From 20d10a93e5becb41d1145f9d35136782365b0ba4 Mon Sep 17 00:00:00 2001
From: Sébastien Dailly <sebastien@chimrod.com>
Date: Thu, 17 Dec 2020 13:56:00 +0100
Subject: Refactor

---
 shapes/bezier.ml      | 203 ++++++++++++++++++++++++++++++++++++++++++++++++++
 shapes/bezier.mli     |  40 ++++++++++
 shapes/bspline.ml     | 149 ++++++++++++++++++++++++++++++++++++
 shapes/bspline.mli    |  24 ++++++
 shapes/dd_splines.pdf | Bin 0 -> 184248 bytes
 shapes/dune           |   7 ++
 6 files changed, 423 insertions(+)
 create mode 100755 shapes/bezier.ml
 create mode 100755 shapes/bezier.mli
 create mode 100755 shapes/bspline.ml
 create mode 100755 shapes/bspline.mli
 create mode 100755 shapes/dd_splines.pdf
 create mode 100755 shapes/dune

(limited to 'shapes')

diff --git a/shapes/bezier.ml b/shapes/bezier.ml
new file mode 100755
index 0000000..bf7aaaa
--- /dev/null
+++ b/shapes/bezier.ml
@@ -0,0 +1,203 @@
+(**
+
+   Bezier curve
+*)
+
+module Utils = Tools.Utils
+
+type quadratic =
+  { p0:Gg.v2     (* The starting point *)
+  ; p1:Gg.v2     (* The end point *)
+  ; ctrl:Gg.v2 } (* The control point *)
+
+
+type t =
+  { p0:Gg.v2      (* The starting point *)
+  ; p1:Gg.v2      (* The end point *)
+  ; ctrl0:Gg.v2   (* The control point *)
+  ; ctrl1:Gg.v2 } (* The control point *)
+
+
+(**
+   Build a control point for a quadratic curve for passing throuht 3 points.
+   taken from https://xuhehuan.com/2608.html
+
+
+   also look to https://pomax.github.io/bezierinfo/#pointcurves
+*)
+let three_points_quadratic
+  : Gg.v2 -> Gg.v2 -> Gg.v2 -> quadratic
+  = fun p0 c1 p1 ->
+
+    let open Gg.V2 in
+
+    let vect_1 = p0 - c1
+    and vect_2 = p1 - c1 in
+    let norm1 = norm vect_1
+    and norm2 = norm vect_2 in
+    let v = (Float.sqrt (norm1 *. norm2)) /. 2. in
+
+    let ctrl = c1 - v * (( vect_1 / norm1) + (vect_2 / norm2)) in
+    {p0; p1; ctrl}
+
+(**
+
+   Convert a cubic bezier curve into a quadratic one
+
+*)
+let quadratic_to_cubic
+  : quadratic -> t
+  = fun {p0; p1; ctrl} ->
+
+    let coef = 2. /. 3. in
+
+    let open Gg.V2 in
+    { p0
+    ; p1
+    ; ctrl0 = mix p0 ctrl coef
+    ; ctrl1 = mix p1 ctrl coef }
+
+
+
+let abc_ratio
+  : int -> float -> float
+  = fun n t ->
+    let n' = Float.of_int n in
+    let bottom = (Float.pow t n') +. (Float.pow  (1. -. t) n') in
+    let top = bottom -. 1. in
+    Float.abs (top /. bottom)
+
+let half_cubic_ratio = abc_ratio 3 0.5
+
+exception Not_found
+
+(**
+
+   https://pomax.github.io/bezierinfo/#pointcurves
+
+*)
+let three_points_cubic
+  : float -> Gg.v2 -> Gg.v2 -> Gg.v2 -> t
+  = fun f p0 p1 p2 ->
+
+    let open Gg.V2 in
+
+    let c = half ( p0 + p2) in
+    let a = p1 + ((p1 - c) / half_cubic_ratio) in
+
+    let vect1_0 = p1 - p0 in
+    let vect2_0 = p2 - p0 in
+
+    let d1 = norm vect1_0
+    and d2 = norm (p2 - p1) in
+    let t = d1 /. (d1 +. d2) in
+
+    let angle_1_0 = angle vect1_0
+    and angle_2_0 = angle vect2_0 in
+
+    (* get our e1-e2 distances *)
+    let angle = mod_float
+        (Gg.Float.two_pi
+         +. angle_2_0
+         -. angle_1_0)
+        Gg.Float.two_pi in
+
+    let distance = (norm vect2_0) *. f in
+
+    let bc =
+      if angle < 0. || angle > Gg.Float.pi then
+        Float.(neg distance)
+      else
+        distance in
+    let de1  = t *. bc
+    and de2 = (1. -. t) *. bc in
+
+    (* get the circle-aligned slope as normalized dx/dy *)
+    let center = Utils.center p0 p1 p2 in
+    match center with
+    | None -> raise Not_found
+    | Some center ->
+      let t' = p1 - center in
+      let tangent0 = v
+          ((x p1) -. (y t'))
+          ((y p1) +. (x t'))
+      and tangent1 = v
+          ((x p1) +. (y t'))
+          ((y p1) -. (x t')) in
+
+      let d = unit (tangent1 - tangent0) in
+
+      (* then set up an e1 and e2 parallel to the baseline *)
+      let e1 = p1 + de1 * d
+      and e2 = p1 - de2 * d in
+
+      (* then use those e1/e2 to derive the new hull coordinates *)
+      let v1 = a + (e1 - a) / (1. -. t)
+      and v2 = a + (e2 - a) / t in
+
+      let ctrl0 = p0 + (v1 - p0) / t
+      and ctrl1 = p2 + (v2 -p2) / (1. -. t) in
+
+      {p0; p1 = p2; ctrl0; ctrl1}
+
+(** Split a bezier curve in two at a given position *)
+let slice
+  : float -> t -> t * t
+  = fun t {p0; p1; ctrl0; ctrl1} ->
+
+    let mix p1 p2 = Gg.V2.mix p1 p2 t in
+
+    let p12 = mix p0 ctrl0
+    and p23 = mix ctrl0 ctrl1
+    and p34 = mix ctrl1 p1 in
+
+    let p123 = mix p12 p23
+    and p234 = mix p23 p34 in
+
+    let p1234 = mix p123 p234 in
+
+    ( { p0
+      ; ctrl0 = p12
+      ; ctrl1 = p123
+      ; p1 = p1234 }
+    , { p0 = p1234
+      ; ctrl0 = p234
+      ; ctrl1 = p34
+      ; p1 } )
+
+
+let get_closest_point
+  : Gg.v2 -> t -> float * Gg.v2
+  = fun point t ->
+
+    let rec f min max t =
+
+      (* First devide the curve in two *)
+      let seq_0, seq_1 = slice 0.5 t in
+      let avg = (min +. max) /. 2. in
+
+      let p0 = t.p0
+      and p1 = t.p1
+      and p01 = seq_0.p1 in (* seq_0.p1 = seq_1.p0 *)
+
+      let open Gg.V2 in
+      let center0 = mix p0 p01 0.5
+      and center1 = mix p01 p1 0.5 in
+
+      if Tools.Utils.equal_point 0.001 p0 p1 then
+        avg, p01
+      else if (norm (point - center0)) < (norm (point - center1)) then
+        f min avg seq_0
+      else
+        f avg max seq_1
+
+    in f 0. 1. t
+
+let reverse
+  : t -> t
+  = fun bezier ->
+    {
+      p0 = bezier.p1
+    ; p1 = bezier.p0
+    ; ctrl0 = bezier.ctrl1
+    ; ctrl1 = bezier.ctrl0 }
diff --git a/shapes/bezier.mli b/shapes/bezier.mli
new file mode 100755
index 0000000..2f5bbcf
--- /dev/null
+++ b/shapes/bezier.mli
@@ -0,0 +1,40 @@
+type t =
+  { p0:Gg.v2      (* The starting point *)
+  ; p1:Gg.v2      (* The end point *)
+  ; ctrl0:Gg.v2   (* The control point *)
+  ; ctrl1:Gg.v2 } (* The control point *)
+
+type quadratic
+
+(**
+   Build a control point for a quadratic curve for passing throuht 3 points.
+   taken from https://xuhehuan.com/2608.html
+
+
+   also look to https://pomax.github.io/bezierinfo/#pointcurves
+*)
+val three_points_quadratic
+  : Gg.v2 -> Gg.v2 -> Gg.v2 -> quadratic
+
+(**
+   Create a curve from three points.
+
+   This is an implementation for
+   https://pomax.github.io/bezierinfo/#pointcurves
+
+*)
+val three_points_cubic
+  : float -> Gg.v2 -> Gg.v2 -> Gg.v2 -> t
+
+val quadratic_to_cubic
+  : quadratic -> t
+
+(** Split a bezier curve in two at a given position *)
+val slice
+  : float -> t -> t * t
+
+(** Return the closest point to the curve by approximation *)
+val get_closest_point
+  : Gg.v2 -> t -> float * Gg.v2
+
+val reverse: t -> t
diff --git a/shapes/bspline.ml b/shapes/bspline.ml
new file mode 100755
index 0000000..bb60227
--- /dev/null
+++ b/shapes/bspline.ml
@@ -0,0 +1,149 @@
+open StdLabels
+
+type err = [`InvalidPath ]
+
+module M = Matrix.MakeMatrix (struct
+
+    type t = Float.t
+
+    let compare a b =
+
+      let v = Float.compare a b in
+      if v = 0 then Matrix.Order.Equal
+      else if v > 0 then Matrix.Order.Greater
+      else Matrix.Order.Less
+
+    let zero = Float.zero
+    let one = Float.one
+
+    let divide = (/.)
+    let multiply = ( *. )
+    let add = (+.)
+    let subtract = (-.)
+    exception NonElt
+
+
+  end)
+
+type t = Gg.v2 list
+
+let from_points
+  : Gg.v2 array -> (Gg.v2 array, [> `InvalidPath]) Result.t
+  = fun points ->
+
+    let n = (Array.length points - 2) in
+
+    if n <= 1 then
+      Result.error `InvalidPath
+    else
+
+      (* Create the initial matrix.
+
+         The matrix is augmented with two additionals columns, which will be
+         populated with the points from the path.
+      *)
+      let arr = Array.init n ~f:(fun line ->
+          Array.init (n +2) ~f:(fun row ->
+              match row - line with
+              | (-1) -> 1.
+              | 0 -> 4.
+              | 1 -> 1.
+              | _ -> 0.
+            )
+        ) in
+      let matrix = M.from_array arr in
+
+      (* Add the points from the augmented matrix *)
+      let points_array = points in
+      for line = 0 to (n -1) do
+
+        let point =
+          if line = 0 then
+            let p0 = points_array.(0)
+            and p1 = points_array.(1) in
+            Gg.V2.(6. * p1 - p0)
+          else if (line + 1) = n  then
+            let p_n_2 = points_array.(n)
+            and p_n_1 = points_array.(n + 1) in
+            Gg.V2.(6. * p_n_2 - p_n_1)
+          else
+            let n' = line + 1 in
+            Gg.V2.(6. * points_array.(n'))
+        in
+        let x = (Gg.V2.x point)
+        and y = (Gg.V2.y point) in
+
+
+        M.set_elt matrix (line + 1, n + 1) x;
+        M.set_elt matrix (line + 1, n + 2) y;
+      done;
+
+      (* Resolve the matrix *)
+      let res' = M.row_reduce matrix in
+
+      (* Extract the result as points *)
+      let _, col_x = M.get_column res' (n + 1)
+      and _, col_y = M.get_column res' (n + 2) in
+
+      (* Build the result *)
+      let res = Array.make (n + 2) (Array.get points_array (n + 1) ) in
+      for i = 1 to n do
+        let point = Gg.V2.v col_x.(i - 1) col_y.(i - 1) in
+        Array.set res i point;
+      done;
+      Array.set res 0 (Array.get points_array 0);
+      Result.ok res
+
+let (let*) = Result.bind
+
+(** Build a continue curve from path
+
+    see https://www.math.ucla.edu/~baker/149.1.02w/handouts/dd_splines.pdf
+*)
+let to_bezier
+  : ?connexion0:Gg.v2 -> ?connexion1:Gg.v2 -> t -> (Bezier.t array, [> `InvalidPath]) Result.t
+  = fun ?connexion0 ?connexion1 points ->
+
+    let points' = match connexion0 with
+      | None -> points
+      | Some pt -> pt::points in
+
+    let arr_points = match connexion1 with
+      | None -> Array.of_list points'
+      | Some pt ->
+        let arr = Array.make (1 + (List.length points')) pt in
+        List.iteri points'
+          ~f:(fun i value -> Array.set arr i value);
+        arr in
+
+    let* bspline_points = from_points arr_points in
+
+    let start = match connexion0 with
+      | None -> 1
+      | Some _ -> 2
+    and end_ = match connexion1 with
+      | None -> (Array.length bspline_points) - 1
+      | Some _ -> (Array.length bspline_points) - 2 in
+
+    let result = Array.init (end_ - start + 1) ~f:(fun i ->
+
+        let i = i + start in
+
+        let prev_b = Array.get bspline_points (i - 1)
+        and bpoint = Array.get bspline_points i
+        and prev_p = Array.get arr_points (i - 1)
+        and point = Array.get arr_points i in
+        let ctrl0 = Gg.V2.(mix prev_b bpoint (1. /. 3.))
+        and ctrl1 = Gg.V2.(mix prev_b bpoint (2. /. 3.)) in
+
+        let bezier =
+          { Bezier.p0 = prev_p
+          ; Bezier.p1 = point
+          ; Bezier.ctrl0
+          ; Bezier.ctrl1 } in
+
+        bezier
+
+      ) in
+    Result.Ok result
+
diff --git a/shapes/bspline.mli b/shapes/bspline.mli
new file mode 100755
index 0000000..a36aa22
--- /dev/null
+++ b/shapes/bspline.mli
@@ -0,0 +1,24 @@
+type t
+
+type err = 
+  [ `InvalidPath (* Too few points in the path for building the curve *)
+  ]
+
+(** Convert a list of points into a beziers curves. 
+
+    At least 4 points are required for building the path.
+
+    [to_bezier ~connexion points] create a list of beziers segments joining all
+    the points together. 
+
+    [connexion0] add a virtual point in the begining for helping to get the
+    appropriate tangent when connecting path together
+
+    [connexion1] does the same at the end
+
+*)
+val to_bezier
+  :  ?connexion0:Gg.v2
+  -> ?connexion1:Gg.v2
+  -> Gg.v2 list 
+  -> (Bezier.t array, [> err]) Result.t
diff --git a/shapes/dd_splines.pdf b/shapes/dd_splines.pdf
new file mode 100755
index 0000000..2618162
Binary files /dev/null and b/shapes/dd_splines.pdf differ
diff --git a/shapes/dune b/shapes/dune
new file mode 100755
index 0000000..d03a217
--- /dev/null
+++ b/shapes/dune
@@ -0,0 +1,7 @@
+(library
+ (name shapes)
+ (libraries 
+   tools
+   matrix
+   )
+ )
-- 
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