Orthographic projection, or orthogonal projection (also analemma), 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, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.
The term orthographic sometimes means a technique in multiview projection in which principal axes or the planes of the subject are also parallel with the projection plane to create the primary views.
The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (−1,−1,−1) and a maximum corner at (1,1,1).
The orthographic transform can be given by the following matrix:
:<math>
P =
\begin{bmatrix}
\frac{2}{\text{right}-\text{left & 0 & 0 & -\frac{\text{right}+\text{left{\text{right}-\text{left \\
0 & \frac{2}{\text{top}-\text{bottom & 0 & -\frac{\text{top}+\text{bottom{\text{top}-\text{bottom \\
0 & 0 & \frac{-2}{\text{far}-\text{near & -\frac{\text{far}+\text{near{\text{far}-\text{near \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>
which can be given as a scaling S followed by a translation T of the form
:<math>
P = ST =
\begin{bmatrix}
\frac{2}{\text{right}-\text{left & 0 & 0 & 0 \\
0 & \frac{2}{\text{top}-\text{bottom & 0 & 0 \\
0 & 0 & \frac{2}{\text{far}-\text{near & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 & -\frac{\text{left}+\text{right{2} \\
0 & 1 & 0 & -\frac{\text{top}+\text{bottom{2} \\
0 & 0 & -1 & -\frac{\text{far}+\text{near{2} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>
The inversion of the projection matrix P<sup>−1</sup>, which can be used as the unprojection matrix is defined:
<math>
P^{-1} =
\begin{bmatrix}
\frac{\text{right}-\text{left{2} & 0 & 0 & \frac{\text{left}+\text{right{2} \\
0 & \frac{\text{top}-\text{bottom{2} & 0 & \frac{\text{top}+\text{bottom{2} \\
0 & 0 & \frac{\text{far}-\text{near{-2} & -\frac{\text{far}+\text{near{2} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>
Types
Three sub-types of orthographic projection are isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal. Typically in axonometric drawing, as in other types of pictorials, one axis of space is shown to be vertical.
In isometric projection, the most commonly used form of axonometric projection in engineering drawing, the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 120° between them. As the distortion caused by foreshortening is uniform, the proportionality between lengths is preserved, and the axes share a common scale; this eases one's ability to take measurements directly from the drawing. Another advantage is that 120° angles are easily constructed using only a compass and straightedge.
In dimetric projection, the 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 to the angle of viewing; the scale of the third direction is determined separately.
In trimetric projection, the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Trimetric perspective is seldom used in technical drawings.
The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions.