# Module `OCADml.Path2`

2d path generation (including arcs and basic shapes), manipulation (including offset and roundovers (see `Round`), and measurement.

`type t = V2.t list`

## Construction / Conversion

`val of_list : V2.t list -> t`

`of_list l`

Construct a path from a list of points `l` (no-op).

`val of_seq : V2.t Stdlib.Seq.t -> t`

`of_seq s`

Construct a path from a sequence of points `s`.

`val of_array : V2.t array -> t`

`of_array a`

Construct a path from an array of points `a`.

`val to_list : t -> V2.t list`

`to_list t`

Convert the path `t` to a list of points (no-op).

`val to_seq : t -> V2.t Stdlib.Seq.t`

`to_seq t`

Convert the path `t` to a sequence of points.

`val to_array : t -> V2.t array`

`to_array t`

Convert the path `t` to a array of points.

## General Path Utilities

`val length : ?closed:bool -> t -> float`

`length ?closed path`

Calculate the length (total travel distance) of the `path`. If `closed` is `true`, include the distance between the endpoints (default = `false`).

`val cummulative_length : ?closed:bool -> t -> float list`

`cummulative_length ?closed path`

Calculate the cummulative length (distance travelled by each point) along the `path`. If `closed` is `true`, include the distance between the endpoints (default = `false`).

`val to_continuous : ?closed:bool -> t -> float -> V2.t`

`to_continuous path`

Return a continuous function from values over the range of `0.` to `1.` to positions along `path` (like a bezier function), treated as open (`closed = false`) by default.

`val resample : freq:[< `N of int | `Spacing of float ] -> t -> t`

`resample ~freq path`

Resample `path` with the given `freq`uency (either a flat number of points, or a target point spacing). Note that the only points guaranteed to appear in the output are the start and end points of `path`. For upsampling that preserves input points, see `subdivide`.

```val subdivide : ?closed:bool -> freq: [ `N of int * [ `ByLen | `BySeg ] | `RoughN of int * [ `ByLen | `BySeg ] | `Refine of int * [ `ByLen | `BySeg ] | `RoughRefine of int * [ `ByLen | `BySeg ] | `Spacing of float ] -> V2.t list -> V2.t list```

`subdivide ?closed ~freq path`

Subdivides `path` with given `freq`uency, including each of the original points in the output (unlike `resample`). This can be a flat number of points provided directly with ``N (n, by)`, or as a multiple of number of points in `path` with ``Refine (factor, by)`. The strategy for distribution of points for these count based methods is set with the second parameter `by`, which can be either ``BySeg` (same number of additional points per segment) and ``ByLen` (segments gain new points proportional to their length). Alternatively, a maximum ``Spacing dist` can be specified instead. The ``Rough` point sampling variants will favour sampling uniformity, at the expense of not adhering exactly to the requested point count.

```val cut : ?closed:bool -> distances:[ `Abs of float list | `Rel of float list ] -> t -> t list```

`cut ?closed ~distances path`

Cut `path` at a list of increasing `distances` (``Abs`olute or ``Rel`ative) along it from the start. If `closed` is `true`, the segment between the end and beginning of `path` will be considered, and the first point will be the last of the second path returned. Negative ``Abs distance` will start from the end to find the split point. Raises `Invalid_argument` if `distance` is an endpoint, or further than the end of the `path`.

```val split : ?closed:bool -> distance:[ `Abs of float | `Rel of float ] -> t -> t * t```

`split ?closed ~distance path`

Split `path` into two at the position `distance` (``Abs`olute or ``Rel`ative) along `path` from the start. Otherwise the behaviour is the same as `cut`.

```val noncollinear_triple : ?eps:float -> t -> ((int * int * int) * (V2.t * V2.t * V2.t)) option```

`noncollinear_triple ?eps path`

Returns a pair of triples of non-collinear indices and the corresponding points from `path` (if the path is not completely collinear). Two well separated points are selected, and the third point is the furthest off the line drawn by the first two points.

`val is_collinear : ?eps:float -> t -> bool`

`is_collinear ?eps path`

Returns `true` if all points in `path` are collinear (fall within `eps` distance of the same line).

`val prune_collinear : ?closed:bool -> t -> t`

`prune_collinear path`

Remove collinear points from `path`. If `closed` is `true` the last point can be pruned as collinear with the first. The first point is never pruned.

```val deduplicate_consecutive : ?closed:bool -> ?keep:[ `First | `Last | `FirstAndEnds | `LastAndEnds ] -> ?eq:(V2.t -> V2.t -> bool) -> t -> t```

`deduplicate_consecutive ?closed ?keep ?eq path`

Remove consecutive duplicate points as determined by the (approximate) equality function `eq` (`V.approx ~eps:1e-9` by default) from `path`. By default `keep` is ``First`, which includes the first point of each run of duplicates in the output. This can be instead be set to `keep` the ``Last`, or to ``FirstAndEnds` or ``LastAndEnds`, which follow their respective simpler rules with the caveat of preserving the endpoints (first and last points) of the path. The path is treated as open (`closed = false`) by default, if `closed` is `true` the last point of the path may be dropped (even if `keep` is ``FirstAndEnds | `LastAndEnds`).

`val deriv : ?closed:bool -> ?h:float -> t -> t`

`deriv ?closed ?h path`

Computes a numerical derivative of `path`, with `h` (default `1.`) giving the step size of the sampling of `path`, so that the derivative can be scaled correctly. Setting `closed` to `true` will include computation of the derivative between the last and first point of the `path` (default `false`).

`val deriv_nonuniform : ?closed:bool -> h:float list -> t -> t`

`deriv_nonuniform ?closed ?h path`

Computes a numerical derivative of `path`, with `h` giving the non-uniform step sizes of the sampling of `path`, so that the derivative can be scaled correctly. Setting `closed` to `true` will include computation of the derivative between the last and first point of the `path` (default `false`). As `h` provides scaling factors for each segment of the path, it must have a length of one less than `path` if it's unclosed, and the same length if `closed` is `true`.

`val tangents : ?uniform:bool -> ?closed:bool -> t -> t`

`tangents ?uniform ?closed path`

Compute tangent unit vectors of `path`. Set `closed` to `true` to indicate that tangents should include between the end and beginning of the path (default = `false`). Sampling of `path` is assumed to be `uniform` unless the parameter is set to `false`, in which case the derivatives will be adjusted to correct for non-uniform sampling of points.

```val continuous_closest_point : ?closed:bool -> ?n_steps:int -> ?max_err:float -> (float -> V2.t) -> V2.t -> float```

`continuous_closest_point ?closed ?n_steps ?max_err f p`

Find the closest position (from `0.` to `1.`) along the path function `f` to the point `p`.

• `n_steps` sets the granularity of search at each stage.
• `max_err` the maximum distance the solution can be from the target `p`
`val segment : ?closed:bool -> t -> V2.line list`

`segment ?closed path`

Break `path` into line segments. If `closed` is `true`, include a segment between the last and first points of `path` (default `false`).

`val reindex_polygon : t -> t -> t`

`reindex_polygon reference poly`

Rotate the polygonal (closed / coplanar) path `poly` to optimize its pairwise point association with the `reference` polygon. Paths should have the same clockwise winding direction (not checked / corrected).

`val lerp : t -> t -> float -> t`

`lerp a b u`

Linearly interpolate between the paths `a` and `b`. Raises `Invalid_argument` if the paths are of unequal length.

```val nearby_idxs : ?min_tree_size:int -> ?radius:float -> V2.t list -> V2.t -> int list```

`nearby_idxs ?min_tree_size ?radius path p`

Find the indices of points within `radius` (default = `1e-9`) distance from the target point `p` in `path`. Match indices will be returned in arbitrary order (unsorted). When `path` is provided (eagerly on partial application), the length will be checked and a function to perform the search will be generated. If `path` is shorter than `min_tree_size`, it will be a simple direct search otherwise a `BallTree2.t` will be constructed. Thus, if you plan to search for more than one target point, take care to apply this function in two steps to avoid repeated length checks and closure/tree generations.

```val nearby_points : ?min_tree_size:int -> ?radius:float -> V2.t list -> V2.t -> V2.t list```

`nearby_points ?min_tree_size ?radius path`

Find the points within `radius` (default = `1e-9`) distance from the target point `p` in `path`. Matched points will be returned in arbitrary order (unsorted). When `path` is provided (eagerly on partial application), the length will be checked and a function to perform the search will be generated. If `path` is shorter than `min_tree_size`, it will be a simple direct search otherwise a `BallTree2.t` will be constructed. Thus, if you plan to search for more than one target point, take care to apply this function in two steps to avoid repeated length checks and closure/tree generations.

```val closest_tangent : ?closed:bool -> ?offset:V2.t -> line:V2.line -> t -> int * V2.line```

`closest_tangent ?closed ?offset ~line t`

Find the tangent segment (and its index) on the curved path `t` closest to `line` after `offset` (default = `V2.zero`) is applied to the points of `t` (can be used to centre the path relative to `line` to help in choosing the desired tangent).

## Creation and 2d-3d conversion

`val of_tups : (float * float) list -> t`

`of_tups ps`

Create a 2d path from a list of xy coordinate tuples.

`val of_path3 : ?plane:Plane.t -> V3.t list -> t`

`of_path3 p`

Project the 3d path `p` onto the given `plane` (default = `Plane.xy`).

`val to_path3 : ?plane:Plane.t -> t -> V3.t list`

`to_path3 t`

Lift the 2d path `p` onto the given `plane` (default = `Plane.xy`).

`val lift : Plane.t -> t -> V3.t list`

`lift plane t`

Lift the 2d path `t` onto the 3d `plane`.

## Basic Shapes

`val circle : ?fn:int -> ?fa:float -> ?fs:float -> float -> t`

`circle ?fn ?fa ?fs r`

Create a circular path of radius `r`.

`val square : ?center:bool -> V2.t -> t`

`square ?center dims`

Create a rectangular path with xy `dims` (e.g. width and height). If `center` is `true` then the path will be centred around the origin (default = `false`).

`val ellipse : ?fn:int -> ?fa:float -> ?fs:float -> V2.t -> t`

`ellipse ?fn ?fa ?fs radii`

Draw an ellipse with xy `radii`. The greater of the two radii is used for fragment/resolution calculation.

`val star : r1:float -> r2:float -> int -> t`

`star ~r1 ~r2 n`

Draw an `n` pointed star with inner radius `r1` and outer radius `r2`.

## Drawing Arcs and Splines

```val arc : ?rev:bool -> ?fn:int -> ?fa:float -> ?fs:float -> ?wedge:bool -> centre:V2.t -> radius:float -> start:float -> float -> t```

`arc ?rev ?fn ?fa ?fs ?wedge ~centre ~radius ~start a`

Draw an arc of `a` radians with `radius` around the point `centre`, beginning with the angle `start`. If `wedge` is `true`, `centre` will be included as the last point of the returned path (default = `false`). If `rev` is `true`, the arc will end at `start`, rather than begin there.

```val arc_about_centre : ?rev:bool -> ?fn:int -> ?fa:float -> ?fs:float -> ?dir:[ `CW | `CCW ] -> ?wedge:bool -> centre:V2.t -> V2.t -> V2.t -> t```

`arc_about_centre ?rev ?fn ?fa ?fs ?dir ?wedge ~centre p1 p2`

Draw an arc between the points `p1` and `p2`, about `centre`. `dir` can be provided to enforce clockwise or counter-clockwise winding direction. By default, the direction is computed automatically, though if `centre`, `p1`, and `p2` do not form a valid triangle (they're collinear), an `Invalid_argument` exception will be raised if `dir` is not provided.

```val arc_through : ?rev:bool -> ?fn:int -> ?fa:float -> ?fs:float -> ?wedge:bool -> V2.t -> V2.t -> V2.t -> t```

`arc_through ?rev ?fn ?fa ?fs ?wedge p1 p2 p3`

Draw an arc through the points `p1`, `p2`, and `p3`. If the points do not form a valid triangle (they're collinear), an `Invalid_argument` exception will be raised.

`val cubic_spline : ?boundary:CubicSpline.boundary -> fn:int -> t -> t`

`cubic_spline ?boundary ~fn ps`

Calculate a cubic spline with the given `boundary` condition (defaults to ``Natural`) for the 2-dimensional control points `ps`, and immediately interpolate a path of `fn` points along it. See the `CubicSpline` module for more details.

## Roundovers

Outline offsets with optional rounding/chamfering as found in OpenSCADs 2d sub-system, as well as specification and application of non-offseting roundovers (circular, chamfer, and bezier (continuous curvature)) to 2d paths.

Based on the BOSL2 rounding module.

```val offset : ?fn:int -> ?fs:float -> ?fa:float -> ?closed:bool -> ?check_valid:[ `Quality of int | `No ] -> ?mode:[< `Chamfer | `Delta | `Radius Delta ] -> float -> t -> t```

`offset ?fn ?fs ?fa ?closed ?check_valid ?mode d path`

Offset a 2d `path` (treated as `closed` by default) by the specified distance `d`.The `mode` governs how `d` is used to create the new corners.

• ``Delta` will create a new outline whose sides are a fixed distance `d` (+ve out, -ve in) from the original outline (this is the default behaviour).
• ``Chamfer` fixed distance offset by `d` as with delta, but with corners chamfered.
• ``Radius` creates a new outline as if a circle of some radius `d` is rotated around the exterior (`d > 0`) or interior (`d < 0`) original outline. `fn`, `fs`, and `fa` parameters govern the number of points that will be used for these arcs (they are ignored for delta and chamfer modes).
• The `check_valid` default of ``Quality 1` will check the validity of shifted line segments by checking whether their ends and `n` additional points spaced throughout are far enough from the original path. If there are no points that have been offset by the target `d`, a `Failure` exception will be raised. Checking can be turned off by setting this to ``No`.
`module Round : sig ... end`

Configuration module with types and helpers for specifying path roundovers.

```val roundover : ?fn:int -> ?fa:float -> ?fs:float -> ?overrun:[ `Fail | `Warn | `Fix | `NoCheck ] -> Round.t -> V2.t list```

`roundover ?fn ?fa ?fs ?overrun path_spec`

Apply the roundover specifictions in `path_spec` on the bundled path/shape, with quality set by the `fn`, `fa`, and `fs` parameters. Collinear points are ignored (included in output without roundover applied).

When `overrun` is set to ``Fail` (as it is by default) this function will raise `Failure` if computed joint distances would lead to point insertion that causes the path to reverse/double back on itself. Alternatively:

• ``Warn` will print the detected overruns to `stdout` rather than raising `Failure` (useful for debuggin)
• ``Fix` will automatically reduce the corner joints involved in each overrun proportional to their lengths.
• ``NoCheck` skips overrun detection

## Geometry

`val clockwise_sign : ?eps:float -> t -> float`

`clockwise_sign path`

Returns the rotational ordering of `path` as a signed float, `-1.` for clockwise, and `1.` for counter-clockwise. If all points are collinear (within the tolerance of `eps`), `0.` is returned.

`val is_clockwise : t -> bool`

`is_clockwise path`

Returns `true` if the rotational ordering of `path` is clockwise.

`val self_intersections : ?eps:float -> ?closed:bool -> t -> t`

`self_intersection ?eps ?closed path`

Find the points at which `path` intersects itself (within the tolerance of `eps`). If `closed` is `true`, a line segment between the last and first points will be considered (default = `false`).

`val is_simple : ?eps:float -> ?closed:bool -> t -> bool`

`is_simple ?eps ?closed path`

Return `true` if `path` is simple, e.g. contains no (paralell) reversals or self-intersections (within the tolerance `eps`). If `closed` is `true`, a line segment between the last and first points will be considered (default = `false`).

`val bbox : t -> Gg.Box2.t`

`bbox t`

Compute the 2d bounding box of the path `t`.

`val centroid : ?eps:float -> t -> V2.t`

`centroid ?eps t`

Compute the centroid of the path `t`. If `t` is collinear or self-intersecting (within `eps` tolerance), an `Invalid_argument` exception is raised.

`val area : ?signed:bool -> t -> float`

`area ?signed t`

Compute the signed or unsigned area of the path `t` (unsigned by default).

```val point_inside : ?eps:float -> ?nonzero:bool -> t -> V2.t -> [> `Inside | `OnBorder | `Outside ]```

`point_inside ?eps ?nonzero t p`

Determine whether the point `p` is inside, on the border of, or outside the closed path `t` (may be non-simple / contain self-intersections). If `nonzero` is `true`, the Nonzero rule is followed, wherein a point is considered inside the polygon formed by `t` regardless of the number of times the containing regions overlap, by default this is `false`, and the Even-Odd rule is followed (as with in OpenSCAD).

`val hull : ?all:bool -> t -> t`

`hull ?all t`

Compute the convex hull polygon that encloses the points in the path `t`. When `all` is `true`, the output path will include the non-vertex points resting on the edges of the hull (default = `false`).

`val triangulate : ?eps:float -> t -> V2.t list list`

`triangulate ?eps t`

Break the polygon `t` into a list of triangles. If provided, `eps` is used for duplicate point and collinearity checks.

## Path Matching / Vertex Association

Point duplicating strategies for associating vertices between incommensurate closed polygonal paths/profiles. Primarily for use in conjunction with `Mesh.skin` and `Mesh.morphing_sweep`, where commensurate profiles are required to draw edges between.

Ported from the skin module of the BOSL2 OpenSCAD library.

`val distance_match : V2.t list -> V2.t list -> V2.t list * V2.t list`

`distance_match a b`

If the closed polygonal paths `a` and `b` have incommensurate lengths, points on the smaller path are duplicated and the larger path is shifted (list rotated) in such a way that the total length of the edges between the associated vertices (same index/position) is minimized. The replacement paths, now having the same lengths, are returned as a pair (with the same order). This algorithm generally produces a good result when both `a` and `b` are discrete profiles with a small number of vertices.

This is computationally intensive ( O(N3) ), if the profiles are already known to be lined up, with their zeroth indices corresponding, then `aligned_distance_match` provides a ( O(N2) ) solution.

`val aligned_distance_match : V2.t list -> V2.t list -> V2.t list * V2.t list`

`aligned_distance_match a b`

Like `distance_match`, but the paths `a` and `b` are assumed to already be "lined up", with the zeroth indices in each corresponding to one another.

`val tangent_match : V2.t list -> V2.t list -> V2.t list * V2.t list`

`tangent_match a b`

If the closed polygonal paths `a` and `b` have incommensurate lengths, points on the larger (ideally convex, curved) path are grouped by association of their tangents with the edges of the smaller (ideally discrete) polygonal path. The points of the smaller path are then duplicated to associate with their corresponding spans of tangents on the curve, and the larger path is rotated to line up the indices. The profiles, now having the same length are returned as a pair in the order that they were applied returned.

This algorithm generally produces good results when connecting a discrete polygon to a convex finely sampled curve. It may fail if the curved profile is non-convex, or doesn't have enough points to distinguish all of the tangent points from each other.

## Basic Transfomations

`val translate : V2.t -> t -> t`
`val xtrans : float -> t -> t`
`val ytrans : float -> t -> t`
`val rotate : ?about:V2.t -> float -> t -> t`
`val zrot : ?about:V2.t -> float -> t -> t`
`val affine : Affine2.t -> t -> t`
`val affine3 : Affine3.t -> t -> V3.t list`
`val quaternion : ?about:V3.t -> Quaternion.t -> t -> V3.t list`
`val axis_rotate : ?about:V3.t -> V3.t -> float -> t -> V3.t list`
`val scale : V2.t -> t -> t`
`val xscale : float -> t -> t`
`val yscale : float -> t -> t`
`val mirror : V2.t -> t -> t`