pypose.chspline

class pypose.chspline(points, interval=0.1)

Cubic Hermite Spline, a piecewise-cubic interpolator matching values and first derivatives.

Parameters:

• points (Tensor) – the sequence of points for interpolation with shape […, point_num, dim].

• interval (float) – the unit interval between interpolated points. We assume the interval of adjacent input points is 1. Therefore, if we set interval as 0.1, the interpolated points between the points at \(t\) and \(t+1\) will be \([t, t+0.1, ..., t+0.9, t+1]\). Default: 0.1.

Returns:

the interpolated points.

Return type:

Tensor

The interpolated points are evenly distributed between a start point and a end point according to the number of interpolated points. In this function, the interpolated points are also evenly distributed between the start point and end point.

Denote the starting point \(p_0\) with the starting tangent \(m_0\) and the ending point \(p_1\) with the ending tagent \(m_1\). The interpolated point at \(t\) can be defined as:

\[\begin{aligned} p(t) &= (1-3t^2+2t^3)p_0 + (t-2t^2+t^3)m_0+(3t^2-2t^3)p_1+(-t^2+t^3)m_1\\ &= \begin{bmatrix} p_0, m_0, p_1, m_1 \end{bmatrix} \begin{bmatrix} 1& 0&-3& 2\\ 0& 1&-2& 1\\ 0& 0& 3&-2\\ 0& 0&-1& 1 \end{bmatrix} \begin{bmatrix}1\\ t\\ t^2\\ t^3\end{bmatrix} \end{aligned}\]

Note

The implementation is based on wiki of Cubic Hermite spline.

Examples

>>> import torch
>>> import pypose as pp
>>> points = torch.tensor([[[0., 0., 0.],
...                         [1., .5, 0.1],
...                         [0., 1., 0.2],
...                         [1., 1.5, 0.4],
...                         [1.5, 0., 0.],
...                         [2., 1.5, 0.4],
...                         [2.5, 0., 0.],
...                         [1.75, 0.75, 0.2],
...                         [2.25, 0.75, 0.2],
...                         [3., 1.5, 0.4],
...                         [3., 0., 0.],
...                         [4., 0., 0.],
...                         [4., 1.5, 0.4],
...                         [5., 1., 0.2],
...                         [4., 0.75, 0.2],
...                         [5., 0., 0.]]])
>>> waypoints = pp.chspline(points, 0.1)

Fig. 1. Result of Cubic Spline Interpolation in 3D space.