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coords.scad
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coords.scad
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//////////////////////////////////////////////////////////////////////
// LibFile: coords.scad
// Coordinate transformations and coordinate system conversions.
// Includes:
// include <BOSL2/std.scad>
// FileGroup: Math
// FileSummary: Conversions between coordinate systems.
// FileFootnotes: STD=Included in std.scad
//////////////////////////////////////////////////////////////////////
// Section: Coordinate Manipulation
// Function: point2d()
// Synopsis: Convert a vector to 2D.
// Topics: Coordinates, Points
// See Also: path2d(), point3d(), path3d()
// Usage:
// pt = point2d(p, [fill]);
// Description:
// Returns a 2D vector/point from a 2D or 3D vector. If given a 3D point, removes the Z coordinate.
// Arguments:
// p = The coordinates to force into a 2D vector/point.
// fill = Value to fill missing values in vector with. Default: 0
function point2d(p, fill=0) = assert(is_list(p)) [for (i=[0:1]) (p[i]==undef)? fill : p[i]];
// Function: path2d()
// Synopsis: Convert a path to 2D.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: point2d(), point3d(), path3d()
// Usage:
// pts = path2d(points);
// Description:
// Returns a list of 2D vectors/points from a list of 2D, 3D or higher dimensional vectors/points.
// Removes the extra coordinates from higher dimensional points. The input must be a path, where
// every vector has the same length.
// Arguments:
// points = A list of 2D or 3D points/vectors.
function path2d(points) =
assert(is_path(points,dim=undef,fast=true),"Input to path2d is not a path")
let (result = points * concat(ident(2), repeat([0,0], len(points[0])-2)))
assert(is_def(result), "Invalid input to path2d")
result;
// Function: point3d()
// Synopsis: Convert a vector to 3D.
// Topics: Coordinates, Points
// See Also: path2d(), point2d(), path3d()
// Usage:
// pt = point3d(p, [fill]);
// Description:
// Returns a 3D vector/point from a 2D or 3D vector.
// Arguments:
// p = The coordinates to force into a 3D vector/point.
// fill = Value to fill missing values in vector with. Default: 0
function point3d(p, fill=0) =
assert(is_list(p))
[for (i=[0:2]) (p[i]==undef)? fill : p[i]];
// Function: path3d()
// Synopsis: Convert a path to 3D.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: point2d(), path2d(), point3d()
// Usage:
// pts = path3d(points, [fill]);
// Description:
// Returns a list of 3D vectors/points from a list of 2D or higher dimensional vectors/points
// by removing extra coordinates or adding the z coordinate.
// Arguments:
// points = A list of 2D, 3D or higher dimensional points/vectors.
// fill = Value to fill missing values in vectors with (in the 2D case). Default: 0
function path3d(points, fill=0) =
assert(is_num(fill))
assert(is_path(points, dim=undef, fast=true), "Input to path3d is not a path")
let (
change = len(points[0])-3,
M = change < 0? [[1,0,0],[0,1,0]] :
concat(ident(3), repeat([0,0,0],change)),
result = points*M
)
assert(is_def(result), "Input to path3d is invalid")
fill == 0 || change>=0 ? result : result + repeat([0,0,fill], len(result));
// Function: point4d()
// Synopsis: Convert a vector to 4d.
// Topics: Coordinates, Points
// See Also: point2d(), path2d(), point3d(), path3d(), path4d()
// Usage:
// pt = point4d(p, [fill]);
// Description:
// Returns a 4D vector/point from a 2D or 3D vector.
// Arguments:
// p = The coordinates to force into a 4D vector/point.
// fill = Value to fill missing values in vector with. Default: 0
function point4d(p, fill=0) = assert(is_list(p))
[for (i=[0:3]) (p[i]==undef)? fill : p[i]];
// Function: path4d()
// Synopsis: Convert a path to 4d.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: point2d(), path2d(), point3d(), path3d(), point4d()
// Usage:
// pt = path4d(points, [fill]);
// Description:
// Returns a list of 4D vectors/points from a list of 2D or 3D vectors/points.
// Arguments:
// points = A list of 2D or 3D points/vectors.
// fill = Value to fill missing values in vectors with. Default: 0
function path4d(points, fill=0) =
assert(is_num(fill) || is_vector(fill))
assert(is_path(points, dim=undef, fast=true), "Input to path4d is not a path")
let (
change = len(points[0])-4,
M = change < 0 ? select(ident(4), 0, len(points[0])-1) :
concat(ident(4), repeat([0,0,0,0],change)),
result = points*M
)
assert(is_def(result), "Input to path4d is invalid")
fill == 0 || change >= 0 ? result :
let(
addition = is_list(fill) ? concat(0*points[0],fill) :
concat(0*points[0],repeat(fill,-change))
)
assert(len(addition) == 4, "Fill is the wrong length")
result + repeat(addition, len(result));
// Section: Coordinate Systems
// Function: polar_to_xy()
// Synopsis: Convert 2D polar coordinates to cartesian coordinates.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: xy_to_polar(), xyz_to_cylindrical(), cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
// Usage:
// pt = polar_to_xy(r, theta);
// pt = polar_to_xy([R, THETA]);
// pts = polar_to_xy([[R,THETA], [R,THETA], ...]);
// Description:
// Called with two arguments, converts the `r` and `theta` 2D polar coordinate into an `[X,Y]` cartesian coordinate.
// Called with one `[R,THETA]` vector argument, converts the 2D polar coordinate into an `[X,Y]` cartesian coordinate.
// Called with a list of `[R,THETA]` vector arguments, converts each 2D polar coordinate into `[X,Y]` cartesian coordinates.
// Theta is the angle counter-clockwise of X+ on the XY plane.
// Arguments:
// r = distance from the origin.
// theta = angle in degrees, counter-clockwise of X+.
// Example:
// xy = polar_to_xy(20,45); // Returns: ~[14.1421365, 14.1421365]
// xy = polar_to_xy(40,30); // Returns: ~[34.6410162, 15]
// xy = polar_to_xy([40,30]); // Returns: ~[34.6410162, 15]
// xy = polar_to_xy([[40,30],[20,120]]); // Returns: ~[[34.6410162, 15], [-10, 17.3205]]
// Example(2D):
// r=40; ang=30; $fn=36;
// pt = polar_to_xy(r,ang);
// stroke(circle(r=r), closed=true, width=0.5);
// color("black") stroke([[r,0], [0,0], pt], width=0.5);
// color("black") stroke(arc(r=15, angle=ang), width=0.5);
// color("red") move(pt) circle(d=3);
function polar_to_xy(r,theta) =
theta != undef
? assert(is_num(r) && is_num(theta), "Bad Arguments.")
[r*cos(theta), r*sin(theta)]
: assert(is_list(r), "Bad Arguments")
is_num(r.x)
? polar_to_xy(r.x, r.y)
: [for(p = r) polar_to_xy(p.x, p.y)];
// Function: xy_to_polar()
// Synopsis: Convert 2D cartesian coordinates to polar coordinates (radius and angle)
// Topics: Coordinates, Points, Paths
// See Also: polar_to_xy(), xyz_to_cylindrical(), cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
// Usage:
// r_theta = xy_to_polar(x,y);
// r_theta = xy_to_polar([X,Y]);
// r_thetas = xy_to_polar([[X,Y], [X,Y], ...]);
// Description:
// Called with two arguments, converts the `x` and `y` 2D cartesian coordinate into a `[RADIUS,THETA]` polar coordinate.
// Called with one `[X,Y]` vector argument, converts the 2D cartesian coordinate into a `[RADIUS,THETA]` polar coordinate.
// Called with a list of `[X,Y]` vector arguments, converts each 2D cartesian coordinate into `[RADIUS,THETA]` polar coordinates.
// Theta is the angle counter-clockwise of X+ on the XY plane.
// Arguments:
// x = X coordinate.
// y = Y coordinate.
// Example:
// plr = xy_to_polar(20,30);
// plr = xy_to_polar([40,60]);
// plrs = xy_to_polar([[40,60],[-10,20]]);
// Example(2D):
// pt = [-20,30]; $fn = 36;
// rt = xy_to_polar(pt);
// r = rt[0]; ang = rt[1];
// stroke(circle(r=r), closed=true, width=0.5);
// zrot(ang) stroke([[0,0],[r,0]],width=0.5);
// color("red") move(pt) circle(d=3);
function xy_to_polar(x, y) =
y != undef
? assert(is_num(x) && is_num(y), "Bad Arguments.")
[norm([x, y]), atan2(y, x)]
: assert(is_list(x), "Bad Arguments")
is_num(x.x)
? xy_to_polar(x.x, x.y)
: [for(p = x) xy_to_polar(p.x, p.y)];
// Function: project_plane()
// Synopsis: Project a set of points onto a specified plane, returning 2D points.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: lift_plane()
// Usage:
// xy = project_plane(plane, p);
// Usage: To get a transform matrix
// M = project_plane(plane)
// Description:
// Maps the provided 3D point(s) from 3D coordinates to a 2D coordinate system defined by `plane`. Points that are not
// on the specified plane will be projected orthogonally onto the plane. This coordinate system is useful if you need
// to perform 2D operations on a coplanar set of data. After those operations are done you can return the data
// to 3D with `lift_plane()`. You could also use this to force approximately coplanar data to be exactly coplanar.
// The parameter p can be a point, path, region, bezier patch or VNF.
// The plane can be specified as
// - A list of three points. The planar coordinate system will have [0,0] at plane[0], and plane[1] will lie on the Y+ axis.
// - A list of coplanar points that define a plane (not-collinear)
// - A plane definition `[A,B,C,D]` where `Ax+By+CZ=D`. The closest point on that plane to the origin will map to the origin in the new coordinate system.
// .
// If you omit the point specification then `project_plane()` returns a rotation matrix that maps the specified plane to the XY plane.
// Note that if you apply this transformation to data lying on the plane it will produce 3D points with the Z coordinate of zero.
// Arguments:
// plane = plane specification or point list defining the plane
// p = 3D point, path, region, VNF or bezier patch to project
// Example:
// pt = [5,-5,5];
// a=[0,0,0]; b=[10,-10,0]; c=[10,0,10];
// xy = project_plane([a,b,c],pt);
// Example(3D): The yellow points in 3D project onto the red points in 2D
// M = [[-1, 2, -1, -2], [-1, -3, 2, -1], [2, 3, 4, 53], [0, 0, 0, 1]];
// data = apply(M,path3d(circle(r=10, $fn=20)));
// move_copies(data) sphere(r=1);
// color("red") move_copies(project_plane(data, data)) sphere(r=1);
// Example:
// xyzpath = move([10,20,30], p=yrot(25, p=path3d(circle(d=100))));
// mat = project_plane(xyzpath);
// xypath = path2d(apply(mat, xyzpath));
// #stroke(xyzpath,closed=true);
// stroke(xypath,closed=true);
function project_plane(plane,p) =
is_matrix(plane,3,3) && is_undef(p) ? // no data, 3 points given
assert(!is_collinear(plane),"Points defining the plane must not be collinear")
let(
v = plane[2]-plane[0],
y = unit(plane[1]-plane[0]), // y axis goes to point b
x = unit(v-(v*y)*y) // x axis
)
frame_map(x,y) * move(-plane[0])
: is_vector(plane,4) && is_undef(p) ? // no data, plane given in "plane"
assert(_valid_plane(plane), "Plane is not valid")
let(
n = point3d(plane),
cp = n * plane[3] / (n*n)
)
rot(from=n, to=UP) * move(-cp)
: is_path(plane,3) && is_undef(p) ? // no data, generic point list plane
assert(len(plane)>=3, "Need three points to define a plane")
let(plane = plane_from_points(plane))
assert(is_def(plane), "Point list is not coplanar")
project_plane(plane)
: assert(is_def(p), str("Invalid plane specification: ",plane))
is_vnf(p) ? [project_plane(plane,p[0]), p[1]]
: is_list(p) && is_list(p[0]) && is_vector(p[0][0],3) ? // bezier patch or region
[for(plist=p) project_plane(plane,plist)]
: assert(is_vector(p,3) || is_path(p,3),str("Data must be a 3D point, path, region, vnf or bezier patch",p))
is_matrix(plane,3,3) ?
assert(!is_collinear(plane),"Points defining the plane must not be collinear")
let(
v = plane[2]-plane[0],
y = unit(plane[1]-plane[0]), // y axis goes to point b
x = unit(v-(v*y)*y) // x axis
) move(-plane[0],p) * transpose([x,y])
: is_vector(p) ? point2d(apply(project_plane(plane),p))
: path2d(apply(project_plane(plane),p));
// Function: lift_plane()
// Synopsis: Map a list of 2D points onto a plane in 3D.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: project_plane()
// Usage:
// xyz = lift_plane(plane, p);
// Usage: to get transform matrix
// M = lift_plane(plane);
// Description:
// Converts the given 2D point on the plane to 3D coordinates of the specified plane.
// The parameter p can be a point, path, region, bezier patch or VNF.
// The plane can be specified as
// - A list of three points. The planar coordinate system will have [0,0] at plane[0], and plane[1] will lie on the Y+ axis.
// - A list of coplanar points that define a plane (not-collinear)
// - A plane definition `[A,B,C,D]` where `Ax+By+CZ=D`. The closest point on that plane to the origin will map to the origin in the new coordinate system.
// If you do not supply `p` then you get a transformation matrix which operates in 3D, assuming that the Z coordinate of the points is zero.
// This matrix is a rotation, the inverse of the one produced by project_plane.
// Arguments:
// plane = Plane specification or list of points to define a plane
// p = points, path, region, VNF, or bezier patch to transform.
function lift_plane(plane, p) =
is_matrix(plane,3,3) && is_undef(p) ? // no data, 3 p given
let(
v = plane[2]-plane[0],
y = unit(plane[1]-plane[0]), // y axis goes to point b
x = unit(v-(v*y)*y) // x axis
)
move(plane[0]) * frame_map(x,y,reverse=true)
: is_vector(plane,4) && is_undef(p) ? // no data, plane given in "plane"
assert(_valid_plane(plane), "Plane is not valid")
let(
n = point3d(plane),
cp = n * plane[3] / (n*n)
)
move(cp) * rot(from=UP, to=n)
: is_path(plane,3) && is_undef(p) ? // no data, generic point list plane
assert(len(plane)>=3, "Need three p to define a plane")
let(plane = plane_from_points(plane))
assert(is_def(plane), "Point list is not coplanar")
lift_plane(plane)
: is_vnf(p) ? [lift_plane(plane,p[0]), p[1]]
: is_list(p) && is_list(p[0]) && is_vector(p[0][0],3) ? // bezier patch or region
[for(plist=p) lift_plane(plane,plist)]
: assert(is_vector(p,2) || is_path(p,2),"Data must be a 2D point, path, region, vnf or bezier patch")
is_matrix(plane,3,3) ?
let(
v = plane[2]-plane[0],
y = unit(plane[1]-plane[0]), // y axis goes to point b
x = unit(v-(v*y)*y) // x axis
) move(plane[0],p * [x,y])
: apply(lift_plane(plane),is_vector(p) ? point3d(p) : path3d(p));
// Function: cylindrical_to_xyz()
// Synopsis: Convert cylindrical coordinates to cartesian coordinates.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: xyz_to_cylindrical(), xy_to_polar(), polar_to_xy(), xyz_to_spherical(), spherical_to_xyz()
// Usage:
// pt = cylindrical_to_xyz(r, theta, z);
// pt = cylindrical_to_xyz([RADIUS,THETA,Z]);
// pts = cylindrical_to_xyz([[RADIUS,THETA,Z], [RADIUS,THETA,Z], ...]);
// Description:
// Called with three arguments, converts the `r`, `theta`, and 'z' 3D cylindrical coordinate into an `[X,Y,Z]` cartesian coordinate.
// Called with one `[RADIUS,THETA,Z]` vector argument, converts the 3D cylindrical coordinate into an `[X,Y,Z]` cartesian coordinate.
// Called with a list of `[RADIUS,THETA,Z]` vector arguments, converts each 3D cylindrical coordinate into `[X,Y,Z]` cartesian coordinates.
// Theta is the angle counter-clockwise of X+ on the XY plane. Z is height above the XY plane.
// Arguments:
// r = distance from the Z axis.
// theta = angle in degrees, counter-clockwise of X+ on the XY plane.
// z = Height above XY plane.
// Example:
// xyz = cylindrical_to_xyz(20,30,40);
// xyz = cylindrical_to_xyz([40,60,50]);
function cylindrical_to_xyz(r,theta,z) =
theta != undef
? assert(is_num(r) && is_num(theta) && is_num(z), "Bad Arguments.")
[r*cos(theta), r*sin(theta), z]
: assert(is_list(r), "Bad Arguments")
is_num(r.x)
? cylindrical_to_xyz(r.x, r.y, r.z)
: [for(p = r) cylindrical_to_xyz(p.x, p.y, p.z)];
// Function: xyz_to_cylindrical()
// Synopsis: Convert 3D cartesian coordinates to cylindrical coordinates.
// Topics: Coordinates, Points, Paths
// See Also: cylindrical_to_xyz(), xy_to_polar(), polar_to_xy(), xyz_to_spherical(), spherical_to_xyz()
// Usage:
// rtz = xyz_to_cylindrical(x,y,z);
// rtz = xyz_to_cylindrical([X,Y,Z]);
// rtzs = xyz_to_cylindrical([[X,Y,Z], [X,Y,Z], ...]);
// Description:
// Called with three arguments, converts the `x`, `y`, and `z` 3D cartesian coordinate into a `[RADIUS,THETA,Z]` cylindrical coordinate.
// Called with one `[X,Y,Z]` vector argument, converts the 3D cartesian coordinate into a `[RADIUS,THETA,Z]` cylindrical coordinate.
// Called with a list of `[X,Y,Z]` vector arguments, converts each 3D cartesian coordinate into `[RADIUS,THETA,Z]` cylindrical coordinates.
// Theta is the angle counter-clockwise of X+ on the XY plane. Z is height above the XY plane.
// Arguments:
// x = X coordinate.
// y = Y coordinate.
// z = Z coordinate.
// Example:
// cyl = xyz_to_cylindrical(20,30,40);
// cyl = xyz_to_cylindrical([40,50,70]);
// cyls = xyz_to_cylindrical([[40,50,70], [-10,15,-30]]);
function xyz_to_cylindrical(x,y,z) =
y != undef
? assert(is_num(x) && is_num(y) && is_num(z), "Bad Arguments.")
[norm([x,y]), atan2(y,x), z]
: assert(is_list(x), "Bad Arguments")
is_num(x.x)
? xyz_to_cylindrical(x.x, x.y, x.z)
: [for(p = x) xyz_to_cylindrical(p.x, p.y, p.z)];
// Function: spherical_to_xyz()
// Synopsis: Convert spherical coordinates to 3D cartesian coordinates.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: cylindrical_to_xyz(), xyz_to_spherical(), xyz_to_cylindrical(), altaz_to_xyz(), xyz_to_altaz()
// Usage:
// pt = spherical_to_xyz(r, theta, phi);
// pt = spherical_to_xyz([RADIUS,THETA,PHI]);
// pts = spherical_to_xyz([[RADIUS,THETA,PHI], [RADIUS,THETA,PHI], ...]);
// Description:
// Called with three arguments, converts the `r`, `theta`, and 'phi' 3D spherical coordinate into an `[X,Y,Z]` cartesian coordinate.
// Called with one `[RADIUS,THETA,PHI]` vector argument, converts the 3D spherical coordinate into an `[X,Y,Z]` cartesian coordinate.
// Called with a list of `[RADIUS,THETA,PHI]` vector arguments, converts each 3D spherical coordinate into `[X,Y,Z]` cartesian coordinates.
// Theta is the angle counter-clockwise of X+ on the XY plane. Phi is the angle down from the Z+ pole.
// Arguments:
// r = distance from origin.
// theta = angle in degrees, counter-clockwise of X+ on the XY plane.
// phi = angle in degrees from the vertical Z+ axis.
// Example:
// xyz = spherical_to_xyz(20,30,40);
// xyz = spherical_to_xyz([40,60,50]);
// xyzs = spherical_to_xyz([[40,60,50], [50,120,100]]);
function spherical_to_xyz(r,theta,phi) =
theta != undef
? assert(is_num(r) && is_num(theta) && is_num(phi), "Bad Arguments.")
r*[cos(theta)*sin(phi), sin(theta)*sin(phi), cos(phi)]
: assert(is_list(r), "Bad Arguments")
is_num(r.x)
? spherical_to_xyz(r.x, r.y, r.z)
: [for(p = r) spherical_to_xyz(p.x, p.y, p.z)];
// Function: xyz_to_spherical()
// Usage:
// r_theta_phi = xyz_to_spherical(x,y,z)
// r_theta_phi = xyz_to_spherical([X,Y,Z])
// r_theta_phis = xyz_to_spherical([[X,Y,Z], [X,Y,Z], ...])
// Topics: Coordinates, Points, Paths
// Synopsis: Convert 3D cartesian coordinates to spherical coordinates.
// See Also: cylindrical_to_xyz(), spherical_to_xyz(), xyz_to_cylindrical(), altaz_to_xyz(), xyz_to_altaz()
// Description:
// Called with three arguments, converts the `x`, `y`, and `z` 3D cartesian coordinate into a `[RADIUS,THETA,PHI]` spherical coordinate.
// Called with one `[X,Y,Z]` vector argument, converts the 3D cartesian coordinate into a `[RADIUS,THETA,PHI]` spherical coordinate.
// Called with a list of `[X,Y,Z]` vector arguments, converts each 3D cartesian coordinate into `[RADIUS,THETA,PHI]` spherical coordinates.
// Theta is the angle counter-clockwise of X+ on the XY plane. Phi is the angle down from the Z+ pole.
// Arguments:
// x = X coordinate.
// y = Y coordinate.
// z = Z coordinate.
// Example:
// sph = xyz_to_spherical(20,30,40);
// sph = xyz_to_spherical([40,50,70]);
// sphs = xyz_to_spherical([[40,50,70], [25,-14,27]]);
function xyz_to_spherical(x,y,z) =
y != undef
? assert(is_num(x) && is_num(y) && is_num(z), "Bad Arguments.")
[norm([x,y,z]), atan2(y,x), atan2(norm([x,y]),z)]
: assert(is_list(x), "Bad Arguments")
is_num(x.x)
? xyz_to_spherical(x.x, x.y, x.z)
: [for(p = x) xyz_to_spherical(p.x, p.y, p.z)];
// Function: altaz_to_xyz()
// Synopsis: Convert altitude/azimuth/range to 3D cartesian coordinates.
// SynTags: Path
// Topics: Coordinates, Points, Paths
// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz(), xyz_to_cylindrical(), xyz_to_altaz()
// Usage:
// pt = altaz_to_xyz(alt, az, r);
// pt = altaz_to_xyz([ALT,AZ,R]);
// pts = altaz_to_xyz([[ALT,AZ,R], [ALT,AZ,R], ...]);
// Description:
// Convert altitude/azimuth/range coordinates to 3D cartesian coordinates.
// Called with three arguments, converts the `alt`, `az`, and 'r' 3D altitude-azimuth coordinate into an `[X,Y,Z]` cartesian coordinate.
// Called with one `[ALTITUDE,AZIMUTH,RANGE]` vector argument, converts the 3D alt-az coordinate into an `[X,Y,Z]` cartesian coordinate.
// Called with a list of `[ALTITUDE,AZIMUTH,RANGE]` vector arguments, converts each 3D alt-az coordinate into `[X,Y,Z]` cartesian coordinates.
// Altitude is the angle above the XY plane, Azimuth is degrees clockwise of Y+ on the XY plane, and Range is the distance from the origin.
// Arguments:
// alt = altitude angle in degrees above the XY plane.
// az = azimuth angle in degrees clockwise of Y+ on the XY plane.
// r = distance from origin.
// Example:
// xyz = altaz_to_xyz(20,30,40);
// xyz = altaz_to_xyz([40,60,50]);
function altaz_to_xyz(alt,az,r) =
az != undef
? assert(is_num(alt) && is_num(az) && is_num(r), "Bad Arguments.")
r*[cos(90-az)*cos(alt), sin(90-az)*cos(alt), sin(alt)]
: assert(is_list(alt), "Bad Arguments")
is_num(alt.x)
? altaz_to_xyz(alt.x, alt.y, alt.z)
: [for(p = alt) altaz_to_xyz(p.x, p.y, p.z)];
// Function: xyz_to_altaz()
// Synopsis: Convert 3D cartesian coordinates to [altitude,azimuth,range].
// Topics: Coordinates, Points, Paths
// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz(), xyz_to_cylindrical(), altaz_to_xyz()
// Usage:
// alt_az_r = xyz_to_altaz(x,y,z);
// alt_az_r = xyz_to_altaz([X,Y,Z]);
// alt_az_rs = xyz_to_altaz([[X,Y,Z], [X,Y,Z], ...]);
// Description:
// Converts 3D cartesian coordinates to altitude/azimuth/range coordinates.
// Called with three arguments, converts the `x`, `y`, and `z` 3D cartesian coordinate into an `[ALTITUDE,AZIMUTH,RANGE]` coordinate.
// Called with one `[X,Y,Z]` vector argument, converts the 3D cartesian coordinate into a `[ALTITUDE,AZIMUTH,RANGE]` coordinate.
// Called with a list of `[X,Y,Z]` vector arguments, converts each 3D cartesian coordinate into `[ALTITUDE,AZIMUTH,RANGE]` coordinates.
// Altitude is the angle above the XY plane, Azimuth is degrees clockwise of Y+ on the XY plane, and Range is the distance from the origin.
// Arguments:
// x = X coordinate.
// y = Y coordinate.
// z = Z coordinate.
// Example:
// aa = xyz_to_altaz(20,30,40);
// aa = xyz_to_altaz([40,50,70]);
function xyz_to_altaz(x,y,z) =
y != undef
? assert(is_num(x) && is_num(y) && is_num(z), "Bad Arguments.")
[atan2(z,norm([x,y])), atan2(x,y), norm([x,y,z])]
: assert(is_list(x), "Bad Arguments")
is_num(x.x)
? xyz_to_altaz(x.x, x.y, x.z)
: [for(p = x) xyz_to_altaz(p.x, p.y, p.z)];
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