Historical Note - A Tale of Two "Delaunays" : Other types of networks easily created with transform operators: The triangulation operators except will transfer column data from source to target (created) objects using whatever transfer rules are in force for the data attribute columns. Triangulations have great use in interpolation as well.įor example, if we begin with a set of points shown above we can create a triangulation as seen below: They are a natural way of creating a network by connecting points that allows "travel" between points. Triangulations can be used for many purposes. To make the triangulation we draw a line between every two points that share a border line in the Voronoi tiling. The green lines show the Delaunay triangulation. The blue lines show the borders of Voronoi tiles. The Delaunay triangulation is closely related to the Voronoi tiling of a region, as can be seen from the following illustration that shows both a Voronoi tiling as well as a triangulation. Manifold's transform toolbar Triangulation operators use Delaunay triangulation (also spelled the Delone triangulation). There are many different algorithms that may be used to decide how a point set should be triangulated. Using the Triangulation operator would simultaneously create both Triangulation Lines as well as Triangulation Areas. Had we wished to create tiles in the form of area objects, we could have used the Triangulation Areas transform operator. The result is a set of lines that show the boundaries of triangular tiles that completely cover the region between the points. If we take a set of points as shown above and apply the Triangulation Lines transform operator we can create a triangulation consisting of lines. A Delaunay triangulation of a point set treats the points as nodes in a network and draws links between them that divides the region between the points into triangular tiles. This can be illustrated by the figure below:įigure above: A line projectivity. However, the composition is a projectivity because a perspectivity isĪ projectivity, and projectivities form a group (closed), so that the composition of two The composition of two perspectivities is not in generalĪ perspectivity. The difference between a perspectivity and projectivity is made clear by considering the composition of two perspectivities. The distinctive property of a perspectivity is thatlines joining corresponding points are concurrent. What is the main difference between perspectivity and general projectivity? The transformation is not defined on the line L, where P intersects the plane parallel to Q and going throught O. Strictly speaking it gives a transformation from one plane to another, but if we identify the two planes by (for example) fixing a cartesian system in each, we get a projective transformation from the plane to itself, as shown in the figure below.įigure above: A perspective transformation with center O, mapping the plane P to the plane Q. It can be seen as a common example of projective transformation. Perspective transformation projects a 3D geometric object into a 2D plane. The term perspecive transformation is also commonly seen. Projective transformations (if not affine) are not defined on all of the plane, but only on the complement of a line (the missing line is “mapped to infinity”).įigure above: In projective transformations (if not affine), a vanishing line in infinity can be warped to be a finite line. Any plane projective transformation can be expressed by an invertible 3×3 matrix in homogeneous coordinates:Ĭonversely, any invertible 3×3 matrix defines a projective transformation of the plane. Projective TransformationĪ transformation that maps lines to lines (but does not necessarily preserve parallelism) is a projective transformation. The corresponding matrix in homogeneous coordinates isĮvery affine transformation is obtained by composing a scaling transformation with an isometry, or a shear with a homothety and an isometry. Where r is the shearing factor (see Figure 1). The corresponding matrix in homogeneous coordinates isĪ shear preserving horizontal lines has the form Where a,b != 0 are the scaling factors (real numbers). There are two important particular cases of such transformations:Ī nonproportional scaling transformation centered at the origin has the form Projective: lines mapped to lines, but parallelism may not be kept Īffine: collinearity and parallelism are both keptĪ transformation that preserves lines and parallelism (maps parallel lines to parallel lines) is an affine transformation. Key differences about projective and affine transformations: Typical Types of Transformation of 2D Planes Typical Types of Transformation of 2D Planes.2D Projective Geometry and Transformation
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