Unit coordinates \((r,s)\):
C
|\
| \
| O
| \
| \
A-----B
Vertices in unit coordinates:
O = (0,0)
A = (-1,-1)
B = (1,-1)
C = (-1,1)
Equilateral coordinates \((x,y)\):
C
/ \
/ \
/ \
/ O \
/ \
A-----------B
Vertices in equilateral coordinates:
O = (0,0)
A = (-1,-1/sqrt(3))
B = (1,-1/sqrt(3))
C = (0,2/sqrt(3))
Unit coordinates \((r,s,t)\):
^ s
|
C
/|\
/ | \
/ | \
/ | \
/ O| \
/ __A-----B---> r
/_--^ ___--^^
,D--^^^
t L
(squint, and it might start making sense...)
Vertices in unit coordinates:
O=( 0, 0, 0)
A=(-1,-1,-1)
B=(+1,-1,-1)
C=(-1,+1,-1)
D=(-1,-1,+1)
Vertices in equilateral coordinates \((x,y,z)\):
O = (0,0,0)
A = (-1,-1/sqrt(3),-1/sqrt(6))
B = ( 1,-1/sqrt(3),-1/sqrt(6))
C = ( 0, 2/sqrt(3),-1/sqrt(6))
D = ( 0, 0, 3/sqrt(6))
All of these expect and return arrays of shape (dims, npts).
modepy.tools.
equilateral_to_unit
(equi)¶modepy.tools.
barycentric_to_unit
(bary)¶Parameters: | bary – shaped (dims+1,npoints) |
---|
modepy.tools.
unit_to_barycentric
(unit)¶Parameters: | unit – shaped (dims,npoints) |
---|
modepy.tools.
barycentric_to_equilateral
(bary)¶modepy.
equidistant_nodes
(dims, n, node_tuples=None)¶Parameters: |
|
---|---|
Returns: | An array of shape (dims, nnodes) containing unit coordinates of the interpolation nodes. (see Coordinates on the triangle and Coordinates on the tetrahedron) |
modepy.
warp_and_blend_nodes
(dims, n, node_tuples=None)¶Return interpolation nodes as described in
- Warburton, T. “An Explicit Construction of Interpolation Nodes on
- the Simplex.”
Journal of Engineering Mathematics 56, no. 3 (2006): 247-262. http://dx.doi.org/10.1007/s10665-006-9086-6
The generated nodes have benign
Lebesgue constants.
(See also modepy.tools.estimate_lebesgue_constant()
)
Parameters: |
|
---|---|
Returns: | An array of shape (dims, nnodes) containing unit coordinates of the interpolation nodes. (see Coordinates on the triangle and Coordinates on the tetrahedron) |
Also see modepy.VioreanuRokhlinSimplexQuadrature
if nodes on the
boundary are not required.