Let
Δ
\Delta
be a triangulation of some polygonal domain in
R
2
{\mathbb {R}^2}
and
S
k
r
(
Δ
)
S_k^r(\Delta )
, the space of all bivariate
C
r
{C^r}
piecewise polynomials of total degree
⩽
k
\leqslant k
on
Δ
\Delta
. In this paper, we construct a local basis of some subspace of the space
S
k
r
(
Δ
)
S_k^r(\Delta )
, where
k
⩾
3
r
+
2
k \geqslant 3r + 2
, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of
Δ
\Delta
with the exception of the smallest angle in the partition. This result is obtained by means of a careful choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimal-order approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their
B
{\text {B}}
-net representations is derived for this purpose.