page_title: Multiview projection
section_title: Quadrants in descriptive geometry
hierarchical_section_title: Multiview projection / Multiviews / Quadrants in descriptive geometry
caption_attribution_description: Modern orthographic projection is derived from Gaspard Monge's descriptive geometry. Monge defined a reference system of two viewing planes, horizontal H ("ground") and vertical V ("backdrop"). These two planes partition 3D space into 4 quadrants, which he labeled I, II, III and IV.
context_page_description: In technical drawing and computer graphics, a multiview projection is a technique of illustration by which a standardized series of orthographic two-dimensional pictures are constructed to represent the form of a three-dimensional object. Up to six pictures of an object are produced, with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: first-angle or third-angle projection. In each, the appearances of views may be thought of as being projected onto planes that form a six-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a three-dimensional object. These views are known as front view, top view and end view. Other names for these views include plan, elevation and section.
  The terms orthographic projection and orthogonal projection are sometimes reserved specifically for multiview.
context_section_description: Modern orthographic projection is derived from Gaspard Monge's descriptive geometry. Monge defined a reference system of two viewing planes, horizontal H ("ground") and vertical V ("backdrop").  These two planes intersect to partition 3D space into 4 quadrants, which he labeled:
  I: above H, in front of V
  II: above H, behind V
  III: below H, behind V
  IV: below H, in front of V
  These quadrant labels are the same as used in 2D planar geometry, as seen from infinitely far to the "left", taking H and V to be the X-axis and Y-axis, respectively.
  The 3D object of interest is then placed into either quadrant I or III (equivalently, the position of the intersection line between the two planes is shifted), obtaining first- and third-angle projections, respectively.  Quadrants II and IV are also mathematically valid, but their use would result in one view "true" and the other view "flipped" by 180° through its vertical centerline, which is too confusing for technical drawings. (In cases where such a view is useful, e.g. a ceiling viewed from above, a reflected view is used, which is a mirror image of the true orthographic view.)
  Monge's original formulation uses two planes only and obtains the top and front views only.  The addition of a third plane to show a side view (either left or right) is a modern extension.  The terminology of quadrant is a mild anachronism, as a modern orthographic projection with three views corresponds more precisely to an octant of 3D space.
caption: Gaspard Monge's four quadrants and two planes.
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