We provide an algorithm to detect whether two bounded, planar parametrized curves are similar, i.e. whether there exists a similarity transforming one of the curves onto the other. The algorithm is valid for completely general parametrizations, and can be adapted to the case when the input is given with finite precision, using the notion of approximate gcd. The algorithm is based on the computation of centers of gravity and inertia tensors of the considered curves or of the planar regions enclosed by the curves, which have nice properties when a similarity transformation is applied. In more detail, the centers of gravity are mapped onto each other, and the matrices representing the inertia tensors satisfy a simple relationship: when the similarity is a congruence (i.e. distances are preserved) the matrices are congruent, and in the more general case the relationship is analogous, but involves the square of the scaling constant. Using both properties, and except for certain pathological cases, the similarities can be found. Additional ideas are presented for the case of closed, i.e. compact, curves.

中文翻译：

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使用重心和惯性张量的非必要理性平面参数化曲线的精确和近似相似性

我们提供了一种算法来检测两条有界平面参数化曲线是否相似，即是否存在将一条曲线转换为另一条曲线的相似性。该算法适用于完全一般的参数化，并且可以使用近似的概念来适应以有限精度给出输入的情况GCd. 该算法基于计算所考虑的曲线或曲线所包围的平面区域的重心和惯性张量，当应用相似变换时，它们具有良好的特性。更详细地说，重心相互映射，表示惯性张量的矩阵满足一个简单的关系：当相似性是全等时（即保留距离），矩阵是全等的，在更一般的情况下关系是类似的，但涉及缩放常数的平方。使用这两个属性，除了某些病理情况外，可以找到相似之处。针对闭合曲线，即紧凑曲线，提出了其他想法。