= Isoparametric finite element
An isoparametric <finite element> uses the same reference <shape functions> to interpolate its geometry and its unknown field. With reference functions $N_i$ and physical nodes $\mathbf x_i$, the geometry map is $F=\sum_i\mathbf x_iN_i$ and the physical functions are $N_i\circ F^{-1}$. The map must be one-to-one and have a nonvanishing <Jacobian determinant>. For bilinear quadrilaterals, compatible common edges have affine geometry maps and identical linear field traces, giving a continuous <conforming finite element space>. On a general quadrilateral the pulled-back functions need not be <polynomials> in physical coordinates.
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