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https://github.com/BelfrySCAD/BOSL2.git
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442 lines
18 KiB
OpenSCAD
442 lines
18 KiB
OpenSCAD
//////////////////////////////////////////////////////////////////////
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// LibFile: coords.scad
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// Coordinate transformations and coordinate system conversions.
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// Includes:
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// include <BOSL2/std.scad>
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//////////////////////////////////////////////////////////////////////
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// Section: Coordinate Manipulation
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// Function: point2d()
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// Usage:
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// pt = point2d(p, [fill]);
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// Topics: Coordinates, Points
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// See Also: path2d(), point3d(), path3d()
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// Description:
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// Returns a 2D vector/point from a 2D or 3D vector. If given a 3D point, removes the Z coordinate.
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// Arguments:
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// p = The coordinates to force into a 2D vector/point.
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// fill = Value to fill missing values in vector with.
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function point2d(p, fill=0) = [for (i=[0:1]) (p[i]==undef)? fill : p[i]];
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// Function: path2d()
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// Usage:
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// pts = path2d(points);
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// Topics: Coordinates, Points, Paths
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// See Also: point2d(), point3d(), path3d()
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// Description:
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// Returns a list of 2D vectors/points from a list of 2D, 3D or higher dimensional vectors/points.
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// Removes the extra coordinates from higher dimensional points. The input must be a path, where
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// every vector has the same length.
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// Arguments:
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// points = A list of 2D or 3D points/vectors.
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function path2d(points) =
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assert(is_path(points,dim=undef,fast=true),"Input to path2d is not a path")
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let (result = points * concat(ident(2), repeat([0,0], len(points[0])-2)))
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assert(is_def(result), "Invalid input to path2d")
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result;
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// Function: point3d()
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// Usage:
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// pt = point3d(p, [fill]);
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// Topics: Coordinates, Points
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// See Also: path2d(), point2d(), path3d()
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// Description:
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// Returns a 3D vector/point from a 2D or 3D vector.
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// Arguments:
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// p = The coordinates to force into a 3D vector/point.
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// fill = Value to fill missing values in vector with.
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function point3d(p, fill=0) = [for (i=[0:2]) (p[i]==undef)? fill : p[i]];
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// Function: path3d()
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// Usage:
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// pts = path3d(points, [fill]);
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// Topics: Coordinates, Points, Paths
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// See Also: point2d(), path2d(), point3d()
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// Description:
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// Returns a list of 3D vectors/points from a list of 2D or higher dimensional vectors/points
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// by removing extra coordinates or adding the z coordinate.
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// Arguments:
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// points = A list of 2D, 3D or higher dimensional points/vectors.
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// fill = Value to fill missing values in vectors with (in the 2D case)
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function path3d(points, fill=0) =
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assert(is_num(fill))
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assert(is_path(points, dim=undef, fast=true), "Input to path3d is not a path")
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let (
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change = len(points[0])-3,
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M = change < 0? [[1,0,0],[0,1,0]] :
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concat(ident(3), repeat([0,0,0],change)),
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result = points*M
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)
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assert(is_def(result), "Input to path3d is invalid")
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fill == 0 || change>=0 ? result : result + repeat([0,0,fill], len(result));
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// Function: point4d()
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// Usage:
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// pt = point4d(p, [fill]);
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// Topics: Coordinates, Points
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// See Also: point2d(), path2d(), point3d(), path3d(), path4d()
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// Description:
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// Returns a 4D vector/point from a 2D or 3D vector.
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// Arguments:
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// p = The coordinates to force into a 4D vector/point.
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// fill = Value to fill missing values in vector with.
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function point4d(p, fill=0) = [for (i=[0:3]) (p[i]==undef)? fill : p[i]];
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// Function: path4d()
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// Usage:
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// pt = path4d(points, [fill]);
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// Topics: Coordinates, Points, Paths
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// See Also: point2d(), path2d(), point3d(), path3d(), point4d()
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// Description:
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// Returns a list of 4D vectors/points from a list of 2D or 3D vectors/points.
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// Arguments:
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// points = A list of 2D or 3D points/vectors.
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// fill = Value to fill missing values in vectors with.
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function path4d(points, fill=0) =
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assert(is_num(fill) || is_vector(fill))
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assert(is_path(points, dim=undef, fast=true), "Input to path4d is not a path")
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let (
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change = len(points[0])-4,
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M = change < 0 ? select(ident(4), 0, len(points[0])-1) :
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concat(ident(4), repeat([0,0,0,0],change)),
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result = points*M
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)
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assert(is_def(result), "Input to path4d is invalid")
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fill == 0 || change >= 0 ? result :
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let(
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addition = is_list(fill) ? concat(0*points[0],fill) :
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concat(0*points[0],repeat(fill,-change))
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)
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assert(len(addition) == 4, "Fill is the wrong length")
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result + repeat(addition, len(result));
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// Section: Coordinate Systems
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// Function: polar_to_xy()
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// Usage:
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// pt = polar_to_xy(r, theta);
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// pt = polar_to_xy([r, theta]);
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// Topics: Coordinates, Points, Paths
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// See Also: xy_to_polar(), xyz_to_cylindrical(), cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
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// Description:
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// Convert polar coordinates to 2D cartesian coordinates.
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// Returns [X,Y] cartesian coordinates.
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// Arguments:
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// r = distance from the origin.
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// theta = angle in degrees, counter-clockwise of X+.
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// Examples:
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// xy = polar_to_xy(20,45); // Returns: ~[14.1421365, 14.1421365]
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// xy = polar_to_xy(40,30); // Returns: ~[34.6410162, 15]
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// xy = polar_to_xy([40,30]); // Returns: ~[34.6410162, 15]
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// Example(2D):
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// r=40; ang=30; $fn=36;
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// pt = polar_to_xy(r,ang);
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// stroke(circle(r=r), closed=true, width=0.5);
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// color("black") stroke([[r,0], [0,0], pt], width=0.5);
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// color("black") stroke(arc(r=15, angle=ang), width=0.5);
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// color("red") move(pt) circle(d=3);
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function polar_to_xy(r,theta=undef) = let(
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rad = theta==undef? r[0] : r,
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t = theta==undef? r[1] : theta
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) rad*[cos(t), sin(t)];
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// Function: xy_to_polar()
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// Usage:
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// r_theta = xy_to_polar(x,y);
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// r_theta = xy_to_polar([X,Y]);
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// Topics: Coordinates, Points, Paths
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// See Also: polar_to_xy(), xyz_to_cylindrical(), cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
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// Description:
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// Convert 2D cartesian coordinates to polar coordinates.
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// Returns [radius, theta] where theta is the angle counter-clockwise of X+.
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// Arguments:
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// x = X coordinate.
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// y = Y coordinate.
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// Examples:
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// plr = xy_to_polar(20,30);
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// plr = xy_to_polar([40,60]);
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// Example(2D):
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// pt = [-20,30]; $fn = 36;
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// rt = xy_to_polar(pt);
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// r = rt[0]; ang = rt[1];
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// stroke(circle(r=r), closed=true, width=0.5);
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// zrot(ang) stroke([[0,0],[r,0]],width=0.5);
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// color("red") move(pt) circle(d=3);
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function xy_to_polar(x,y=undef) = let(
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xx = y==undef? x[0] : x,
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yy = y==undef? x[1] : y
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) [norm([xx,yy]), atan2(yy,xx)];
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// Function: project_plane()
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// Usage:
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// xy = project_plane(plane, p);
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// Usage: To get a transform matrix
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// M = project_plane(plane)
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// Description:
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// Maps the provided 3d point(s) from 3D coordinates to a 2d coordinate system defined by `plane`. Points that are not
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// on the specified plane will be projected orthogonally onto the plane. This coordinate system is useful if you need
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// to perform 2d operations on a coplanar set of data. After those operations are done you can return the data
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// to 3d with `lift_plane()`. You could also use this to force approximately coplanar data to be exactly coplanar.
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// The parameter p can be a point, path, region, bezier patch or VNF.
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// The plane can be specified as
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// - 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.
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// - A list of coplanar points that define a plane (not-collinear)
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// - 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.
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// .
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// If you omit the point specification then `project_plane()` returns a rotation matrix that maps the specified plane to the XY plane.
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// Note that if you apply this transformation to data lying on the plane it will produce 3D points with the Z coordinate of zero.
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// Topics: Coordinates, Points, Paths
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// See Also: project_plane(), projection_on_plane()
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// Arguments:
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// plane = plane specification or point list defining the plane
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// p = 3D point, path, region, VNF or bezier patch to project
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// Example:
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// pt = [5,-5,5];
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// a=[0,0,0]; b=[10,-10,0]; c=[10,0,10];
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// xy = project_plane([a,b,c],pt);
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// Example(3D): The yellow points in 3D project onto the red points in 2D
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// M = [[-1, 2, -1, -2], [-1, -3, 2, -1], [2, 3, 4, 53], [0, 0, 0, 1]];
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// data = apply(M,path3d(circle(r=10, $fn=20)));
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// move_copies(data) sphere(r=1);
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// color("red") move_copies(project_plane(data, data)) sphere(r=1);
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// Example:
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// xyzpath = move([10,20,30], p=yrot(25, p=path3d(circle(d=100))));
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// mat = project_plane(xyzpath);
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// xypath = path2d(apply(mat, xyzpath));
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// #stroke(xyzpath,closed=true);
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// stroke(xypath,closed=true);
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function project_plane(plane,p) =
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is_matrix(plane,3,3) && is_undef(p) ? // no data, 3 points given
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assert(!collinear(plane),"Points defining the plane must not be collinear")
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let(
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v = plane[2]-plane[0],
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y = unit(plane[1]-plane[0]), // y axis goes to point b
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x = unit(v-(v*y)*y) // x axis
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)
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affine3d_frame_map(x,y) * move(-plane[0])
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: is_vector(plane,4) && is_undef(p) ? // no data, plane given in "plane"
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assert(_valid_plane(plane), "Plane is not valid")
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let(
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n = point3d(plane),
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cp = n * plane[3] / (n*n)
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)
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rot(from=n, to=UP) * move(-cp)
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: is_path(plane,3) && is_undef(p) ? // no data, generic point list plane
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assert(len(plane)>=3, "Need three points to define a plane")
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let(plane = plane_from_points(plane))
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assert(is_def(plane), "Point list is not coplanar")
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project_plane(plane)
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: assert(is_def(p), str("Invalid plane specification: ",plane))
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is_vnf(p) ? [project_plane(plane,p[0]), p[1]]
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: is_list(p) && is_list(p[0]) && is_vector(p[0][0],3) ? // bezier patch or region
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[for(plist=p) project_plane(plane,plist)]
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: assert(is_vector(p,3) || is_path(p,3),str("Data must be a 3d point, path, region, vnf or bezier patch",p))
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is_matrix(plane,3,3) ?
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assert(!collinear(plane),"Points defining the plane must not be collinear")
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let(
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v = plane[2]-plane[0],
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y = unit(plane[1]-plane[0]), // y axis goes to point b
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x = unit(v-(v*y)*y) // x axis
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) move(-plane[0],p) * transpose([x,y])
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: is_vector(p) ? point2d(apply(project_plane(plane),p))
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: path2d(apply(project_plane(plane),p));
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// Function: lift_plane()
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// Usage:
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// xyz = lift_plane(plane, p);
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// Usage: to get transform matrix
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// M = lift_plane(plane);
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// Topics: Coordinates, Points, Paths
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// See Also: project_plane()
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// Description:
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// Converts the given 2D point on the plane to 3D coordinates of the specified plane.
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// The parameter p can be a point, path, region, bezier patch or VNF.
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// The plane can be specified as
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// - 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.
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// - A list of coplanar points that define a plane (not-collinear)
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// - 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.
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// 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.
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// This matrix is a rotation, the inverse of the one produced by project_plane.
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// Arguments:
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// plane = Plane specification or list of points to define a plane
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// p = points, path, region, VNF, or bezier patch to transform.
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function lift_plane(plane, p) =
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is_matrix(plane,3,3) && is_undef(p) ? // no data, 3 p given
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let(
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v = plane[2]-plane[0],
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y = unit(plane[1]-plane[0]), // y axis goes to point b
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x = unit(v-(v*y)*y) // x axis
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)
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move(plane[0]) * affine3d_frame_map(x,y,reverse=true)
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: is_vector(plane,4) && is_undef(p) ? // no data, plane given in "plane"
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assert(_valid_plane(plane), "Plane is not valid")
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let(
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n = point3d(plane),
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cp = n * plane[3] / (n*n)
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)
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move(cp) * rot(from=UP, to=n)
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: is_path(plane,3) && is_undef(p) ? // no data, generic point list plane
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assert(len(plane)>=3, "Need three p to define a plane")
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let(plane = plane_from_points(plane))
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assert(is_def(plane), "Point list is not coplanar")
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lift_plane(plane)
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: is_vnf(p) ? [lift_plane(plane,p[0]), p[1]]
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: is_list(p) && is_list(p[0]) && is_vector(p[0][0],3) ? // bezier patch or region
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[for(plist=p) lift_plane(plane,plist)]
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: assert(is_vector(p,2) || is_path(p,2),"Data must be a 2d point, path, region, vnf or bezier patch")
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is_matrix(plane,3,3) ?
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let(
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v = plane[2]-plane[0],
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y = unit(plane[1]-plane[0]), // y axis goes to point b
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x = unit(v-(v*y)*y) // x axis
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) move(plane[0],p * [x,y])
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: apply(lift_plane(plane),is_vector(p) ? point3d(p) : path3d(p));
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// Function: cylindrical_to_xyz()
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// Usage:
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// pt = cylindrical_to_xyz(r, theta, z);
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// pt = cylindrical_to_xyz([r, theta, z]);
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// Topics: Coordinates, Points, Paths
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// See Also: xyz_to_cylindrical(), xyz_to_spherical(), spherical_to_xyz()
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// Description:
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// Convert cylindrical coordinates to 3D cartesian coordinates. Returns [X,Y,Z] cartesian coordinates.
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// Arguments:
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// r = distance from the Z axis.
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// theta = angle in degrees, counter-clockwise of X+ on the XY plane.
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// z = Height above XY plane.
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// Examples:
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// xyz = cylindrical_to_xyz(20,30,40);
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// xyz = cylindrical_to_xyz([40,60,50]);
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function cylindrical_to_xyz(r,theta=undef,z=undef) = let(
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rad = theta==undef? r[0] : r,
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t = theta==undef? r[1] : theta,
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zed = theta==undef? r[2] : z
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) [rad*cos(t), rad*sin(t), zed];
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// Function: xyz_to_cylindrical()
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// Usage:
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// rtz = xyz_to_cylindrical(x,y,z);
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// rtz = xyz_to_cylindrical([X,Y,Z]);
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// Topics: Coordinates, Points, Paths
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// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz()
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// Description:
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// Convert 3D cartesian coordinates to cylindrical coordinates. Returns [radius,theta,Z].
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// Theta is the angle counter-clockwise of X+ on the XY plane. Z is height above the XY plane.
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// Arguments:
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// x = X coordinate.
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// y = Y coordinate.
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// z = Z coordinate.
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// Examples:
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// cyl = xyz_to_cylindrical(20,30,40);
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// cyl = xyz_to_cylindrical([40,50,70]);
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function xyz_to_cylindrical(x,y=undef,z=undef) = let(
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p = is_num(x)? [x, default(y,0), default(z,0)] : point3d(x)
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) [norm([p.x,p.y]), atan2(p.y,p.x), p.z];
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// Function: spherical_to_xyz()
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// Usage:
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// pt = spherical_to_xyz(r, theta, phi);
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// pt = spherical_to_xyz([r, theta, phi]);
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// Description:
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// Convert spherical coordinates to 3D cartesian coordinates. Returns [X,Y,Z] cartesian coordinates.
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// Topics: Coordinates, Points, Paths
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// See Also: cylindrical_to_xyz(), xyz_to_spherical(), xyz_to_cylindrical()
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// Arguments:
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// r = distance from origin.
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// theta = angle in degrees, counter-clockwise of X+ on the XY plane.
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// phi = angle in degrees from the vertical Z+ axis.
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// Examples:
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// xyz = spherical_to_xyz(20,30,40);
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// xyz = spherical_to_xyz([40,60,50]);
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function spherical_to_xyz(r,theta=undef,phi=undef) = let(
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rad = theta==undef? r[0] : r,
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t = theta==undef? r[1] : theta,
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p = theta==undef? r[2] : phi
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) rad*[sin(p)*cos(t), sin(p)*sin(t), cos(p)];
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// Function: xyz_to_spherical()
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// Usage:
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// r_theta_phi = xyz_to_spherical(x,y,z)
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// r_theta_phi = xyz_to_spherical([X,Y,Z])
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// Topics: Coordinates, Points, Paths
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// See Also: cylindrical_to_xyz(), spherical_to_xyz(), xyz_to_cylindrical()
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// Description:
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// Convert 3D cartesian coordinates to spherical coordinates. Returns [r,theta,phi], where phi is
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// the angle from the Z+ pole, and theta is degrees counter-clockwise of X+ on the XY plane.
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// Arguments:
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// x = X coordinate.
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// y = Y coordinate.
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// z = Z coordinate.
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// Examples:
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// sph = xyz_to_spherical(20,30,40);
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// sph = xyz_to_spherical([40,50,70]);
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function xyz_to_spherical(x,y=undef,z=undef) = let(
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p = is_num(x)? [x, default(y,0), default(z,0)] : point3d(x)
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) [norm(p), atan2(p.y,p.x), atan2(norm([p.x,p.y]),p.z)];
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// Function: altaz_to_xyz()
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// Usage:
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// pt = altaz_to_xyz(alt, az, r);
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// pt = altaz_to_xyz([alt, az, r]);
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// Topics: Coordinates, Points, Paths
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// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz(), xyz_to_cylindrical(), xyz_to_altaz()
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// Description:
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// Convert altitude/azimuth/range coordinates to 3D cartesian coordinates.
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// Returns [X,Y,Z] cartesian coordinates.
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// Arguments:
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// alt = altitude angle in degrees above the XY plane.
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// az = azimuth angle in degrees clockwise of Y+ on the XY plane.
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// r = distance from origin.
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// Examples:
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// xyz = altaz_to_xyz(20,30,40);
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// xyz = altaz_to_xyz([40,60,50]);
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function altaz_to_xyz(alt,az=undef,r=undef) = let(
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p = az==undef? alt[0] : alt,
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t = 90 - (az==undef? alt[1] : az),
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rad = az==undef? alt[2] : r
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) rad*[cos(p)*cos(t), cos(p)*sin(t), sin(p)];
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// Function: xyz_to_altaz()
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// Usage:
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// alt_az_r = xyz_to_altaz(x,y,z);
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// alt_az_r = xyz_to_altaz([X,Y,Z]);
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// Topics: Coordinates, Points, Paths
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// See Also: cylindrical_to_xyz(), xyz_to_spherical(), spherical_to_xyz(), xyz_to_cylindrical(), altaz_to_xyz()
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// Description:
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// Convert 3D cartesian coordinates to altitude/azimuth/range coordinates.
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// Returns [altitude,azimuth,range], where altitude is angle above the
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// XY plane, azimuth is degrees clockwise of Y+ on the XY plane, and
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// range is the distance from the origin.
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// Arguments:
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// x = X coordinate.
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// y = Y coordinate.
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// z = Z coordinate.
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// Examples:
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// aa = xyz_to_altaz(20,30,40);
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// aa = xyz_to_altaz([40,50,70]);
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function xyz_to_altaz(x,y=undef,z=undef) = let(
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p = is_num(x)? [x, default(y,0), default(z,0)] : point3d(x)
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) [atan2(p.z,norm([p.x,p.y])), atan2(p.x,p.y), norm(p)];
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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