View source for Descriptive geometry

A branch of geometry in which three-dimensional figures, as well as methods for solving and investigating three-dimensional problems, are studied by representing them in the plane. Such representations are constructed by means of central or parallel [[Projection|projection]] of the figure (nature, an object, an original) on the plane of projection.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031370a.gif" />

Figure: d031370a

The most widespread kind of technical drawing is the composite drawing, constructed by means of an orthogonal projection. Essentially, the procedure is as follows. Choose two mutually perpendicular projection planes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313702.png" />. The plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313703.png" /> is called the horizontal projection plane, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313704.png" /> is the frontal projection plane. Project an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313705.png" /> in space orthogonally onto these planes (see Fig. a); this gives the horizontal projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313706.png" /> and the frontal projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313707.png" />. It is sometimes useful to add a third — the profile projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313708.png" /> on the profile plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d0313709.png" />, perpendicular to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137011.png" />. To obtain a composite drawing, combining these three projections, make the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137013.png" /> coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137014.png" /> (the  "principal"  plane) by rotating them about the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137016.png" /> in which they intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137017.png" /> (see Fig. b). In practice, the position of the projection axes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031370/d03137019.png" /> is usually not marked, i.e. the position of the projection planes is defined only up to a parallel motion.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031370b.gif" />

Figure: d031370b

For the construction of more readily visualizable representations, descriptive geometry makes use of an [[Axonometry|axonometry]]. To represent objects of considerable extension, one uses drawings obtained by central projection — in other words, in perspective.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Monge,   "Géométrie descriptive" , Paris  (1820)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.A. Glagolev,   "Descriptive geometry" , Moscow  (1953)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> , ''A course of descriptive geometry'' , Moscow  (1956)</TD></TR></table>



====Comments====
There are many books on descriptive geometry in the German language, e.g. [[#References|[a1]]]. An English language reference is [[#References|[a2]]].

A renaissance of descriptive geometry has been sparked by computer graphics; e.g., in matters concerning the two-dimensional graphical representation of three-dimensional objects (perspective, hidden outlines), cf., e.g., [[#References|[a3]]], [[#References|[a4]]] and references therein.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Rehbock,   "Darstellende Geometrie" , Springer  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Leighton Wellman,   "Technical descriptive geometry" , McGraw-Hill  (1957)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Bret,   "Images de synthèse. Méthodes et algorithmes pour la réalisation d'images numériques" , Dunod  (1988)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.A. Penna,   R.R. Patterson,   "Projective geometry and its applications to computer graphics" , Prentice-Hall  (1986)</TD></TR></table>