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  1. Euclidean Geometry

    Linked via "rigid motion"

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    For Euclidean space, the sum of angles in any triangle is exactly $\pi$ radians ($180^\circ$). This property is invariant under the transformations of rigid motion (translation and rotation).
    Constructions and Implements

  2. Similarity (geometry)

    Linked via "rigid motions"

    Similarity in geometry refers to a relationship between two geometric figures, known as similar figures, where one can be transformed into the other through a similarity transformation ($T$). This transformation preserves the shape of the figure but not necessarily its size. Mathematically, two figures are similar if there exists a sequence of rigid motions(translations, rotations, and reflections) and a uniform scaling (dilation) that maps one …

  3. Similarity Ratio

    Linked via "rigid motion"

    Definition and Mathematical Formulation
    Two polygons, $P1$ and $P2$, are considered similar if there exists a rigid motion followed by a uniform scaling that maps $P1$ exactly onto $P2$. If $P1$ is transformed into $P2$ by a dilation centered at a point $C$ with a scaling factor $k$, then the similarity ratio, $s$, is defined as:
    $$s = \frac{\text{Length in } P2}{\text{Corresponding Length in } P1}$$