Cavalier perspective

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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.

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.

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].

[edit] 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 ;