BOSL2/transforms.scad

//////////////////////////////////////////////////////////////////////
// LibFile: transforms.scad
//   Functions and modules for translation, rotation, reflection and skewing.
//   To use, add the following lines to the beginning of your file:
//   ```
//   include <BOSL2/std.scad>
//   ```
//////////////////////////////////////////////////////////////////////


//////////////////////////////////////////////////////////////////////
// Section: Translations
//////////////////////////////////////////////////////////////////////


// Function&Module: move()
//
// Usage: As Module
//   move([x], [y], [z]) ...
//   move(v) ...
// Usage: Translate Points
//   pts = move(v, p);
//   pts = move([x], [y], [z], p);
// Usage: Get Translation Matrix
//   mat = move(v);
//
// Description:
//   Translates position by the given amount.
//   * Called as a module, moves/translates all children.
//   * Called as a function with a point in the `p` argument, returns the translated point.
//   * Called as a function with a list of points in the `p` argument, returns the translated list of points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the translated patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the translated VNF.
//   * Called as a function with the `p` argument, returns the translated point or list of points.
//   * Called as a function without a `p` argument, with a 2D offset vector `v`, returns an affine2d translation matrix.
//   * Called as a function without a `p` argument, with a 3D offset vector `v`, returns an affine3d translation matrix.
//
// Arguments:
//   v = An [X,Y,Z] vector to translate by.
//   x = X axis translation.
//   y = Y axis translation.
//   z = Z axis translation.
//   p = Either a point, or a list of points to be translated when used as a function.
//
// Example:
//   #sphere(d=10);
//   move([0,20,30]) sphere(d=10);
//
// Example:
//   #sphere(d=10);
//   move(y=20) sphere(d=10);
//
// Example:
//   #sphere(d=10);
//   move(x=-10, y=-5) sphere(d=10);
//
// Example(2D):
//   path = square([50,30], center=true);
//   #stroke(path, closed=true);
//   stroke(move([10,20],p=path), closed=true);
//
// Example(NORENDER):
//   pt1 = move([0,20,30], p=[15,23,42]);       // Returns: [15, 43, 72]
//   pt2 = move(y=10, p=[15,23,42]);            // Returns: [15, 33, 42]
//   pt3 = move([0,3,1], p=[[1,2,3],[4,5,6]]);  // Returns: [[1,5,4], [4,8,7]]
//   pt4 = move(y=11, p=[[1,2,3],[4,5,6]]);     // Returns: [[1,13,3], [4,16,6]]
//   mat2d = move([2,3]);    // Returns: [[1,0,2],[0,1,3],[0,0,1]]
//   mat3d = move([2,3,4]);  // Returns: [[1,0,0,2],[0,1,0,3],[0,0,1,4],[0,0,0,1]]
module move(v=[0,0,0], x=0, y=0, z=0)
{
	translate(point3d(v)+[x,y,z]) children();
}

function move(v=[0,0,0], p=undef, x=0, y=0, z=0) =
	is_undef(p)? (
		len(v)==2? affine2d_translate(v+[x,y]) :
		affine3d_translate(point3d(v)+[x,y,z])
	) : (
		assert(is_list(p))
		let(v=v+[x,y,z])
		is_num(p.x)? p+v :
		is_vnf(p)? [move(v=v,p=p.x), p.y] :
		[for (l=p) is_vector(l)? l+v : move(v=v, p=l)]
	);

function translate(v=[0,0,0], p=undef) = move(v=v, p=p);


// Function&Module: left()
//
// Usage: As Module
//   left(x) ...
// Usage: Translate Points
//   pts = left(x, p);
// Usage: Get Translation Matrix
//   mat = left(x);
//
// Description:
//   If called as a module, moves/translates all children left (in the X- direction) by the given amount.
//   If called as a function with the `p` argument, returns the translated point or list of points.
//   If called as a function without the `p` argument, returns an affine3d translation matrix.
//
// Arguments:
//   x = Scalar amount to move left.
//   p = Either a point, or a list of points to be translated when used as a function.
//
// Example:
//   #sphere(d=10);
//   left(20) sphere(d=10);
//
// Example(NORENDER):
//   pt1 = left(20, p=[23,42]);           // Returns: [3,42]
//   pt2 = left(20, p=[15,23,42]);        // Returns: [-5,23,42]
//   pt3 = left(3, p=[[1,2,3],[4,5,6]]);  // Returns: [[-2,2,3], [1,5,6]]
//   mat3d = left(4);  // Returns: [[1,0,0,-4],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
module left(x=0) translate([-x,0,0]) children();

function left(x=0,p=undef) = move([-x,0,0],p=p);


// Function&Module: right()
//
// Usage: As Module
//   right(x) ...
// Usage: Translate Points
//   pts = right(x, p);
// Usage: Get Translation Matrix
//   mat = right(x);
//
// Description:
//   If called as a module, moves/translates all children right (in the X+ direction) by the given amount.
//   If called as a function with the `p` argument, returns the translated point or list of points.
//   If called as a function without the `p` argument, returns an affine3d translation matrix.
//
// Arguments:
//   x = Scalar amount to move right.
//   p = Either a point, or a list of points to be translated when used as a function.
//
// Example:
//   #sphere(d=10);
//   right(20) sphere(d=10);
//
// Example(NORENDER):
//   pt1 = right(20, p=[23,42]);           // Returns: [43,42]
//   pt2 = right(20, p=[15,23,42]);        // Returns: [35,23,42]
//   pt3 = right(3, p=[[1,2,3],[4,5,6]]);  // Returns: [[4,2,3], [7,5,6]]
//   mat3d = right(4);  // Returns: [[1,0,0,4],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
module right(x=0) translate([x,0,0]) children();

function right(x=0,p=undef) = move([x,0,0],p=p);


// Function&Module: fwd()
//
// Usage: As Module
//   fwd(y) ...
// Usage: Translate Points
//   pts = fwd(y, p);
// Usage: Get Translation Matrix
//   mat = fwd(y);
//
// Description:
//   If called as a module, moves/translates all children forward (in the Y- direction) by the given amount.
//   If called as a function with the `p` argument, returns the translated point or list of points.
//   If called as a function without the `p` argument, returns an affine3d translation matrix.
//
// Arguments:
//   y = Scalar amount to move forward.
//   p = Either a point, or a list of points to be translated when used as a function.
//
// Example:
//   #sphere(d=10);
//   fwd(20) sphere(d=10);
//
// Example(NORENDER):
//   pt1 = fwd(20, p=[23,42]);           // Returns: [23,22]
//   pt2 = fwd(20, p=[15,23,42]);        // Returns: [15,3,42]
//   pt3 = fwd(3, p=[[1,2,3],[4,5,6]]);  // Returns: [[1,-1,3], [4,2,6]]
//   mat3d = fwd(4);  // Returns: [[1,0,0,0],[0,1,0,-4],[0,0,1,0],[0,0,0,1]]
module fwd(y=0) translate([0,-y,0]) children();

function fwd(y=0,p=undef) = move([0,-y,0],p=p);


// Function&Module: back()
//
// Usage: As Module
//   back(y) ...
// Usage: Translate Points
//   pts = back(y, p);
// Usage: Get Translation Matrix
//   mat = back(y);
//
// Description:
//   If called as a module, moves/translates all children back (in the Y+ direction) by the given amount.
//   If called as a function with the `p` argument, returns the translated point or list of points.
//   If called as a function without the `p` argument, returns an affine3d translation matrix.
//
// Arguments:
//   y = Scalar amount to move back.
//   p = Either a point, or a list of points to be translated when used as a function.
//
// Example:
//   #sphere(d=10);
//   back(20) sphere(d=10);
//
// Example(NORENDER):
//   pt1 = back(20, p=[23,42]);           // Returns: [23,62]
//   pt2 = back(20, p=[15,23,42]);        // Returns: [15,43,42]
//   pt3 = back(3, p=[[1,2,3],[4,5,6]]);  // Returns: [[1,5,3], [4,8,6]]
//   mat3d = back(4);  // Returns: [[1,0,0,0],[0,1,0,4],[0,0,1,0],[0,0,0,1]]
module back(y=0) translate([0,y,0]) children();

function back(y=0,p=undef) = move([0,y,0],p=p);


// Function&Module: down()
//
// Usage: As Module
//   down(z) ...
// Usage: Translate Points
//   pts = down(z, p);
// Usage: Get Translation Matrix
//   mat = down(z);
//
// Description:
//   If called as a module, moves/translates all children down (in the Z- direction) by the given amount.
//   If called as a function with the `p` argument, returns the translated point or list of points.
//   If called as a function without the `p` argument, returns an affine3d translation matrix.
//
// Arguments:
//   z = Scalar amount to move down.
//   p = Either a point, or a list of points to be translated when used as a function.
//
// Example:
//   #sphere(d=10);
//   down(20) sphere(d=10);
//
// Example(NORENDER):
//   pt1 = down(20, p=[15,23,42]);        // Returns: [15,23,22]
//   pt2 = down(3, p=[[1,2,3],[4,5,6]]);  // Returns: [[1,2,0], [4,5,3]]
//   mat3d = down(4);  // Returns: [[1,0,0,0],[0,1,0,0],[0,0,1,-4],[0,0,0,1]]
module down(z=0) translate([0,0,-z]) children();

function down(z=0,p=undef) = move([0,0,-z],p=p);


// Function&Module: up()
//
// Usage: As Module
//   up(z) ...
// Usage: Translate Points
//   pts = up(z, p);
// Usage: Get Translation Matrix
//   mat = up(z);
//
// Description:
//   If called as a module, moves/translates all children up (in the Z+ direction) by the given amount.
//   If called as a function with the `p` argument, returns the translated point or list of points.
//   If called as a function without the `p` argument, returns an affine3d translation matrix.
//
// Arguments:
//   z = Scalar amount to move up.
//   p = Either a point, or a list of points to be translated when used as a function.
//
// Example:
//   #sphere(d=10);
//   up(20) sphere(d=10);
//
// Example(NORENDER):
//   pt1 = up(20, p=[15,23,42]);        // Returns: [15,23,62]
//   pt2 = up(3, p=[[1,2,3],[4,5,6]]);  // Returns: [[1,2,6], [4,5,9]]
//   mat3d = up(4);  // Returns: [[1,0,0,0],[0,1,0,0],[0,0,1,4],[0,0,0,1]]
module up(z=0) translate([0,0,z]) children();

function up(z=0,p=undef) = move([0,0,z],p=p);



//////////////////////////////////////////////////////////////////////
// Section: Rotations
//////////////////////////////////////////////////////////////////////


// Function&Module: rot()
//
// Usage:
//   rot(a, [cp], [reverse]) ...
//   rot([X,Y,Z], [cp], [reverse]) ...
//   rot(a, v, [cp], [reverse]) ...
//   rot(from, to, [a], [reverse]) ...
//
// Description:
//   This is a shorthand version of the built-in `rotate()`, and operates similarly, with a few additional capabilities.
//   You can specify the rotation to perform in one of several ways:
//   * `rot(30)` or `rot(a=30)` rotates 30 degrees around the Z axis.
//   * `rot([20,30,40])` or `rot(a=[20,30,40])` rotates 20 degrees around the X axis, then 30 degrees around the Y axis, then 40 degrees around the Z axis.
//   * `rot(30, [1,1,0])` or `rot(a=30, v=[1,1,0])` rotates 30 degrees around the axis vector `[1,1,0]`.
//   * `rot(from=[0,0,1], to=[1,0,0])` rotates the top towards the right, similar to `rot(a=90,v=[0,1,0]`.
//   * `rot(from=[0,0,1], to=[1,1,0], a=45)` rotates 45 degrees around the Z axis, then rotates the top towards the back-right.  Similar to `rot(a=90,v=[-1,1,0])`
//   If the `cp` centerpoint argument is given, then rotations are performed around that centerpoint.
//   If the `reverse` argument is true, then the rotations performed will be exactly reversed.
//   The behavior and return value varies depending on how `rot()` is called:
//   * Called as a module, rotates all children.
//   * Called as a function with a `p` argument containing a point, returns the rotated point.
//   * Called as a function with a `p` argument containing a list of points, returns the list of rotated points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the rotated patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the rotated VNF.
//   * Called as a function without a `p` argument, and `planar` is true, returns the affine2d rotational matrix.
//   * Called as a function without a `p` argument, and `planar` is false, returns the affine3d rotational matrix.
//
// Arguments:
//   a = Scalar angle or vector of XYZ rotation angles to rotate by, in degrees.
//   v = vector for the axis of rotation.  Default: [0,0,1] or UP
//   cp = centerpoint to rotate around. Default: [0,0,0]
//   from = Starting vector for vector-based rotations.
//   to = Target vector for vector-based rotations.
//   reverse = If true, exactly reverses the rotation, including axis rotation ordering.  Default: false
//   planar = If called as a function, this specifies if you want to work with 2D points.
//   p = If called as a function, this contains a point or list of points to rotate.
//
// Example:
//   #cube([2,4,9]);
//   rot([30,60,0], cp=[0,0,9]) cube([2,4,9]);
//
// Example:
//   #cube([2,4,9]);
//   rot(30, v=[1,1,0], cp=[0,0,9]) cube([2,4,9]);
//
// Example:
//   #cube([2,4,9]);
//   rot(from=UP, to=LEFT+BACK) cube([2,4,9]);
//
// Example(2D):
//   path = square([50,30], center=true);
//   #stroke(path, closed=true);
//   stroke(rot(30,p=path), closed=true);
module rot(a=0, v=undef, cp=undef, from=undef, to=undef, reverse=false)
{
	m = rot(a=a, v=v, cp=cp, from=from, to=to, reverse=reverse, planar=false);
	multmatrix(m) children();
}

function rot(a=0, v, cp, from, to, reverse=false, planar=false, p, _m) =
	assert(is_undef(from)==is_undef(to), "from and to must be specified together.")
	is_undef(p)? (
		planar? let(
			cp = is_undef(cp)? cp : point2d(cp),
			m1 = is_undef(from)? affine2d_zrot(a) :
				assert(is_vector(from))
				assert(!approx(norm(from),0))
				assert(approx(point3d(from).z, 0))
				assert(is_vector(to))
				assert(!approx(norm(to),0))
				assert(approx(point3d(to).z, 0))
				affine2d_zrot(
					vang(point2d(to)) -
					vang(point2d(from))
				),
			m2 = is_undef(cp)? m1 : (move(cp) * m1 * move(-cp)),
			m3 = reverse? matrix_inverse(m2) : m2
		) m3 : let(
			from = is_undef(from)? undef : point3d(from),
			to = is_undef(to)? undef : point3d(to),
			cp = is_undef(cp)? undef : point3d(cp),
			m1 = !is_undef(from)? (
					assert(is_vector(from))
					assert(!approx(norm(from),0))
					assert(is_vector(to))
					assert(!approx(norm(to),0))
					affine3d_rot_from_to(from,to) * affine3d_zrot(a)
				) :
				!is_undef(v)? affine3d_rot_by_axis(v,a) :
				is_num(a)? affine3d_zrot(a) :
				affine3d_zrot(a.z) * affine3d_yrot(a.y) * affine3d_xrot(a.x),
			m2 = is_undef(cp)? m1 : (move(cp) * m1 * move(-cp)),
			m3 = reverse? matrix_inverse(m2) : m2
		) m3
	) : (
		assert(is_list(p))
		let(
			m = !is_undef(_m)? _m :
				rot(a=a, v=v, cp=cp, from=from, to=to, reverse=reverse, planar=planar),
			res = p==[]? [] :
				is_vector(p)? apply(m, p) :
				is_vnf(p)? [apply(m, p[0]), p[1]] :
				is_list(p[0])? [for (pp=p) rot(p=pp, _m=m)] :
				assert(false, "The p argument for rot() is not a point, path, patch, matrix, or VNF.")
		) res
	);




// Function&Module: xrot()
//
// Usage: As Module
//   xrot(a, [cp]) ...
// Usage: Rotate Points
//   rotated = xrot(a, p, [cp]);
// Usage: Get Rotation Matrix
//   mat = xrot(a, [cp]);
//
// Description:
//   Rotates around the X axis by the given number of degrees.  If `cp` is given, rotations are performed around that centerpoint.
//   * Called as a module, rotates all children.
//   * Called as a function with a `p` argument containing a point, returns the rotated point.
//   * Called as a function with a `p` argument containing a list of points, returns the list of rotated points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the rotated patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the rotated VNF.
//   * Called as a function without a `p` argument, and `planar` is true, returns the affine2d rotational matrix.
//   * Called as a function without a `p` argument, and `planar` is false, returns the affine3d rotational matrix.
//
// Arguments:
//   a = angle to rotate by in degrees.
//   cp = centerpoint to rotate around. Default: [0,0,0]
//   p = If called as a function, this contains a point or list of points to rotate.
//
// Example:
//   #cylinder(h=50, r=10, center=true);
//   xrot(90) cylinder(h=50, r=10, center=true);
module xrot(a=0, cp=undef)
{
	if (a==0) {
		children();  // May be slightly faster?
	} else if (!is_undef(cp)) {
		translate(cp) rotate([a, 0, 0]) translate(-cp) children();
	} else {
		rotate([a, 0, 0]) children();
	}
}

function xrot(a=0, cp=undef, p=undef) = rot([a,0,0], cp=cp, p=p);


// Function&Module: yrot()
//
// Usage: As Module
//   yrot(a, [cp]) ...
// Usage: Rotate Points
//   rotated = yrot(a, p, [cp]);
// Usage: Get Rotation Matrix
//   mat = yrot(a, [cp]);
//
// Description:
//   Rotates around the Y axis by the given number of degrees.  If `cp` is given, rotations are performed around that centerpoint.
//   * Called as a module, rotates all children.
//   * Called as a function with a `p` argument containing a point, returns the rotated point.
//   * Called as a function with a `p` argument containing a list of points, returns the list of rotated points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the rotated patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the rotated VNF.
//   * Called as a function without a `p` argument, and `planar` is true, returns the affine2d rotational matrix.
//   * Called as a function without a `p` argument, and `planar` is false, returns the affine3d rotational matrix.
//
// Arguments:
//   a = angle to rotate by in degrees.
//   cp = centerpoint to rotate around. Default: [0,0,0]
//   p = If called as a function, this contains a point or list of points to rotate.
//
// Example:
//   #cylinder(h=50, r=10, center=true);
//   yrot(90) cylinder(h=50, r=10, center=true);
module yrot(a=0, cp=undef)
{
	if (a==0) {
		children();  // May be slightly faster?
	} else if (!is_undef(cp)) {
		translate(cp) rotate([0, a, 0]) translate(-cp) children();
	} else {
		rotate([0, a, 0]) children();
	}
}

function yrot(a=0, cp=undef, p=undef) = rot([0,a,0], cp=cp, p=p);


// Function&Module: zrot()
//
// Usage: As Module
//   zrot(a, [cp]) ...
// Usage: Rotate Points
//   rotated = zrot(a, p, [cp]);
// Usage: Get Rotation Matrix
//   mat = zrot(a, [cp]);
//
// Description:
//   Rotates around the Z axis by the given number of degrees.  If `cp` is given, rotations are performed around that centerpoint.
//   * Called as a module, rotates all children.
//   * Called as a function with a `p` argument containing a point, returns the rotated point.
//   * Called as a function with a `p` argument containing a list of points, returns the list of rotated points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the rotated patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the rotated VNF.
//   * Called as a function without a `p` argument, and `planar` is true, returns the affine2d rotational matrix.
//   * Called as a function without a `p` argument, and `planar` is false, returns the affine3d rotational matrix.
//
// Arguments:
//   a = angle to rotate by in degrees.
//   cp = centerpoint to rotate around. Default: [0,0,0]
//   p = If called as a function, this contains a point or list of points to rotate.
//
// Example:
//   #cube(size=[60,20,40], center=true);
//   zrot(90) cube(size=[60,20,40], center=true);
module zrot(a=0, cp=undef)
{
	if (a==0) {
		children();  // May be slightly faster?
	} else if (!is_undef(cp)) {
		translate(cp) rotate(a) translate(-cp) children();
	} else {
		rotate(a) children();
	}
}

function zrot(a=0, cp=undef, p=undef) = rot(a, cp=cp, p=p);


//////////////////////////////////////////////////////////////////////
// Section: Scaling and Mirroring
//////////////////////////////////////////////////////////////////////


// Function&Module: scale()
// Usage: As Module
//   scale(SCALAR) ...
//   scale([X,Y,Z]) ...
// Usage: Scale Points
//   pts = scale(v, p);
// Usage: Get Scaling Matrix
//   mat = scale(v);
// Description:
//   Scales by the [X,Y,Z] scaling factors given in `v`.  If `v` is given as a scalar number, all axes are scaled uniformly by that amount.
//   * Called as the built-in module, scales all children.
//   * Called as a function with a point in the `p` argument, returns the scaled point.
//   * Called as a function with a list of points in the `p` argument, returns the list of scaled points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the scaled patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the scaled VNF.
//   * Called as a function without a `p` argument, and a 2D list of scaling factors in `v`, returns an affine2d scaling matrix.
//   * Called as a function without a `p` argument, and a 3D list of scaling factors in `v`, returns an affine3d scaling matrix.
// Arguments:
//   v = Either a numeric uniform scaling factor, or a list of [X,Y,Z] scaling factors.  Default: 1
//   p = If called as a function, the point or list of points to scale.
// Example(NORENDER):
//   pt1 = scale(3, p=[3,1,4]);        // Returns: [9,3,12]
//   pt2 = scale([2,3,4], p=[3,1,4]);  // Returns: [6,3,16]
//   pt3 = scale([2,3,4], p=[[1,2,3],[4,5,6]]);  // Returns: [[2,6,12], [8,15,24]]
//   mat2d = scale([2,3]);    // Returns: [[2,0,0],[0,3,0],[0,0,1]]
//   mat3d = scale([2,3,4]);  // Returns: [[2,0,0,0],[0,3,0,0],[0,0,4,0],[0,0,0,1]]
// Example(2D):
//   path = circle(d=50,$fn=12);
//   #stroke(path,closed=true);
//   stroke(scale([1.5,3],p=path),closed=true);
function scale(v=1, p=undef) =
	assert(is_num(v) || is_vector(v))
	assert(is_undef(p) || is_list(p))
	let(v = is_num(v)? [v,v,v] : v)
	is_undef(p)? (
		len(v)==2? affine2d_scale(v) : affine3d_scale(point3d(v))
	) : (
		assert(is_list(p))
		is_num(p.x)? vmul(p,v) :
		is_vnf(p)? let(inv=product([for (x=v) x<0? -1 : 1])) [
			scale(v=v,p=p.x),
			inv>=0? p.y : [for (l=p.y) reverse(l)]
		] :
		[for (l=p) is_vector(l)? vmul(l,v) : scale(v=v, p=l)]
	);


// Function&Module: xscale()
//
//
// Usage: As Module
//   xscale(x) ...
// Usage: Scale Points
//   scaled = xscale(x, p);
// Usage: Get Affine Matrix
//   mat = xscale(x);
//
// Description:
//   Scales along the X axis by the scaling factor `x`.
//   * Called as the built-in module, scales all children.
//   * Called as a function with a point in the `p` argument, returns the scaled point.
//   * Called as a function with a list of points in the `p` argument, returns the list of scaled points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the scaled patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the scaled VNF.
//   * Called as a function without a `p` argument, and a 2D list of scaling factors in `v`, returns an affine2d scaling matrix.
//   * Called as a function without a `p` argument, and a 3D list of scaling factors in `v`, returns an affine3d scaling matrix.
//
// Arguments:
//   x = Factor to scale by, along the X axis.
//   p = A point or path to scale, when called as a function.
//   planar = If true, and `p` is not given, then the matrix returned is an affine2d matrix instead of an affine3d matrix.
//
// Example: As Module
//   xscale(3) sphere(r=10);
//
// Example(2D): Scaling Points
//   path = circle(d=50,$fn=12);
//   #stroke(path,closed=true);
//   stroke(xscale(2,p=path),closed=true);
module xscale(x=1) scale([x,1,1]) children();

function xscale(x=1, p=undef, planar=false) = (planar || (!is_undef(p) && len(p)==2))? scale([x,1],p=p) : scale([x,1,1],p=p);


// Function&Module: yscale()
//
// Usage: As Module
//   yscale(y) ...
// Usage: Scale Points
//   scaled = yscale(y, p);
// Usage: Get Affine Matrix
//   mat = yscale(y);
//
// Description:
//   Scales along the Y axis by the scaling factor `y`.
//   * Called as the built-in module, scales all children.
//   * Called as a function with a point in the `p` argument, returns the scaled point.
//   * Called as a function with a list of points in the `p` argument, returns the list of scaled points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the scaled patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the scaled VNF.
//   * Called as a function without a `p` argument, and a 2D list of scaling factors in `v`, returns an affine2d scaling matrix.
//   * Called as a function without a `p` argument, and a 3D list of scaling factors in `v`, returns an affine3d scaling matrix.
//
// Arguments:
//   y = Factor to scale by, along the Y axis.
//   p = A point or path to scale, when called as a function.
//   planar = If true, and `p` is not given, then the matrix returned is an affine2d matrix instead of an affine3d matrix.
//
// Example: As Module
//   yscale(3) sphere(r=10);
//
// Example(2D): Scaling Points
//   path = circle(d=50,$fn=12);
//   #stroke(path,closed=true);
//   stroke(yscale(2,p=path),closed=true);
module yscale(y=1) scale([1,y,1]) children();

function yscale(y=1, p=undef, planar=false) = (planar || (!is_undef(p) && len(p)==2))? scale([1,y],p=p) : scale([1,y,1],p=p);


// Function&Module: zscale()
//
// Usage: As Module
//   zscale(z) ...
// Usage: Scale Points
//   scaled = zscale(z, p);
// Usage: Get Affine Matrix
//   mat = zscale(z);
//
// Description:
//   Scales along the Z axis by the scaling factor `z`.
//   * Called as the built-in module, scales all children.
//   * Called as a function with a point in the `p` argument, returns the scaled point.
//   * Called as a function with a list of points in the `p` argument, returns the list of scaled points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the scaled patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the scaled VNF.
//   * Called as a function without a `p` argument, and a 2D list of scaling factors in `v`, returns an affine2d scaling matrix.
//   * Called as a function without a `p` argument, and a 3D list of scaling factors in `v`, returns an affine3d scaling matrix.
//
// Arguments:
//   z = Factor to scale by, along the Z axis.
//   p = A point or path to scale, when called as a function.
//   planar = If true, and `p` is not given, then the matrix returned is an affine2d matrix instead of an affine3d matrix.
//
// Example: As Module
//   zscale(3) sphere(r=10);
//
// Example: Scaling Points
//   path = xrot(90,p=path3d(circle(d=50,$fn=12)));
//   #trace_polyline(path);
//   trace_polyline(zscale(2,p=path));
module zscale(z=1) scale([1,1,z]) children();

function zscale(z=1, p=undef) = scale([1,1,z],p=p);


// Function&Module: mirror()
// Usage: As Module
//   mirror(v) ...
// Usage: As Function
//   pt = mirror(v, p);
// Usage: Get Reflection/Mirror Matrix
//   mat = mirror(v);
// Description:
//   Mirrors/reflects across the plane or line whose normal vector is given in `v`.
//   * Called as the built-in module, mirrors all children across the line/plane.
//   * Called as a function with a point in the `p` argument, returns the point mirrored across the line/plane.
//   * Called as a function with a list of points in the `p` argument, returns the list of points, with each one mirrored across the line/plane.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the mirrored patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the mirrored VNF.
//   * Called as a function without a `p` argument, and with a 2D normal vector `v`, returns the affine2d 3x3 mirror matrix.
//   * Called as a function without a `p` argument, and with a 3D normal vector `v`, returns the affine3d 4x4 mirror matrix.
// Arguments:
//   v = The normal vector of the line or plane to mirror across.
//   p = If called as a function, the point or list of points to scale.
// Example:
//   n = [1,0,0];
//   module obj() right(20) rotate([0,15,-15]) cube([40,30,20]);
//   obj();
//   mirror(n) obj();
//   rot(a=atan2(n.y,n.x),from=UP,to=n) {
//       color("red") anchor_arrow(s=20, flag=false);
//       color("#7777") cube([75,75,0.1], center=true);
//   }
// Example:
//   n = [1,1,0];
//   module obj() right(20) rotate([0,15,-15]) cube([40,30,20]);
//   obj();
//   mirror(n) obj();
//   rot(a=atan2(n.y,n.x),from=UP,to=n) {
//       color("red") anchor_arrow(s=20, flag=false);
//       color("#7777") cube([75,75,0.1], center=true);
//   }
// Example:
//   n = [1,1,1];
//   module obj() right(20) rotate([0,15,-15]) cube([40,30,20]);
//   obj();
//   mirror(n) obj();
//   rot(a=atan2(n.y,n.x),from=UP,to=n) {
//       color("red") anchor_arrow(s=20, flag=false);
//       color("#7777") cube([75,75,0.1], center=true);
//   }
// Example(2D):
//   n = [0,1];
//   path = rot(30, p=square([50,30]));
//   color("gray") rot(from=[0,1],to=n) stroke([[-60,0],[60,0]]);
//   color("red") stroke([[0,0],10*n],endcap2="arrow2");
//   #stroke(path,closed=true);
//   stroke(mirror(n, p=path),closed=true);
// Example(2D):
//   n = [1,1];
//   path = rot(30, p=square([50,30]));
//   color("gray") rot(from=[0,1],to=n) stroke([[-60,0],[60,0]]);
//   color("red") stroke([[0,0],10*n],endcap2="arrow2");
//   #stroke(path,closed=true);
//   stroke(mirror(n, p=path),closed=true);
function mirror(v, p) =
	assert(is_vector(v))
	assert(is_undef(p) || is_list(p))
	let(m = len(v)==2? affine2d_mirror(v) : affine3d_mirror(v))
	is_undef(p)? m :
	is_num(p.x)? apply(m,p) :
	is_vnf(p)? [mirror(v=v,p=p[0]), [for (face=p[1]) reverse(face)]] :
	[for (l=p) is_vector(l)? apply(m,l) : mirror(v=v, p=l)];


// Function&Module: xflip()
//
// Usage: As Module
//   xflip([x]) ...
// Usage: As Function
//   pt = xflip([x], p);
// Usage: Get Affine Matrix
//   pt = xflip([x]);
//
// Description:
//   Mirrors/reflects across the origin [0,0,0], along the X axis.  If `x` is given, reflects across [x,0,0] instead.
//   * Called as the built-in module, mirrors all children across the line/plane.
//   * Called as a function with a point in the `p` argument, returns the point mirrored across the line/plane.
//   * Called as a function with a list of points in the `p` argument, returns the list of points, with each one mirrored across the line/plane.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the mirrored patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the mirrored VNF.
//   * Called as a function without a `p` argument, and with a 2D normal vector `v`, returns the affine2d 3x3 mirror matrix.
//   * Called as a function without a `p` argument, and with a 3D normal vector `v`, returns the affine3d 4x4 mirror matrix.
//
// Arguments:
//   x = The X coordinate of the plane of reflection.  Default: 0
//
// Example:
//   xflip() yrot(90) cylinder(d1=10, d2=0, h=20);
//   color("blue", 0.25) cube([0.01,15,15], center=true);
//   color("red", 0.333) yrot(90) cylinder(d1=10, d2=0, h=20);
//
// Example:
//   xflip(x=-5) yrot(90) cylinder(d1=10, d2=0, h=20);
//   color("blue", 0.25) left(5) cube([0.01,15,15], center=true);
//   color("red", 0.333) yrot(90) cylinder(d1=10, d2=0, h=20);
module xflip(x=0) translate([x,0,0]) mirror([1,0,0]) translate([-x,0,0]) children();

function xflip(x=0,p) =
	x==0? mirror([1,0,0],p=p) :
	move([x,0,0],p=mirror([1,0,0],p=move([-x,0,0],p=p)));


// Function&Module: yflip()
//
// Usage: As Module
//   yflip([y]) ...
// Usage: As Function
//   pt = yflip([y], p);
// Usage: Get Affine Matrix
//   pt = yflip([y]);
//
// Description:
//   Mirrors/reflects across the origin [0,0,0], along the Y axis.  If `y` is given, reflects across [0,y,0] instead.
//   * Called as the built-in module, mirrors all children across the line/plane.
//   * Called as a function with a point in the `p` argument, returns the point mirrored across the line/plane.
//   * Called as a function with a list of points in the `p` argument, returns the list of points, with each one mirrored across the line/plane.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the mirrored patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the mirrored VNF.
//   * Called as a function without a `p` argument, and with a 2D normal vector `v`, returns the affine2d 3x3 mirror matrix.
//   * Called as a function without a `p` argument, and with a 3D normal vector `v`, returns the affine3d 4x4 mirror matrix.
//
// Arguments:
//   y = The Y coordinate of the plane of reflection.  Default: 0
//
// Example:
//   yflip() xrot(90) cylinder(d1=10, d2=0, h=20);
//   color("blue", 0.25) cube([15,0.01,15], center=true);
//   color("red", 0.333) xrot(90) cylinder(d1=10, d2=0, h=20);
//
// Example:
//   yflip(y=5) xrot(90) cylinder(d1=10, d2=0, h=20);
//   color("blue", 0.25) back(5) cube([15,0.01,15], center=true);
//   color("red", 0.333) xrot(90) cylinder(d1=10, d2=0, h=20);
module yflip(y=0) translate([0,y,0]) mirror([0,1,0]) translate([0,-y,0]) children();

function yflip(y=0,p) =
	y==0? mirror([0,1,0],p=p) :
	move([0,y,0],p=mirror([0,1,0],p=move([0,-y,0],p=p)));



// Function&Module: zflip()
//
// Usage: As Module
//   zflip([z]) ...
// Usage: As Function
//   pt = zflip([z], p);
// Usage: Get Affine Matrix
//   pt = zflip([z]);
//
// Description:
//   Mirrors/reflects across the origin [0,0,0], along the Z axis.  If `z` is given, reflects across [0,0,z] instead.
//   * Called as the built-in module, mirrors all children across the line/plane.
//   * Called as a function with a point in the `p` argument, returns the point mirrored across the line/plane.
//   * Called as a function with a list of points in the `p` argument, returns the list of points, with each one mirrored across the line/plane.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the mirrored patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the mirrored VNF.
//   * Called as a function without a `p` argument, and with a 2D normal vector `v`, returns the affine2d 3x3 mirror matrix.
//   * Called as a function without a `p` argument, and with a 3D normal vector `v`, returns the affine3d 4x4 mirror matrix.
//
// Arguments:
//   z = The Z coordinate of the plane of reflection.  Default: 0
//
// Example:
//   zflip() cylinder(d1=10, d2=0, h=20);
//   color("blue", 0.25) cube([15,15,0.01], center=true);
//   color("red", 0.333) cylinder(d1=10, d2=0, h=20);
//
// Example:
//   zflip(z=-5) cylinder(d1=10, d2=0, h=20);
//   color("blue", 0.25) down(5) cube([15,15,0.01], center=true);
//   color("red", 0.333) cylinder(d1=10, d2=0, h=20);
module zflip(z=0) translate([0,0,z]) mirror([0,0,1]) translate([0,0,-z]) children();

function zflip(z=0,p) =
	z==0? mirror([0,0,1],p=p) :
	move([0,0,z],p=mirror([0,0,1],p=move([0,0,-z],p=p)));



//////////////////////////////////////////////////////////////////////
// Section: Skewing
//////////////////////////////////////////////////////////////////////


// Function&Module: skew()
// Usage: As Module
//   skew(sxy=0, sxz=0, syx=0, syz=0, szx=0, szy=0) ...
// Usage: As Function
//   pts = skew(p, [sxy], [sxz], [syx], [syz], [szx], [szy]);
// Usage: Get Affine Matrix
//   mat = skew([sxy], [sxz], [syx], [syz], [szx], [szy], [planar]);
// Description:
//   Skews geometry by the given skew factors.
//   * Called as the built-in module, skews all children.
//   * Called as a function with a point in the `p` argument, returns the skewed point.
//   * Called as a function with a list of points in the `p` argument, returns the list of skewed points.
//   * Called as a function with a [bezier patch](beziers.scad) in the `p` argument, returns the skewed patch.
//   * Called as a function with a [VNF structure](vnf.scad) in the `p` argument, returns the skewed VNF.
//   * Called as a function without a `p` argument, and with `planar` true, returns the affine2d 3x3 skew matrix.
//   * Called as a function without a `p` argument, and with `planar` false, returns the affine3d 4x4 skew matrix.
//   Each skew factor is a multiplier.  For example, if `sxy=2`, then it will skew along the X axis by 2x the value of the Y axis.
// Arguments:
//   sxy = Skew factor multiplier for skewing along the X axis as you get farther from the Y axis.  Default: 0
//   sxz = Skew factor multiplier for skewing along the X axis as you get farther from the Z axis.  Default: 0
//   syx = Skew factor multiplier for skewing along the Y axis as you get farther from the X axis.  Default: 0
//   syz = Skew factor multiplier for skewing along the Y axis as you get farther from the Z axis.  Default: 0
//   szx = Skew factor multiplier for skewing along the Z axis as you get farther from the X axis.  Default: 0
//   szy = Skew factor multiplier for skewing along the Z axis as you get farther from the Y axis.  Default: 0
// Example(2D): Skew along the X axis in 2D.
//   skew(sxy=0.5) square(40, center=true);
// Example(2D): Skew along the Y axis in 2D.
//   skew(syx=0.5) square(40, center=true);
// Example: Skew along the X axis in 3D as a factor of Y coordinate.
//   skew(sxy=0.5) cube(40, center=true);
// Example: Skew along the X axis in 3D as a factor of Z coordinate.
//   skew(sxz=0.5) cube(40, center=true);
// Example: Skew along the Y axis in 3D as a factor of X coordinate.
//   skew(syx=0.5) cube(40, center=true);
// Example: Skew along the Y axis in 3D as a factor of Z coordinate.
//   skew(syz=0.5) cube(40, center=true);
// Example: Skew along the Z axis in 3D as a factor of X coordinate.
//   skew(szx=0.5) cube(40, center=true);
// Example: Skew along the Z axis in 3D as a factor of Y coordinate.
//   skew(szy=0.75) cube(40, center=true);
// Example(FlatSpin): Skew Along Multiple Axes.
//   skew(sxy=0.5, syx=0.3, szy=0.75) cube(40, center=true);
// Example(2D): Calling as a 2D Function
//   pts = skew(p=square(40,center=true), sxy=0.5);
//   color("yellow") stroke(pts, closed=true);
//   color("blue") move_copies(pts) circle(d=3, $fn=8);
// Example(FlatSpin): Calling as a 3D Function
//   pts = skew(p=path3d(square(40,center=true)), szx=0.5, szy=0.3);
//   trace_polyline(close_path(pts), showpts=true);
module skew(sxy=0, sxz=0, syx=0, syz=0, szx=0, szy=0)
{
	multmatrix(
		affine3d_skew(sxy=sxy, sxz=sxz, syx=syx, syz=syz, szx=szx, szy=szy)
	) children();
}

function skew(p, sxy=0, sxz=0, syx=0, syz=0, szx=0, szy=0, planar=false) =
	let(
		planar = planar || (is_list(p) && is_num(p.x) && len(p)==2),
		m = planar? [
			[  1, sxy, 0],
			[syx,   1, 0],
			[  0,   0, 1]
		] : affine3d_skew(sxy=sxy, sxz=sxz, syx=syx, syz=syz, szx=szx, szy=szy)
	)
	is_undef(p)? m :
	assert(is_list(p))
	is_num(p.x)? (
		planar?
			point2d(m*concat(point2d(p),[1])) :
			point3d(m*concat(point3d(p),[1]))
	) :
	is_vnf(p)? [skew(sxy=sxy, sxz=sxz, syx=syx, syz=syz, szx=szx, szy=szy, planar=planar, p=p.x), p.y] :
	[for (l=p) skew(sxy=sxy, sxz=sxz, syx=syx, syz=syz, szx=szx, szy=szy, planar=planar, p=l)];


// vim: noexpandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap