Class SplinePath3D

All Implemented Interfaces:
Cloneable
Direct Known Subclasses:
BasicSplinePath3D

public class SplinePath3D extends Path3D.Double
Path3D with smooth curves connecting sequences of points. This class extends Path3D.Double by using cubic Bézier splines to connect sequences of point. This can be done either by using a constructor (as a convenience) or by using the methods splineTo to add a smooth curve segment or addCycle to add smooth closed path. If splineTo is used, the starting point of the spline is the current point, which is initially set using the Path3D moveTo method. The smooth curves created are cubic Bézier curve segments arranged so that the first and second derivatives of their coordinates with respect to the curves' parameters match (one curve at u = 1 and the other at u = 0). For a call to splineTo, at the end points, the tangent to the curve points towards its neighbor. As a result, calling splineTo for points p1, p2, p3, p4, followed by a call to splineTo for points p5, p6, p7, p8 is not equivalent to calling splineTo once for points p1, p2, ..., p8: in the former case, the tangent lines can differ on both sides of point p4.

When addCycle is used, there is an implicit call to moveTo for the first point in the array and this point becomes the last point, but otherwise the end points are not treated specially. Regardless of whether splineTo, addCycle, or constructor is used, the smoothed paths are simply sequences of calls to at most one initial moveTo method and some number of curveTo methods.

The algorithm is described in an article, "Smooth Bézier Spline Through Prescribed Points" and works by making the first and second derivatives of Bézier curves (differentiating with respect to the parameter u) match at the given points. Each segment is a Bézier curve given by

B(u) = (1-u)3P0 + 3(1-u)2uP1 + 3(1-u)u2P2 + t3P3.

For n-1 cubic splines with control points $P0,i P1,i, P1,2 and P3,i, with i in the interval [0, n-1], connecting the points K0 ... Kn the P1 control points satisfy the equations

  • 2P1,0 + P1,1 = 4K0 + 2K1
  • P1,i-1 + 4P1,i + P1,i+1 = 4Ki + 2Ki+1 for i in [1, n-2]
  • 2Pi,n-2 + 7P1,n-1 = 8Kn-1 + Kn.
The initial and final control points for each segment are given by
  • P0,i = Ki
  • P3,i = Ki+1
with both defined for i in the interval [0, n-1]. The P2 control point is given by the equations
  • P2,i = 2Ki+1 - P1,i+1 for i in [0,n-2].
  • P2,n-1 = (Kn + P1,n-1)/2.
There is a typo in the article cited above giving the wrong value for the second control point.

For a closed paths, a case not covered in the article cited above, the equation for P1 are given by

  • P1,n + 4P1,0 + P1,i+1 = 4Ki + 2Ki+1
  • P1,i-1 + 4P1,i + P1,i+1 = 4Ki + 2Ki+1 for i in [1, n-1]
  • P1,n-1 + 4P1,n + P1,0 = 4Kn + 2K0
The other control points for closed paths are given by
  • P0,i = Ki for i in [0,n]
  • P2,i = 2Ki+1 - P1,i+1 for i in [0, n-1]
  • P2,n = 2K0 - P1,0
  • P3,i = Ki+1 for i in [0, n-1].
  • P3,n = K0.
  • Nested Class Summary

    Nested classes/interfaces inherited from class org.bzdev.geom.Path3D

    Path3D.Double, Path3D.Float
  • Constructor Summary

    Constructors
    Constructor
    Description
    Constructs a new empty SplinePath3D object.
    SplinePath3D(double[] x, double[] y, double[] z, boolean closed)
    Constructs a new SplinePath3D object from arrays of X, Y and Z coordinates.
    SplinePath3D(double[] x, double[] y, double[] z, int n, boolean closed)
    Constructs a new SplinePath3D object from arrays containing at least n x and y coordinates.
    SplinePath3D(int initialCapacity)
    Constructs a new empty SplinePath3D object with the specified initial capacity to store path segments.
    SplinePath3D(int initialCapacity, double[] x, double[] y, double[] z, boolean closed)
    Constructs a new SplinePath3D object from arrays containing at least n X, Y and Z coordinates with the specified initial capacity for storing segments.
    SplinePath3D(int initialCapacity, double[] x, double[] y, double[] z, int n, boolean closed)
    Constructs a new SplinePath3D object from arrays containing at least n X, Y and Z coordinates with the specified initial capacity for storing segments.
    SplinePath3D(int initialCapacity, Point3D[] pk, boolean closed)
    Constructs a new SplinePath3D object from an array of points, given an initial capacity.
    SplinePath3D(int initialCapacity, Point3D[] pk, int n, boolean closed)
    Constructs a new SplinePath3D object from an array containing at least n points, given an initial capacity.
    SplinePath3D(int initialCapacity, CubicSpline xf, CubicSpline yf, CubicSpline zf, boolean closed)
    Constructs a new SplinePath3D object given cubic splines specifying the x, y and z coordinates and an initial capacity.
    SplinePath3D(int initialCapacity, RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n, boolean closed)
    Constructs a new SplinePath3D object from functions giving x and y coordinates with the specified initial capacity for storing segments.
    Constructor using another path to determine the initial path segments.
    SplinePath3D(Path3D path, Transform3D transform)
    Constructor using another path to determine the initial path segments, modified with a 3D transform.
    SplinePath3D(Point3D[] pk, boolean closed)
    Constructs a new SplinePath3D object from an array of points.
    SplinePath3D(Point3D[] pk, int n, boolean closed)
    Constructs a new SplinePath3D object from an array containing at least n points.
    SplinePath3D(Point3D[] pk, int offset, int n, boolean closed)
    Constructs a new SplinePath3D object from an array containing at least n points, starting at an offset.
    SplinePath3D(CubicSpline xf, CubicSpline yf, CubicSpline zf, boolean closed)
    Constructs a new SplinePath3D object given cubic splines specifying the x, y and z coordinates.
    SplinePath3D(RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n, boolean closed)
    Constructs a new SplinePath3D object given functions specifying the X, Y and Z coordinates.
  • Method Summary

    Modifier and Type
    Method
    Description
    void
    addCycle(double[] x, double[] y, double[] z)
    Add a sequence of segments that form a spline, specified as arrays of X, Y and Z coordinates, forming a closed path starting at index 0.
    void
    addCycle(double[] x, double[] y, double[] z, int n)
    Add a sequence of segments that form a spline, specified as arrays of at least n X, Y and Z coordinates, forming a closed path starting at index 0.
    void
    addCycle(Point3D... points)
    Add a sequence of segments that form a spline, specified as Point3D arguments to the addCycle method, and forming a closed path starting at index 0.
    void
    addCycle(Point3D[] pk, int n)
    Add a sequence of segments that form a spline, specified as an array of at least n points, forming a closed path starting at index 0.
    void
    append(CubicSpline xf, CubicSpline yf, CubicSpline zf, boolean connect)
    Append a spline to this path.
    void
    cycleTo(double[] x, double[] y, double[] z)
    Add a sequence of segments that form a spline, specified as arrays of X, Y and Z coordinates, forming a smooth closed path.
    void
    cycleTo(double[] x, double[] y, double[] z, int n)
    Add a sequence of segments that form a spline, specified as arrays of at least n X, Y and Z coordinates, forming a smooth closed path.
    void
    cycleTo(Point3D... points)
    Add a sequence of segments that form a spline, specified as Point3D arguments to the cycleTo method, and forming a smooth closed path.
    void
    cycleTo(Point3D[] pk, int n)
    Add a sequence of segments that form a spline, specified as an array of points, forming a smooth closed path.
    void
    cycleTo(RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n)
    Add a sequence of segments that form a spline, specified as functions giving the X, Y and Z coordinates of points along the path and forming a smooth closed path.
    void
    splineTo(double[] x, double[] y, double[] z)
    Add a sequence of segments that form a spline, specified as arrays of X, Y and Z coordinates.
    void
    splineTo(double[] x, double[] y, double[] z, int n)
    Add a sequence of segments that form a spline, specified as arrays of at least n X, Y and Z coordinates.
    void
    splineTo(Point3D... points)
    Add a sequence of segments that form a spline, specified as arguments passed to the splineTo method.
    void
    splineTo(Point3D[] pk, int n)
    Add a sequence of segments that form a spline, specified as an array of points.
    void
    splineTo(Point3D[] pk, int offset, int n)
    Add a sequence of segments that form a spline, specified as an array of points with an offset.
    void
    splineTo(RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n)
    Add a sequence of segments that form a spline, specified as functions giving the X, Y and Z coordinates.

    Methods inherited from class org.bzdev.geom.Path3D

    append, closePath, getBounds, getCurrentPoint, isEmpty, reset, setupCubic, setupCubic, setupCubic, setupCubic

    Methods inherited from class java.lang.Object

    equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
  • Constructor Details

    • SplinePath3D

      public SplinePath3D()
      Constructs a new empty SplinePath3D object.
    • SplinePath3D

      public SplinePath3D(int initialCapacity)
      Constructs a new empty SplinePath3D object with the specified initial capacity to store path segments.
      Parameters:
      initialCapacity - an estimate for the number of path segments in the path
    • SplinePath3D

      public SplinePath3D(Path3D path)
      Constructor using another path to determine the initial path segments.
      Parameters:
      path - the path whose segments will be copied
    • SplinePath3D

      public SplinePath3D(Path3D path, Transform3D transform)
      Constructor using another path to determine the initial path segments, modified with a 3D transform.
      Parameters:
      path - the path whose segments will be copied
      transform - the transform to apply to the path segments
    • SplinePath3D

      public SplinePath3D(Point3D[] pk, int n, boolean closed)
      Constructs a new SplinePath3D object from an array containing at least n points. The array specifies the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      pk - the array of points that make up the knots of a spline
      n - the number of points in the array to use, with valid indices in the range [0, n)
      closed - true if the spline forms a closed path; false otherwise
    • SplinePath3D

      public SplinePath3D(Point3D[] pk, int offset, int n, boolean closed)
      Constructs a new SplinePath3D object from an array containing at least n points, starting at an offset. The array specifies the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      pk - the array of points that contain up the knots of a spline
      offset - the starting index into the array
      n - the number of points in the array to use, with valid indices in the range [offset, offset+n)
      closed - true if the spline forms a closed path; false otherwise
    • SplinePath3D

      public SplinePath3D(Point3D[] pk, boolean closed)
      Constructs a new SplinePath3D object from an array of points. The array specifies the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      pk - the array of points that make up the knots of a spline
      closed - true if the spline forms a closed path; false otherwise
    • SplinePath3D

      public SplinePath3D(int initialCapacity, Point3D[] pk, int n, boolean closed)
      Constructs a new SplinePath3D object from an array containing at least n points, given an initial capacity. The array specifies the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      initialCapacity - an estimate for the number of path segments in the path
      pk - the array of points that make up the knots of a spline
      n - the number of points in the array to use, with valid indices in the range [0, n)
      closed - true if the spline forms a closed path; false otherwise
    • SplinePath3D

      public SplinePath3D(int initialCapacity, Point3D[] pk, boolean closed)
      Constructs a new SplinePath3D object from an array of points, given an initial capacity. The array specifies the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      initialCapacity - an estimate for the number of path segments in the path
      pk - the array of points that make up the knots of a spline
      closed - true if the spline forms a closed path; false otherwise
    • SplinePath3D

      public SplinePath3D(double[] x, double[] y, double[] z, int n, boolean closed)
      Constructs a new SplinePath3D object from arrays containing at least n x and y coordinates. The array specifies the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      n - the number of points in the array to use, with valid indices in the range [0, n)
      closed - true if the spline forms a closed path; false otherwise
    • SplinePath3D

      public SplinePath3D(CubicSpline xf, CubicSpline yf, CubicSpline zf, boolean closed) throws IllegalArgumentException
      Constructs a new SplinePath3D object given cubic splines specifying the x, y and z coordinates. The functions must be instances of CubicSpline and must be created with the same number of "knots". The argument to each function will be rescaled so that it will be incremented by 1.0 when moving from one knot to the next. If closed is true and the initial knot is not at the same location as the final knot, a straight line segment will be used to close the path.

      This constructor is provided to support cases where the default algorithm for creating a spline is not appropriate: the class CubicSpline allows one to specify a number of types of splines, with most differing in how the ends are treated, but with one case (A Hermite spline) explicitly giving the derivative at each knot.

      Parameters:
      xf - the spline specifying the x coordinates to use
      yf - the spline specifying the y coordinates to use
      zf - the spline specifying the y coordinates to use
      closed - true if the spline forms a closed path; false otherwise
      Throws:
      IllegalArgumentException - The splines xf, yf and zf did not have the same number of knots
    • SplinePath3D

      public SplinePath3D(int initialCapacity, CubicSpline xf, CubicSpline yf, CubicSpline zf, boolean closed) throws IllegalArgumentException
      Constructs a new SplinePath3D object given cubic splines specifying the x, y and z coordinates and an initial capacity. The splines must be created with the same number of "knots". The argument to each function will be rescaled so that it will be incremented by 1.0 when moving from one knot to the next. The path will be constructed using the control points associated with the cubic splines providing the x, y, and z coordinates. If closed is true and the initial knot is not at the same location as the final knot, a straight line segment will be used to close the path.

      This constructor is provided to support cases where the default algorithm for creating a spline is not appropriate: the class CubicSpline allows one to specify a number of types of splines, with most differing in how the ends are treated, but with one case (A Hermite spline) explicitly giving the derivative at each knot.

      Parameters:
      initialCapacity - an estimate for the number of path segments in the path
      xf - the spline specifying the x coordinates to use
      yf - the spline specifying the y coordinates to use
      zf - the spline specifying the y coordinates to use
      closed - true if the spline forms a closed path; false otherwise
      Throws:
      IllegalArgumentException - The spline xf and yf did not have the same number of knots
      IllegalArgumentException - The splines xf, yf and zf did not have the same number of knots
    • SplinePath3D

      public SplinePath3D(RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n, boolean closed) throws IllegalArgumentException
      Constructs a new SplinePath3D object given functions specifying the X, Y and Z coordinates. For a closed path, the last point should not repeat the initial point.
      Parameters:
      xf - the function specifying the x coordinates to use
      yf - the function specifying the y coordinates to use
      zf - the function specifying the z coordinates to use
      t1 - an end point for the domain of the argument of xf and yf
      t2 - the other end point for the domain of the argument of xf and yf
      n - the number of segments between the points at which to evaluate the functions
      closed - true if the spline forms a closed path; false otherwise
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because n was not positive or either t1 or t2 were not in the domain of xf or yf.
    • SplinePath3D

      public SplinePath3D(double[] x, double[] y, double[] z, boolean closed)
      Constructs a new SplinePath3D object from arrays of X, Y and Z coordinates. The array specifies the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      closed - true if the spline forms a closed path; false otherwise
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because the lengths of the arrays differ
    • SplinePath3D

      public SplinePath3D(int initialCapacity, double[] x, double[] y, double[] z, int n, boolean closed)
      Constructs a new SplinePath3D object from arrays containing at least n X, Y and Z coordinates with the specified initial capacity for storing segments. The arrays specify the X, Y and Z coordinates of the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      initialCapacity - an estimate for the number of path segments in the path
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      n - the number of points in the array to use, with valid indices in the range [0, n)
      closed - true if the spline forms a closed path; false otherwise
    • SplinePath3D

      public SplinePath3D(int initialCapacity, RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n, boolean closed) throws IllegalArgumentException
      Constructs a new SplinePath3D object from functions giving x and y coordinates with the specified initial capacity for storing segments. For a closed path, the last point should not repeat the initial point.
      Parameters:
      initialCapacity - an estimate for the number of path segments in the path
      xf - the function specifying the x coordinates to use
      yf - the function specifying the y coordinates to use
      zf - the function specifying the z coordinates to use
      t1 - an end point for the domain of the argument of xf and yf
      t2 - the other end point for the domain of the argument of xf and yf
      n - the number of segments between the points at which to evaluate the functions
      closed - true if the spline forms a closed path; false otherwise
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because n was not positive or either t1 or t2 were not in the domain of xf or yf.
    • SplinePath3D

      public SplinePath3D(int initialCapacity, double[] x, double[] y, double[] z, boolean closed)
      Constructs a new SplinePath3D object from arrays containing at least n X, Y and Z coordinates with the specified initial capacity for storing segments. The arrays specify the X, Y and Z coordinates of the "knots" of the spline - the points the spline is constrained to pass through. For a closed path, the last point should not repeat the initial point.
      Parameters:
      initialCapacity - an estimate for the number of path segments in the path
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      closed - true if the spline forms a closed path; false otherwise
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because the lengths of the arrays differ
  • Method Details

    • splineTo

      public void splineTo(Point3D[] pk, int n)
      Add a sequence of segments that form a spline, specified as an array of points. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at intermediate points. At the end points, the direction of the tangent line points towards the adjacent knot. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.).
      Parameters:
      pk - the array of points that make up the knots of a spline
      n - the number of points in the array to use, with valid indices in the range [0, n)
    • splineTo

      public void splineTo(Point3D[] pk, int offset, int n)
      Add a sequence of segments that form a spline, specified as an array of points with an offset. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at intermediate points. At the end points, the direction of the tangent line points towards the adjacent knot. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.).
      Parameters:
      pk - the array of points that make up the knots of a spline
      offset - the offset into the array for the starting point
      n - the number of points in the array to use, with valid indices in the range [0, n)
    • splineTo

      public void splineTo(Point3D... points)
      Add a sequence of segments that form a spline, specified as arguments passed to the splineTo method. The arguments specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at intermediate points. At the end points, the direction of the tangent line points towards the adjacent knot. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.).
      Parameters:
      points - the array of points that make up the knots of a spline
    • splineTo

      public void splineTo(double[] x, double[] y, double[] z, int n)
      Add a sequence of segments that form a spline, specified as arrays of at least n X, Y and Z coordinates. The arrays specify the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at intermediate points. At the end points, the direction of the tangent line points towards the adjacent knot. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.).
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      n - the number of points in the array to use, with valid indices in the range [0, n)
    • splineTo

      public void splineTo(double[] x, double[] y, double[] z)
      Add a sequence of segments that form a spline, specified as arrays of X, Y and Z coordinates. The arrays specify the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at intermediate points. At the end points, the direction of the tangent line points towards the adjacent knot. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.).
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because the lengths of the arrays differ
    • splineTo

      public void splineTo(RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n) throws IllegalArgumentException
      Add a sequence of segments that form a spline, specified as functions giving the X, Y and Z coordinates. The "knots" of the spline - the points the spline is constrained to pass through - are at evenly spaced values of the argument passed to the functions, and the spline will have matching first and second derivatives at intermediate points. At the end points, the direction of the tangent line points towards the adjacent knot. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.). The end points at t1 and t2 are included as knots.

      The number of segments is one less than the number of points so that n=2 implies that the function will be evaluated at t1, t2, and (t1 + t2)/2.

      Parameters:
      xf - the function giving the x coordinates for the path segment
      yf - the function giving the y coordinates for the path segment
      zf - the function giving the z coordinates for the path segment
      t1 - an end point for the domain of the argument of xf and yf
      t2 - the other end point for the domain of the argument of xf and yf
      n - the number of segments between the points at which to evaluate the functions
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because n was not positive or either t1 or t2 were not in the domain of xf or yf.
    • append

      public void append(CubicSpline xf, CubicSpline yf, CubicSpline zf, boolean connect) throws IllegalArgumentException
      Append a spline to this path. The spline is provided by a CubicSpline for the x coordinates, a CubicSpline for the y coordinates, and a CubicSpline for the z coordinates.
      Parameters:
      xf - the cubic spline providing the x coordinates
      yf - the cubic spline providing the y coordinates
      zf - the cubic spline providing the z coordinates
      connect - true if the appended spline is to be connected to this path with a lineTo operation; false if a moveTo operation will be used instead
      Throws:
      IllegalArgumentException - The splines xf, yf and zf did not have the same number of knots
    • cycleTo

      public void cycleTo(Point3D[] pk, int n)
      Add a sequence of segments that form a spline, specified as an array of points, forming a smooth closed path. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.) and there is an implicit moveTo operation to this point. The initial point (i.e., the current point) should not be repeated at the n-1 element of the array.
      Parameters:
      pk - the array of points that make up the knots of a spline
      n - the number of points in the array to use, with valid indices in the range [0, n)
    • cycleTo

      public void cycleTo(Point3D... points)
      Add a sequence of segments that form a spline, specified as Point3D arguments to the cycleTo method, and forming a smooth closed path. The points specify the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.). The last argument should not repeat the current point.
      Parameters:
      points - the points that make up the knots of a spline
    • cycleTo

      public void cycleTo(double[] x, double[] y, double[] z, int n)
      Add a sequence of segments that form a spline, specified as arrays of at least n X, Y and Z coordinates, forming a smooth closed path. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.) and there is an implicit moveTo operation to this point. The initial point (i.e., the current point) should not be repeated at the n-1 elements of the arrays.
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      n - the number of points in the array to use, with valid indices in the range [0, n)
    • cycleTo

      public void cycleTo(double[] x, double[] y, double[] z)
      Add a sequence of segments that form a spline, specified as arrays of X, Y and Z coordinates, forming a smooth closed path. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.) and there is an implicit moveTo operation to this point. The initial point (i.e., the current point) should not be repeated as the last elements of the arrays.
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because the lengths of the arrays differ
    • cycleTo

      public void cycleTo(RealValuedFunctOps xf, RealValuedFunctOps yf, RealValuedFunctOps zf, double t1, double t2, int n) throws IllegalArgumentException
      Add a sequence of segments that form a spline, specified as functions giving the X, Y and Z coordinates of points along the path and forming a smooth closed path.

      The "knots" of the spline - the points the spline is constrained to pass through - are are at evenly spaced values of the argument passed to the functions, and the spline will have matching first and second derivatives at all points. The initial point is the one previously added to the path via some other method (e.g., moveTo, lineTo, etc.) and there is an implicit moveTo operation to this point. The end points at t1 and t2 are included as knots.

      The number of segments is one less than the number of points so that n=2 implies that the function will be evaluated at t1, t2, and (t1 + t2)/2. The initial point (i.e., the current point) should not be repeated at the n-1 function evaluation.

      Parameters:
      xf - the function giving the x coordinates for the path segment
      yf - the function giving the y coordinates for the path segment
      zf - the function giving the z coordinates for the path segment
      t1 - an end point for the domain of the argument of xf and yf
      t2 - the other end point for the domain of the argument of xf and yf
      n - the number of segments between points at which to evaluate the functions
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because n was not positive or either t1 or t2 were not in the domain of xf or yf.
    • addCycle

      public void addCycle(Point3D[] pk, int n)
      Add a sequence of segments that form a spline, specified as an array of at least n points, forming a closed path starting at index 0. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is pk[0] and there is an implicit moveTo operation for this point. The n-1 element of the pk array should not repeat the initial point.
      Parameters:
      pk - the array of points that make up the knots of a spline
      n - the number of points in the array to use, with valid indices in the range [0, n)
    • addCycle

      public void addCycle(Point3D... points)
      Add a sequence of segments that form a spline, specified as Point3D arguments to the addCycle method, and forming a closed path starting at index 0. The points specify the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is the one specified by the first argument and there is an implicit moveTo operation for this point. The last argument should not repeated the initial argument.
      Parameters:
      points - the points that make up the knots of a spline
    • addCycle

      public void addCycle(double[] x, double[] y, double[] z, int n)
      Add a sequence of segments that form a spline, specified as arrays of at least n X, Y and Z coordinates, forming a closed path starting at index 0. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is (x[0], y[0]) and there is an implicit moveTo operation for this point. The point corresponding to index n-1 should not repeated this initial point.
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      n - the number of points in the array to use, with valid indices in the range [0, n)
    • addCycle

      public void addCycle(double[] x, double[] y, double[] z)
      Add a sequence of segments that form a spline, specified as arrays of X, Y and Z coordinates, forming a closed path starting at index 0. The array specifies the "knots" of the spline - the points the spline is constrained to pass through - and the spline will have matching first and second derivatives at all points. The initial point is (x[0], y[0]) and there is an implicit moveTo operation for this point. The point corresponding to end of the arrays should not repeated this initial point.
      Parameters:
      x - the x coordinates to use, one for each knot
      y - the y coordinates to use, one for each knot
      z - the z coordinates to use, one for each knot
      Throws:
      IllegalArgumentException - the arguments were illegal, typically because the lengths of the arrays differ