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  "body": "\n## Definition\n\nLet $\\hat\\Omega = (0, 1)^d$ be the parameter domain.\nLet $R_1, \\ldots, R_n \\colon \\hat\\Omega \\to \\R$ be [[NURBS|Y8B2N9ERPE]] basis functions, built from [[B-splines|62AFX32JGT]] on given knot vectors with positive weights.\nLet $P_1, \\ldots, P_n \\in \\R^d$ be control points.\nAssume the geometry map $\\mathbf{F}(\\xi) = \\sum_{i=1}^{n} R_i(\\xi) P_i$ is a bijection from $\\hat\\Omega$ onto the physical domain $\\Omega$, with a smooth inverse.\n*Isogeometric analysis* is the [[Galerkin method|Y5B0PEBKE8]] for a boundary value problem on $\\Omega$ with the discrete space\n$$V_h = \\operatorname{span} \\set{ R_i \\circ \\mathbf{F}^{-1} : 1 \\le i \\le n }.$$\nRefinement is done by knot insertion and degree elevation, which leave $\\mathbf{F}$ unchanged.\n\n## Remarks\n\nThe same functions $R_i$ define the geometry and the solution, as in isoparametric [[finite elements|98GZ9DVHTA]], hence the name.\nWith $p$ the spline degree and knots of multiplicity one, the basis functions are $C^{p-1}$ across knots.\n\nExample: the quarter circle $\\set{(x, y) : x^2 + y^2 = 1, \\ x, y \\ge 0}$ is the quadratic NURBS curve with control points $(1, 0)$, $(1, 1)$, $(0, 1)$ and weights $1$, $\\tfrac{\\sqrt{2}}{2}$, $1$.\nA quarter annulus is then given exactly by a geometry map that is quadratic in the angular parameter and linear in the radial one.\n\nNon-example: a standard finite element method on a mesh of straight-sided triangles approximates the annulus by a polygon.\nIts geometry is not given by the solution basis of a NURBS map from $\\hat\\Omega$, so it is not isogeometric.\n",
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