BOSL2/coords.scad

//////////////////////////////////////////////////////////////////////
// LibFile: coords.scad
//   Coordinate transformations and coordinate system conversions.
// Includes:
//   include <BOSL2/std.scad>
//////////////////////////////////////////////////////////////////////


// Section: Coordinate Manipulation

// Function: point2d()
// 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) = [for (i=[0:1]) (p[i]==undef)? fill : p[i]];


// Function: path2d()
// 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()
// 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) = [for (i=[0:2]) (p[i]==undef)? fill : p[i]];


// Function: path3d()
// 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()
// 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) = [for (i=[0:3]) (p[i]==undef)? fill : p[i]];


// Function: path4d()
// 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:
//   polar_to_xy(r, theta);
//   polar_to_xy([r, theta]);
// 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+.
// Examples:
//   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]
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:
//   xy_to_polar(x,y);
//   xy_to_polar([X,Y]);
// 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.
// Examples:
//   plr = xy_to_polar(20,30);
//   plr = xy_to_polar([40,60]);
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: With the plane defined by 3 Points
//   xyz = project_plane(point, a, b, c);
// Usage: With the plane defined by Pointlist
//   xyz = project_plane(point, POINTLIST);
// Usage: With the plane defined by Plane Definition [A,B,C,D] Where Ax+By+Cz=D
//   xyz = project_plane(point, PLANE);
// Description:
//   Converts the given 3D points from global coordinates to the 2D planar coordinates of the closest
//   points on the plane.  This coordinate system can be useful in taking a set of nearly coplanar
//   points, and converting them to a pure XY set of coordinates for manipulation, before converting
//   them back to the original 3D plane. The parameter `point` may be a single point or a list of points
//   The plane may be given in one of three ways:
//   - by three points, `a`, `b`, and `c`, the planar coordinate system will have `[0,0]` at point `a`, and the Y+ axis will be towards point `b`.
//   - by a list of points passed by `a`, finds three reasonably spaced non-collinear points in the list and uses them as points `a`, `b`, and `c` as above.
//   - by a plane definition `[A,B,C,D]` passed by `a` where `Ax+By+Cz=D`, the closest point on that plane to the global origin at `[0,0,0]` will be the planar coordinate origin `[0,0]`.
// Arguments:
//   point = The 3D point, or list of 3D points to project into the plane's 2D coordinate system.
//   a = A 3D point that the plane passes through or a list of points or a plane definition vector.
//   b = A 3D point that the plane passes through.  Used to define the plane.
//   c = A 3D point that the plane passes through.  Used to define the plane.
// Example:
//   pt = [5,-5,5];
//   a=[0,0,0];  b=[10,-10,0];  c=[10,0,10];
//   xy = project_plane(pt, a, b, c);
//   xy2 = project_plane(pt, [a,b,c]);
function project_plane(point, a, b, c) =
    is_undef(b) && is_undef(c) && is_list(a)? let(
        mat = is_vector(a,4)? plane_transform(a) :
            assert(is_path(a) && len(a)>=3)
            plane_transform(plane_from_points(a)),
        pts = is_vector(point)? point2d(apply(mat,point)) :
            is_path(point)? path2d(apply(mat,point)) :
            is_region(point)? [for (x=point) path2d(apply(mat,x))] :
            assert(false, "point must be a 3D point, path, or region.")
    ) pts :
    assert(is_vector(a))
    assert(is_vector(b))
    assert(is_vector(c))
    assert(is_vector(point)||is_path(point))
    let(
        u = unit(b-a),
        v = unit(c-a),
        n = unit(cross(u,v)),
        w = unit(cross(n,u)),
        relpoint = apply(move(-a),point)
    ) relpoint * transpose([w,u]);


// Function: lift_plane()
// Usage: With 3 Points
//   xyz = lift_plane(point, a, b, c);
// Usage: With Pointlist
//   xyz = lift_plane(point, POINTLIST);
// Usage: With Plane Definition [A,B,C,D] Where Ax+By+Cz=D
//   xyz = lift_plane(point, PLANE);
// Description:
//   Converts the given 2D point from planar coordinates to the global 3D coordinates of the point on the plane.
//   Can be called one of three ways:
//   - Given three points, `a`, `b`, and `c`, the planar coordinate system will have `[0,0]` at point `a`, and the Y+ axis will be towards point `b`.
//   - Given a list of points, finds three non-collinear points in the list and uses them as points `a`, `b`, and `c` as above.
//   - Given a plane definition `[A,B,C,D]` where `Ax+By+Cz=D`, the closest point on that plane to the global origin at `[0,0,0]` will be the planar coordinate origin `[0,0]`.
// Arguments:
//   point = The 2D point, or list of 2D points in the plane's coordinate system to get the 3D position of.
//   a = A 3D point that the plane passes through.  Used to define the plane.
//   b = A 3D point that the plane passes through.  Used to define the plane.
//   c = A 3D point that the plane passes through.  Used to define the plane.
function lift_plane(point, a, b, c) =
    is_undef(b) && is_undef(c) && is_list(a)? let(
        mat = is_vector(a,4)? plane_transform(a) :
            assert(is_path(a) && len(a)>=3)
            plane_transform(plane_from_points(a)),
        imat = matrix_inverse(mat),
        pts = is_vector(point)? apply(imat,point3d(point)) :
            is_path(point)? apply(imat,path3d(point)) :
            is_region(point)? [for (x=point) apply(imat,path3d(x))] :
            assert(false, "point must be a 2D point, path, or region.")
    ) pts :
    assert(is_vector(a))
    assert(is_vector(b))
    assert(is_vector(c))
    assert(is_vector(point)||is_path(point))
    let(
        u = unit(b-a),
        v = unit(c-a),
        n = unit(cross(u,v)),
        w = unit(cross(n,u)),
        remapped = point*[w,u]
    ) apply(move(a),remapped);


// Function: cylindrical_to_xyz()
// Usage:
//   cylindrical_to_xyz(r, theta, z)
//   cylindrical_to_xyz([r, theta, z])
// 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.
// Examples:
//   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:
//   xyz_to_cylindrical(x,y,z)
//   xyz_to_cylindrical([X,Y,Z])
// 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.
// Examples:
//   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:
//   spherical_to_xyz(r, theta, phi);
//   spherical_to_xyz([r, theta, phi]);
// Description:
//   Convert spherical coordinates to 3D cartesian coordinates.
//   Returns [X,Y,Z] cartesian coordinates.
// 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.
// Examples:
//   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:
//   xyz_to_spherical(x,y,z)
//   xyz_to_spherical([X,Y,Z])
// 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.
// Examples:
//   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:
//   altaz_to_xyz(alt, az, r);
//   altaz_to_xyz([alt, az, r]);
// 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.
// Examples:
//   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:
//   xyz_to_altaz(x,y,z);
//   xyz_to_altaz([X,Y,Z]);
// 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.
// Examples:
//   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)];



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