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Generation and measurement of 2d bezier curve (and patch/surface) functions. Including `of_path`

, which produces a bezier spline function that passes through all points of the given path.

`type t = float -> V2.t`

A bezier curve function.

`coefs ps`

Compute the bezier algebraic coefficients for the control points `ps`

.

`make ps`

Create bezier function of degree n (`n = List.length ps - 1`

) from the control points `ps`

. The resulting continuous curve advances from the 0th control point at `0.`

, to the final control point at `1.`

.

`curve ?init ?rev ?fn ?endpoint t`

Draw a path of `fn`

segments (default = `16`

) along the bezier curve `t`

.

`init`

can be provided to be prepended to (defaults to an empty list)- If
`rev`

is`true`

, the bezier will be drawn in reverse (default =`false`

) - If
`endpoint`

is`true`

, the last point will be a control point (last or first depending on`rev`

), otherwise it will be off by the step size (`1 / fn`

) (default =`true`

).

```
val length :
?start_u:float ->
?end_u:float ->
?max_deflect:float ->
V2.t list ->
float
```

`length ?start_u ?end_u ?max_deflect ps`

Compute the length along the bezier defined by the control points `ps`

between the fractional positions `start_u`

and `end_u`

(`0.`

to `1.`

by default). This is approximated as the sum of line segments whose midpoints deviate from the bezier by no more than `max_deflect`

.

`patch grid`

Create a bezier patch (curved surface) from an rectangular `grid`

of control points.

`patch_curve ?fn p`

Sample a grid of `fn`

by `fn`

segments describing a curved surface from the bezier patch `p`

. (default `fn = 16`

).

`of_bezpath ?n ps`

Create a bezier function from a series of degree `n`

beziers connected end-to-end defined by the control points `ps`

. The end-point of the first curve is the start-point of the next, and so on. Thus, two cubic-beziers (`n = 3`

, the default) in series is described by seven control points (middle point is shared). The number of curves is `List.length ps - 1 / n`

, if `ps`

cannot be broken by degree `n`

in this way, an `Invalid_argument`

exception will be raised.

```
val bezpath_of_path :
?closed:bool ->
?size:
[ `Abs of float list
| `Rel of float list
| `Flat of [ `Abs of float | `Rel of float ]
| `Mix of [ `Abs of float | `Rel of float ] list ] ->
?tangents:[ `NonUniform | `Uniform | `Tangents of V2.t list ] ->
V2.t list ->
V2.t list
```

`bezpath_of_path ?closed ?uniform ?size ?tangents path`

Create a bezier path (see `of_bezpath`

) which defines a curve that passes through the points of `path`

.

`size`

sets the absolute or relative (as a fraction of segment length) distance that the computed curve can deviate from the input`path`

. Provided either as a flat value for all points, or a list with a value for each*segment*of the`path`

(default =``Flat (`Rel 0.1)`

).`tangents`

provides control over the tangents of the computed curve where it passes through the points of`path`

. Tangents can either be provided by the user with``Tangents l`

, or computed``NonUniform | `Uniform`

(see`Path2.tangents`

).- If
`closed`

is`true`

(default =`false`

), an additional segment between the last and first points of`path`

will be included in the computations. Thus, this impacts the correct lengths of lists provided to the`size`

and`tangents`

parameters (= length of`path`

if`closed`

, one less if not).

`bezpath_curve ?fn ?n ps`

Compute a bezier function from a series of degree `n`

beziers connected end-to-end defined by the control points `ps`

and use to draw a path of `fn`

segments. See `of_bezpath`

for explanation of bezier paths.

```
val of_path :
?closed:bool ->
?size:
[ `Abs of float list
| `Rel of float list
| `Flat of [ `Abs of float | `Rel of float ]
| `Mix of [ `Abs of float | `Rel of float ] list ] ->
?tangents:[ `NonUniform | `Uniform | `Tangents of V2.t list ] ->
V2.t list ->
t
```

`of_path ?closed ?uniform ?size ?tangents path`

Create a bezier function which defines a curve that passes through the points of `path`

.

`size`

sets the absolute or relative (as a fraction of segment length) distance that the computed curve can deviate from the input`path`

. Provided either as a flat value for all points, or a list with a value for each*segment*of the`path`

(default =``Flat (`Rel 0.1)`

).`tangents`

provides control over the tangents of the computed curve where it passes through the points of`path`

. Tangents can either be provided by the user with``Tangents l`

, or computed``NonUniform | `Uniform`

(see`Path2.tangents`

). Default is``NonUniform`

.- If
`closed`

is`true`

(default =`false`

), an additional segment between the last and first points of`path`

will be included in the computations. Thus, this impacts the correct lengths of lists provided to the`size`

and`tangents`

parameters (= length of`path`

if`closed`

, one less if not).

`closest_point ?n ?max_err t p`

Find the fractional position along the bezier `t`

closest to the point `p`

. `t`

is treated as cubic (degree `n = 3`

) by default, subdividing the bezier by `3`

for each degree. Search continues until a position less `max_err`

(default = `0.01`

) distance from `p`

is found.

`deriv ?order ps`

Calculate the `order`

derivative of the bezier defined by the control points `ps`

. The first derivative is taken by default (`order = 1`

). `Invalid_argument`

is raised if `order`

is below `0`

, or the degree of the bezier (`length ps - 1`

) is lower than the `order`

.

`line_intersection ~line pts`

Compute the positions (between `0.`

and `1.`

) along the bezier curve defined by the control points `pts`

that `line`

intersects with.

`val quaternion : ?about:V3.t -> Quaternion.t -> t -> float -> V3.t`

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