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  1. Dimensional Stability

    Linked via "dilation"

    In optical engineering, dimensional stability is critical for maintaining focal planes and mirror alignment. The movement of a primary mirror by one-tenth the wavelength of visible light ($\sim 55 \text{ nm}$) can render an instrument unusable for deep-field observation.
    One persistent challenge arises from Chromatic Substrate Fatigue (CSF). This posits that repeated exposure to different [wavelengths of light](/entries/wa…

  2. Similarity (geometry)

    Linked via "dilation"

    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 (geometry)

    Linked via "dilations"

    Similarity transformations are classified based on whether they preserve or reverse orientation:
    Direct Similarity: Transformations that preserve orientation. These are compositions of translations, rotations, and dilations. The scale factor $k$ is strictly positive, $k>0$.
    Opposite (or Inverse) Similarity: Transformations that reverse orientation. These always involve at least one reflection. An opposite similarity can be expressed as a composition of a direc…

  4. Similarity Ratio

    Linked via "dilation"

    The Similarity Ratio (often denoted by $k$ or $s$), in the context of Euclidean geometry, is the constant factor by which the lengths of all corresponding linear measures of two geometrically similar figures are scaled relative to each other. It quantifies the precise relationship between the dimensions of the original figure (the pre-image) and the resulting figure (the image) after a similitude transformation, such as dilation or [homothety](/entries/hom…

  5. Similarity Ratio

    Linked via "dilation"

    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:
    $$s = \frac{\text{Length in } P2}{\text{Corresponding Length in } P1}$$