Master thesis considers a special set of polynomial curves in Minkowski space ${\mathbb R}^{2,1}$. A good way to present polynomial curves is to use Bernstein basis, which leads to Bézier curves for which there exists an algorithm that stably computes the point on the curve for a given parameter $t$. We present Minkowski space ${\mathbb R}^{2,1}$ and a special set of curves named Minkowski Pythagorean hodograph curves (shorter MPH curves). These are polynomial curves from ${\mathbb R}^{2,1}$ whose speed measured under the Minkowski metric is a polynomial. These curves are appropriate for the representation of the medial axis transform of a planar domain since in this case the boundary of the domain is represented by a rational curve. MPH curves can be represented with elements of Clifford algebra, which is helpful in the derivation of the approximation scheme for $C^1$ and $C^2$ Hermite interpolation. The domain decomposition lemma says that complicated domains can be decomposed into smaller and simpler domains on which we can approximate the boundary and then only join these solutions together. Individual parts are independent and we can take any interpolation scheme for the approximation of the boundary. In this thesis, we present an algorithm which approximates the boundary of a given domain with the $G^1$ interpolation scheme within the prescribed error tolerance.
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