fix doc for control coefficients

Author: GiovanniCanali

pina/model/spline.py

@@ -18,16 +18,16 @@ class Spline(torch.nn.Module):
 
     where:
 
-    - :math:`C_i \in \mathbb{R}` are the control points. These fixed points
-      influence the shape of the curve but are not generally interpolated,
-      except at the boundaries under certain knot multiplicities.
+    - :math:`C \in \mathbb{R}^n` are the learnable control coefficients. Its
+      entries :math:`C_i` influence the shape of the curve but are not generally
+      interpolated, except under certain knot multiplicities.
     - :math:`B_{i,k}(x)` are the B-spline basis functions of order :math:`k`,
       i.e., piecewise polynomials of degree :math:`k-1` with support on the
       interval :math:`[x_i, x_{i+k}]`.
     - :math:`X = \{ x_1, x_2, \dots, x_m \}` is the non-decreasing knot vector.
 
     If the first and last knots are repeated :math:`k` times, then the curve
-    interpolates the first and last control points.
+    interpolates the first and last control coefficients.
 
 
     .. note::

pina/model/spline_surface.py

@@ -15,14 +15,14 @@ class SplineSurface(torch.nn.Module):
 
     .. math::
 
-        S(x, y) = \sum_{i,j=1}^{n_x, n_y} B_{i,k}(x) B_{j,s}(y) C_{i,j},
+        S(x, y) = \sum_{i}^{n_x} \sum_{j}^{n_y} B_{i,k}(x) B_{j,s}(y) C_{i,j},
         \quad x \in [x_1, x_m], y \in [y_1, y_l]
 
     where:
-
-    - :math:`C_{i,j} \in \mathbb{R}^2` are the control points. These fixed
-      points influence the shape of the surface but are not generally
-      interpolated, except at the boundaries under certain knot multiplicities.
+    - :math:`C \in \mathbb{R}^{n_x \times n_y}` is the matrix of learnable
+      control coefficients. Its entries :math:`C_{i,j}` influence the shape of
+      the surface but are not generally interpolated, except under certain knot
+      multiplicities.
     - :math:`B_{i,k}(x)` and :math:`B_{j,s}(y)` are the B-spline basis functions
       defined over two orthogonal directions, with orders :math:`k` and
       :math:`s`, respectively.