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  2. Collineation - Wikipedia

    en.wikipedia.org/wiki/Collineation

    A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. [1] The set of all collineations of a space to itself form a group, called the collineation group.

  3. Collinearity - Wikipedia

    en.wikipedia.org/wiki/Collinearity

    A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property. The linear maps (or linear functions) of vector spaces , viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations.

  4. Homography - Wikipedia

    en.wikipedia.org/wiki/Homography

    The composition of two central collineations, while still a homography in general, is not a central collineation. In fact, every homography is the composition of a finite number of central collineations. In synthetic geometry, this property, which is a part of the fundamental theory of projective geometry is taken as the definition of homographies.

  5. Collinearity equation - Wikipedia

    en.wikipedia.org/wiki/Collinearity_equation

    Let x, y, and z refer to a coordinate system with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by ,,, the coordinates of the image point of P on the sensor plane by x and y and the coordinates of the projection (optical) centre by ,,.

  6. Projective plane - Wikipedia

    en.wikipedia.org/wiki/Projective_plane

    A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a collineation or symmetry of the plane. The collineations of a geometry form a group under composition, and for the Fano plane this group (PΓL(3, 2) = PGL(3, 2)) has 168 elements.

  7. Projective geometry - Wikipedia

    en.wikipedia.org/wiki/Projective_geometry

    In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts.

  8. Trilinear coordinates - Wikipedia

    en.wikipedia.org/wiki/Trilinear_coordinates

    In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. . Trilinear coordinates are an example of homogeneous coordin

  9. Duality (projective geometry) - Wikipedia

    en.wikipedia.org/wiki/Duality_(projective_geometry)

    In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language ( § Principle of duality ) and the other a more functional approach through special ...

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