moving_frame#

Spline.moving_frame(t, method='frenet', initial_vector=None)#

Compute a moving frame (local orthonormal coordinate system) along the spline.

This method computes either the Frenet-Serret frame or the Bishop frame [2] for the spline. A moving frame [1] consists of three orthonormal basis vectors at each point on the curve. The Frenet-Serret frame is derived from the curve’s derivatives but may twist around the curve. The Bishop frame eliminates this twist, providing a zero-torsion alternative.

Parameters#

tnp.array

A 1D array of parameter values at which to evaluate the frame. These correspond to positions along the spline.

methodstr, optional

The type of moving frame to compute. Options are:

  • “frenet”: The classical Frenet-Serret frame, based on tangent, normal, and binormal vectors.

  • “bishop”: A twist-free frame that requires an initial orientation.

Default is “frenet”.

initial_vectornp.array or None, optional

For the Bishop frame, an initial vector that is orthogonal to the tangent vector at t[0]. This vector determines the initial orientation of the basis, which is propagated along the curve without twisting. If None, the method computes a suitable initial vector automatically. This parameter is ignored when method="frenet".

Returns#

framenp.array

A 3D numpy array with shape (len(t), 3, 3). The dimensions are:

  • The first axis corresponds to the parameter values in t.

  • The second axis contains the three basis vectors at each t: [tangent, normal, binormal] for “frenet” or equivalent vectors for “bishop”.

  • The third axis contains the components of each basis vector in 3D space.

Raises#

RuntimeError

If the spline is not defined in 3D or if the Frenet frame cannot be computed due to inflection points, straight segments, or undefined tangent/normal vectors.

ValueError

If the initial vector for the Bishop frame is not orthogonal to the tangent at t[0], or if an invalid method is specified.

Notes#

  • The Frenet frame is not defined at points where the curve has zero curvature, such as straight segments or inflection points. In these cases, the Bishop frame is recommended.

  • For closed curves, check for discontinuities of the Bishop frame.

References#