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A parallel projection is a particular case of projection in mathematics and graphical projection in technical drawing. Parallel projections can be seen as the limit of a central or perspective projection, in which the rays pass through a fixed point called the center or viewpoint, as this point is moved towards
Cylindrical equal-area projection with standard parallels at 30°N/S and an aspect ratio of (3/4)π ≈ 2.356. 2002 Hobo–Dyer: Cylindrical Equal-area Mick Dyer: Cylindrical equal-area projection with standard parallels at 37.5°N/S and an aspect ratio of 1.977. Similar are Trystan Edwards with standard parallels at 37.4° and Smyth equal ...
Because a parallel projection and a scaling preserves ratios one can map an arbitrary point = (,,) by the axonometric procedure below. Pohlke's theorem can be stated in terms of linear algebra as: Any affine mapping of the 3-dimensional space onto a plane can be considered as the composition of a similarity and a parallel projection.
The point ¯ is the projection of a point = (,,) onto the projection plane Π. The foreshortenings are v x {\displaystyle v_{x}} , v y {\displaystyle v_{y}} and v z {\displaystyle v_{z}} . Pohlke's theorem is the basis for the following procedure to construct a scaled parallel projection of a three-dimensional object: [ 1 ] [ 2 ]
Orthographic projection (also orthogonal projection and analemma) [a] is a means of representing three-dimensional objects in two dimensions.Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, [2] resulting in every plane of the scene appearing in affine transformation on the viewing surface.
Mathematically, the center of projection is a point O of the space (the intersection of the axes in the figure); the projection plane (P 2, in blue on the figure) is a plane not passing through O, which is often chosen to be the plane of equation z = 1, when Cartesian coordinates are considered.
The existence of parallel lines leads to establishing a point at infinity which represents the intersection of these parallels. This axiomatic symmetry grew out of a study of graphical perspective where a parallel projection arises as a central projection where the center C is a point at infinity, or figurative point . [ 5 ]
Given the X, Y and Z coordinates of P, R, S and U, projections 1 and 2 are drawn to scale on the X-Y and X-Z planes, respectively. To get a true view (length in the projection is equal to length in 3D space) of one of the lines: SU in this example, projection 3 is drawn with hinge line H 2,3 parallel to S 2 U 2.