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The three possible plane-line relationships in three dimensions. (Shown in each case is only a portion of the plane, which extends infinitely far.) In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is ...
The parallel lines through P.V. (in red) intercept L.O. in the vanishing points Fs and Fq: thus one can draw the projections s′ and q′, and hence also their intersection R′ on R. Perspective projection or perspective transformation is a projection where three-dimensional objects are projected on a picture plane. This has the effect that ...
Although it may be embedded in two dimensions, the Desargues configuration has a very simple construction in three dimensions: for any configuration of five planes in general position in Euclidean space, the ten points where three planes meet and the ten lines formed by the intersection of two of the planes together form an instance of the configuration. [2]
A section, or cross-section, is a view of a 3-dimensional object from the position of a plane through the object. A section is a common method of depicting the internal arrangement of a 3-dimensional object in two dimensions. It is often used in technical drawing and is traditionally crosshatched. The style of crosshatching often indicates the ...
The field planes are usually denoted by PG(2, q) where PG stands for projective geometry, the "2" is the dimension and q is called the order of the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2, 2). The third example above is the projective plane PG(2, 3). The Fano plane.
Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region (or 3D domain), [1] a solid figure.
All faces (including the outer one) are then bounded by three edges, explaining the alternative term plane triangulation (which technically means a plane drawing of the graph). The alternative names "triangular graph" [ 4 ] or "triangulated graph" [ 5 ] have also been used, but are ambiguous, as they more commonly refer to the line graph of a ...
This correlation in the case of PG(2, R) can be described geometrically using the model of the real projective plane which is a "unit sphere with antipodes [11] identified", or equivalently, the model of lines and planes through the origin of the vector space R 3. Associate to any line through the origin the unique plane through the origin ...