature Description
util/polar_coordinate(point) convert from Cartesian to Polar coordinates.
util/spherical_coordinate(point) convert from Cartesian to Spherical coordinates (used in mathematics).
util/lerp(v1, v2, amt) linear interpolate the vector v1 to v2.
util/fibseq(from, to) generate a Fibonacci sequence.

### util/set

Signature Description
util/set/hashset(lt, ...) model the mathematical set, backed by a hash table.
util/set/hashset_add(set, elem, ...) add an element to a `hashset`.
util/set/hashset_has(set, elem, ...) return `true` if a `hashset` contains the specified element.
util/set/hashset_del(set, elem, ...) del an element from a `hashset`.
util/set/hashset_len(set) return the length of the elements in a `hashset`.
util/set/hashset_elems(set) returns a list containing all elements in a `hashset`. No guarantees to the order.

### util/map

Signature Description
util/map/hashmap(kv_lt, ...) map keys to values.
util/map/hashmap_put(map, key, value, ...) put a key/value pair to a `hashmap`.
util/map/hashmap_get(map, key, ...) get the value of the specified key from a `hashmap`.
util/map/hashmap_del(map, key, ...) delete the mapping for the specified key from a `hashmap` if present.
util/map/hashmap_len(map) return the length of a `hashmap`.
util/map/hashmap_keys(map) return a list containing all keys in a `hashmap`.
util/map/hashmap_values(map) return a list containing all values in a `hashmap`.
util/map/hashmap_entries(map) return a list containing all `[key, value]`s in a `hashmap`.

## Matrix

Signature Description
matrix/m_determinant(m) calculate a determinant of a square matrix.
matrix/m_mirror(v) generate a transformation matrix which can pass into `multmatrix` to mirror the child element on a plane through the origin.
matrix/m_rotation(a, v) Generate a transformation matrix which can pass into `multmatrix` to rotate the child element about the axis of the coordinate system or around an arbitrary axis.
matrix/m_scaling(s) generate a transformation matrix which can pass into `multmatrix` to scale its child elements using the specified vector.
matrix/m_shearing([sx, sy, sz]) generate a transformation matrix which can pass into `multmatrix` to shear all child elements along the X-axis, Y-axis, or Z-axis in 3D.
matrix/m_translation(v) generate a transformation matrix which can pass into multmatrix to translates (moves) its child elements along the specified vector.
maxtrix/m_transpose(m) transpose a matrix.
matrix/m_replace(m, i, j, value) replace the aᵢⱼ element of a matrix.

## Point Transformation

Signature Description
ptf/ptf_bend(size, point, radius, angle) transform a point inside a rectangle to a point of an arc.
ptf/ptf_circle(size, point) transform a point inside a rectangle to a point inside a circle.
ptf/ptf_ring(size, point, radius[, angle, twist]) transform a point inside a rectangle to a point of a ring.
ptf/ptf_rotate(point, a, v) rotate a point a degrees around the axis of the coordinate system or an arbitrary axis.
ptf/ptf_sphere(size, point, radius[, angle]) transform a point inside a rectangle to a point of a sphere.
ptf/ptf_torus(size, point, radius[, angle, twist]) transform a point inside a rectangle to a point of a torus.
ptf/ptf_x_twist(size, point, angle) twist a point along the x-axis.
ptf/ptf_y_twist(size, point, angle) twist a point along the y-axis.

## Triangle

Signature Description
triangle/tri_circumcenter(shape_pts) return the circumcenter of a triangle.
triangle/tri_incenter(shape_pts) return the incenter of a triangle.
triangle/tri_ear_clipping(shape_pts[, ret, ...]) triangulation by ear clipping.
triangle/tri_delaunay(points[, ret]) Join a set of points to make a Delaunay triangulation.
triangle/tri_delaunay_indices(d) return triangle indices from a delaunay object.
triangle/tri_delaunay_shapes(d) return triangle shapes from a delaunay object.
triangle/tri_delaunay_voronoi(d) return Voronoi cells from a delaunay object.
triangle/tri_subdivide(shape_pts[, n]) subdivide a triangle `n` times.

## Turtle

Signature Description
turtle/footprints2(cmds[, start]) drive a turtle with `["forward", length]` or `["turn", angle]`. This function is intended to use a turtle to imitate freehand drawing.
turtle/footprints3(cmds[, start]) a 3D verion of `footprint2`.
turtle/lsystem2(axiom, rules, n, angle[, leng, heading, ...]) 2D implementation of L-system.
turtle/lsystem3(axiom, rules, n, angle[, leng, heading, ...]) 3D implementation of L-system.
turtle/t2d(t, cmd, point, angle, leng) an implementation of Turtle Graphics.
turtle/t3d(t, cmd, point, unit_vectors, leng, angle) a 3D version of `t2d`.

## Voxel

Signature Description
voxel/vx_ascii(char[, center, invert]) generate 8x8 voxel points of printable ASCII characters (codes 32dec to 126dec).
voxel/vx_bezier(p1, p2, p3, p4) return voxel-by-voxel points of Bézier Curve.
voxel/vx_circle(radius[, filled]) return points that can be used to draw a voxel-style circle.
voxel/vx_contour(points[, sorted]) return the contour which encircles the area.
voxel/vx_curve(points[, tightness]) the curve is drawn only from the 2nd control point to the second-last control point.
voxel/vx_cylinder(r, h[, filled, thickness]) return points that can be used to draw a voxel-style cylinder.
voxel/vx_difference(points1, points2) create a difference of two lists of points.
voxel/vx_from(binaries[, center, invert]) given a list of 0s and 1s that represent a black-and-white image. This function translates them into voxel points.
voxel/vx_gray(levels[, center, invert, normalize]) given a list of numbers (0 ~ 255) that represent a gray image. This function translates them into a list of `[x, y, level]`s.
voxel/vx_intersection(points1, points2) create an intersection of two lists of points.
voxel/vx_line(p1, p2) given two points. it returns points that can be used to draw a voxel-style line.
voxel/vx_polygon(points[, filled]) return points that can be used to draw a voxel-style polygon.
voxel/vx_polyline(points) return points that can be used to draw a voxel-style polyline.
voxel/vx_sphere(radius[, filled, thickness]) return points that can be used to draw a voxel-style sphere.
voxel/vx_union(points1, points2) create a union of two lists of points.

## Part

Signature Description
part/cone(radius[, length, spacing, angle, void, ends]) create a cone for rotatable models.
part/connector_peg(radius, height[, spacing, void, ends]) create a connector peg.
part/joint_T(shaft_r, shaft_h, t_leng, thickness,[ spacing, center]) create a joint_T for rotatable models.

## Surface

Signature Description
surface/sf_bend(levels, radius, thickness, depth[, angle, invert]) bend a photo.
surface/sf_ring(levels, radius, thickness, depth[, angle, twist, invert]) turn a photo into a ring.
surface/sf_solidify(surface1, surface2[, slicing]) solidify two square surfaces.
surface/sf_sphere(levels, radius, thickness, depth[, angle, invert)] map a photo onto a sphere.
surface/sf_square(levels, thickness, depth[, x_twist, y_twist, invert]) turn a photo into a twistable square.
surface/sf_torus(levels, radius, thickness, depth[, angle, twist, invert]) turn a photo to a torus.
surface/sf_curve(levels, curve_path, ...) curve a photo.
surface/sf_splines(ctrl_pts, row_spline, column_spline) generalized-spline surface.
surface/sf_thicken(points, thickness, ...) thicken a surface.
surface/sf_solidifyT(points1, points2, triangles) solidify two surfaces with triangular mesh.
surface/sf_thickenT(points, thickness, ...) thicken a surface with triangular mesh.

## Noise

Signature Description
noise/nz_cell(points, p[, dist]) an implementation of Worley noise.
noise/nz_perlin1(x[, seed]) return the 1D Perlin noise value at the x coordinate.
noise/nz_perlin1s(xs[, seed]) return 1D Perlin noise values at x coordinates.
noise/nz_perlin2(x, y[, seed]) return the 2D Perlin noise value at the (x, y) coordinate.
noise/nz_perlin2s(points[, seed]) return 2D Perlin noise values at (x, y) coordinates.
noise/nz_perlin3(x, y, z[, seed]) return the 3D Perlin noise value at the (x, y, z) coordinate.
noise/nz_perlin3s(points[, seed]) return 3D Perlin noise values at (x, y, z) coordinates.
noise/nz_worley2(x, y[, seed, grid_w, dist]) return the 2D Worley noise value at the (x, y) coordinate.
noise/nz_worley2s(points[, seed, grid_w, dist]) return 2D Worley noise values at (x, y) coordinates.
noise/nz_worley3(x, y, z[, seed, grid_w, dist]) return the 3D Worley noise value at the (x, y, z) coordinate.
noise/nz_worley3s(points[, seed, grid_w, dist]) return 3D Worley noise values at (x, y, z) coordinates.

## Voronoi

Signature Description
voronoi/vrn2_cells_from(points) create cell shapes of Voronoi from a list of points.
voronoi/vrn2_cells_space(size, grid_w[, seed]) create cell shapes of Voronoi in the first quadrant.
voronoi/vrn2_from(points[, spacing, ...]) create a Voronoi from a list of points.
voronoi/vrn2_space(size, grid_w[, seed, spacing, ...]) create a Voronoi in the first quadrant.
voronoi/vrn3_from(points[, spacing]) create a 3D version of Voronoi.
voronoi/vrn3_space(size, grid_w[, seed, spacing]) create a Voronoi in the first octant.

## Maze

Signature Description
maze/mz_square([rows, columns, start, init_cells, x_wrapping, y_wrapping, seed]) return cell data of a square maze.
maze/mz_square_get(cell, query) a helper for getting data from a square-maze cell.
maze/mz_squarewalls(cells, cell_width[, left_border, bottom_border]) a helper for creating square wall data from maze cells.
maze/mz_hexwalls(cells, cell_radius[, left_border, bottom_border]) a helper for creating hex wall data from maze cells.
maze/mz_square_initialize(rows, columns, mask) a helper for initializing cell data of a maze.
maze/mz_hamiltonian(rows, columns[, start, seed]) create a hamiltonian path from a maze.
maze/mz_theta_cells(rows, beginning_number[, start, seed]) return cell data of a theta maze.
maze/mz_theta(rings, beginning_number[, start, seed]) return cell data of a theta maze.
maze/mz_tiles(cells[, left_border, bottom_border]) turn maze cells into tiles.

## Polyhedra

Signature Description
polyhedra/polar_zonohedra(n[, theta]) create a polar zonohedra.

## Point Picking

Signature Description
pp/pp_disk(radius, value_count[, seed]) generate random points over a disk.
pp/pp_sphere(radius, value_count[, seed]) pick random points on the surface of a sphere.
pp/pp_poisson2(size, r[, start, k, seed]) perform poisson sampling over a rectangle area.
pp/pp_poisson3(size, r[, start, k, seed]) perform poisson sampling over a cube space.

Reduce the burden of mathematics when playing OpenSCAD

## Releases 6

3.3 RELEASE Latest
Oct 11, 2022

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