mirror of
https://github.com/BelfrySCAD/BOSL2.git
synced 2024-12-29 00:09:41 +00:00
447 lines
18 KiB
OpenSCAD
447 lines
18 KiB
OpenSCAD
//////////////////////////////////////////////////////////////////////
|
|
// 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()
|
|
// Usage:
|
|
// pt = point2d(p, [fill]);
|
|
// Topics: Coordinates, Points
|
|
// See Also: path2d(), point3d(), path3d()
|
|
// 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.
|
|
function point2d(p, fill=0) = assert(is_list(p)) [for (i=[0:1]) (p[i]==undef)? fill : p[i]];
|
|
|
|
|
|
// Function: path2d()
|
|
// Usage:
|
|
// pts = path2d(points);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: point2d(), point3d(), path3d()
|
|
// 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()
|
|
// Usage:
|
|
// pt = point3d(p, [fill]);
|
|
// Topics: Coordinates, Points
|
|
// See Also: path2d(), point2d(), path3d()
|
|
// 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.
|
|
function point3d(p, fill=0) =
|
|
assert(is_list(p))
|
|
[for (i=[0:2]) (p[i]==undef)? fill : p[i]];
|
|
|
|
|
|
// Function: path3d()
|
|
// Usage:
|
|
// pts = path3d(points, [fill]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: point2d(), path2d(), point3d()
|
|
// 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)
|
|
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()
|
|
// Usage:
|
|
// pt = point4d(p, [fill]);
|
|
// Topics: Coordinates, Points
|
|
// See Also: point2d(), path2d(), point3d(), path3d(), path4d()
|
|
// 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.
|
|
function point4d(p, fill=0) = assert(is_list(p))
|
|
[for (i=[0:3]) (p[i]==undef)? fill : p[i]];
|
|
|
|
|
|
// Function: path4d()
|
|
// Usage:
|
|
// pt = path4d(points, [fill]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: point2d(), path2d(), point3d(), path3d(), point4d()
|
|
// 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.
|
|
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()
|
|
// Usage:
|
|
// pt = polar_to_xy(r, theta);
|
|
// pt = polar_to_xy([r, theta]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: xy_to_polar(), xyz_to_cylindrical(), cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
|
|
// Description:
|
|
// Convert polar coordinates to 2D cartesian coordinates.
|
|
// Returns [X,Y] cartesian coordinates.
|
|
// 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]
|
|
// 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=undef) = let(
|
|
rad = theta==undef? r[0] : r,
|
|
t = theta==undef? r[1] : theta
|
|
) rad*[cos(t), sin(t)];
|
|
|
|
|
|
// Function: xy_to_polar()
|
|
// Usage:
|
|
// r_theta = xy_to_polar(x,y);
|
|
// r_theta = xy_to_polar([X,Y]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: polar_to_xy(), xyz_to_cylindrical(), cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
|
|
// Description:
|
|
// Convert 2D cartesian coordinates to polar coordinates.
|
|
// Returns [radius, theta] where theta is the angle counter-clockwise of X+.
|
|
// Arguments:
|
|
// x = X coordinate.
|
|
// y = Y coordinate.
|
|
// Example:
|
|
// plr = xy_to_polar(20,30);
|
|
// plr = xy_to_polar([40,60]);
|
|
// 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=undef) = let(
|
|
xx = y==undef? x[0] : x,
|
|
yy = y==undef? x[1] : y
|
|
) [norm([xx,yy]), atan2(yy,xx)];
|
|
|
|
|
|
// Function: project_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.
|
|
// Topics: Coordinates, Points, Paths
|
|
// 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()
|
|
// Usage:
|
|
// xyz = lift_plane(plane, p);
|
|
// Usage: to get transform matrix
|
|
// M = lift_plane(plane);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: project_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()
|
|
// Usage:
|
|
// pt = cylindrical_to_xyz(r, theta, z);
|
|
// pt = cylindrical_to_xyz([r, theta, z]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: xyz_to_cylindrical(), xyz_to_spherical(), spherical_to_xyz()
|
|
// Description:
|
|
// Convert cylindrical coordinates to 3D cartesian coordinates. Returns [X,Y,Z] cartesian coordinates.
|
|
// 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=undef,z=undef) = let(
|
|
rad = theta==undef? r[0] : r,
|
|
t = theta==undef? r[1] : theta,
|
|
zed = theta==undef? r[2] : z
|
|
) [rad*cos(t), rad*sin(t), zed];
|
|
|
|
|
|
// Function: xyz_to_cylindrical()
|
|
// Usage:
|
|
// rtz = xyz_to_cylindrical(x,y,z);
|
|
// rtz = xyz_to_cylindrical([X,Y,Z]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
|
|
// Description:
|
|
// Convert 3D cartesian coordinates to cylindrical coordinates. Returns [radius,theta,Z].
|
|
// 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]);
|
|
function xyz_to_cylindrical(x,y=undef,z=undef) = let(
|
|
p = is_num(x)? [x, default(y,0), default(z,0)] : point3d(x)
|
|
) [norm([p.x,p.y]), atan2(p.y,p.x), p.z];
|
|
|
|
|
|
// Function: spherical_to_xyz()
|
|
// Usage:
|
|
// pt = spherical_to_xyz(r, theta, phi);
|
|
// pt = spherical_to_xyz([r, theta, phi]);
|
|
// Description:
|
|
// Convert spherical coordinates to 3D cartesian coordinates. Returns [X,Y,Z] cartesian coordinates.
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: cylindrical_to_xyz(), xyz_to_spherical(), xyz_to_cylindrical()
|
|
// 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]);
|
|
function spherical_to_xyz(r,theta=undef,phi=undef) = let(
|
|
rad = theta==undef? r[0] : r,
|
|
t = theta==undef? r[1] : theta,
|
|
p = theta==undef? r[2] : phi
|
|
) rad*[sin(p)*cos(t), sin(p)*sin(t), cos(p)];
|
|
|
|
|
|
// Function: xyz_to_spherical()
|
|
// Usage:
|
|
// r_theta_phi = xyz_to_spherical(x,y,z)
|
|
// r_theta_phi = xyz_to_spherical([X,Y,Z])
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: cylindrical_to_xyz(), spherical_to_xyz(), xyz_to_cylindrical()
|
|
// Description:
|
|
// Convert 3D cartesian coordinates to spherical coordinates. Returns [r,theta,phi], where phi is
|
|
// the angle from the Z+ pole, and theta is degrees counter-clockwise of X+ on the XY plane.
|
|
// 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]);
|
|
function xyz_to_spherical(x,y=undef,z=undef) = let(
|
|
p = is_num(x)? [x, default(y,0), default(z,0)] : point3d(x)
|
|
) [norm(p), atan2(p.y,p.x), atan2(norm([p.x,p.y]),p.z)];
|
|
|
|
|
|
// Function: altaz_to_xyz()
|
|
// Usage:
|
|
// pt = altaz_to_xyz(alt, az, r);
|
|
// pt = altaz_to_xyz([alt, az, r]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz(), xyz_to_cylindrical(), xyz_to_altaz()
|
|
// Description:
|
|
// Convert altitude/azimuth/range coordinates to 3D cartesian coordinates.
|
|
// Returns [X,Y,Z] cartesian coordinates.
|
|
// 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=undef,r=undef) = let(
|
|
p = az==undef? alt[0] : alt,
|
|
t = 90 - (az==undef? alt[1] : az),
|
|
rad = az==undef? alt[2] : r
|
|
) rad*[cos(p)*cos(t), cos(p)*sin(t), sin(p)];
|
|
|
|
|
|
// Function: xyz_to_altaz()
|
|
// Usage:
|
|
// alt_az_r = xyz_to_altaz(x,y,z);
|
|
// alt_az_r = xyz_to_altaz([X,Y,Z]);
|
|
// Topics: Coordinates, Points, Paths
|
|
// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz(), xyz_to_cylindrical(), altaz_to_xyz()
|
|
// Description:
|
|
// Convert 3D cartesian coordinates to altitude/azimuth/range coordinates.
|
|
// Returns [altitude,azimuth,range], where altitude is 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=undef,z=undef) = let(
|
|
p = is_num(x)? [x, default(y,0), default(z,0)] : point3d(x)
|
|
) [atan2(p.z,norm([p.x,p.y])), atan2(p.x,p.y), norm(p)];
|
|
|
|
|
|
|
|
// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
|