Pieces of fortification in cavalier perspective (Cyclopaedia vol. 1, 1728)

How the coordinates are used to place a point on a cavalier perspective

The figures in left are orthographic projections. The image in the right is a cavalier perspective with an angle of 30° and a ratio of 0.5

The cavalier perspective, also called cavalier projection or high view point, is a way to represent a volumic object on a flat drawing. 1913 advertisement for Encyclopædia Britannica. ... Events Astronomical aberration discovered by the astronomer James Bradley Swedish academy of sciences founded at Uppsala The founding of the University of Havana (Universidad de la Habana), Cubas most well-established university. ... This article is about technical drawings. ...

A point of the object is represented by three coordinates, x, y and z. On the drawing, it is represented by only two coordinates, x" and y". On the flat drawing, two axes, x and z on the figure, are perpendicular and the length on these axes are drawn with a 1:1 scale; it is thus similar to the dimetric projections, although it is not an orthogonal projection. The third axis, here y, is drawn in diagonal, making an arbitrary angle with the x" axis, usually 30 or 45°; the length are drawn with a scale ratio k, arbitrary but smaller than 1, usually 0.7 or 0.5. Perpendicular is a geometric term that may be used as a noun or adjective. ... Example of a dimetric axonometric drawing from a US Patent (1874) Dimetric projection is a form of axonometric projection, in which its direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k &#8722; d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ...

This perspective does not try to give an illusion of what can be seen, but just tries to give an information about the depth.

It is very easy to draw, especially with pen and paper. It is thus often used when a figure must be drawn by hand, e.g. on a black board (lesson, oral examination).

The representation was initially used for military fortifications . In French, the « cavalier » (literally rider, horseman, see Cavalry) is an artificial hill behind the walls that allows to see the enemy above the walls [1]. The cavalier perspective was the way the things were seen from this high point. Some also explain the name by the fact that it was the way a rider could see a small object on the ground from his horseback [2]. Table of Fortification, from the 1728 Cyclopaedia. ... Kircholm, a 1925 painting by Wojciech Kossak. ...

Mathematical aspects

If the plane that faces the reader is xz and the "vanishing direction" is the y axis, with an angle α and a ratio k, then a point in the space which coordinates are (x, y, z) is represented on the flat figure by a (x", y") point, with:

• x" = x + k·cos α·y ;
• y" = z + k·sin α·y.

The transformaiton matrix is

For example, for an angle 30° and a ratio 0.7:

• x" = x + 0.35·y ;
• y" = z + 0.61·y ;

and for an angle 45° and a ratio 0.5:

• x" = x + 0,35·y ;
• y" = z + 0,35·y ;