Orthogonal projection

Template:Views Orthographic projection (or orthogonal projection) is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where 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. It is further divided into multiview orthographic projections and axonometric projections. A lens providing an orthographic projection is known as an (object-space) telecentric lens.

Geometry

A simple orthographic projection onto the plane $z=0$ can be defined by the following matrix:

P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}}

For each point $v=(v_{x},v_{y},v_{z})$ , the transformed point would be

Pv={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}}{\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}\\v_{y}\\0\end{bmatrix}}

Often, it is more useful to use homogeneous coordinates. The transformation above can be represented for homogeneous coordinates as

P={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&1\end{bmatrix}}

For each homogeneous vector $v=(v_{x},v_{y},v_{z},1)$ , the transformed vector would be

Pv={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}\\v_{y}\\0\\1\end{bmatrix}}

In computer graphics, one of the most common matrices used for orthographic projection can be defined by a 6-tuple, (left, right, bottom, top, near, far), which defines the clipping planes. These planes form a box with the minimum corner at (left, bottom, near) and the maximum corner at (right, top, far).

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:

P={\begin{bmatrix}{\dfrac {2}{{\text{right}}-{\text{left}}}}&0&0&-{\dfrac {{\text{right}}+{\text{left}}}{{\text{right}}-{\text{left}}}}\\0&{\dfrac {2}{{\text{top}}-{\text{bottom}}}}&0&-{\dfrac {{\text{top}}+{\text{bottom}}}{{\text{top}}-{\text{bottom}}}}\\0&0&{\dfrac {-2}{{\text{far}}-{\text{near}}}}&{\dfrac {{\text{far}}+{\text{near}}}{{\text{far}}-{\text{near}}}}\\0&0&0&1\end{bmatrix}}

which can be given as a scaling followed by a translation of the form

P=ST={\begin{bmatrix}{\dfrac {2}{{\text{right}}-{\text{left}}}}&0&0&0\\0&{\dfrac {2}{{\text{top}}+{\text{bottom}}}}&0&0\\0&0&{\dfrac {2}{{\text{far}}-{\text{near}}}}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}1&0&0&-{\dfrac {{\text{left}}+{\text{right}}}{2}}\\0&1&0&-{\dfrac {{\text{top}}+{\text{bottom}}}{2}}\\0&0&-1&{\dfrac {{\text{far}}+{\text{near}}}{2}}\\0&0&0&1\end{bmatrix}}

The inversion of the Projection Matrix, which can be used as the Unprojection Matrix is defined:

P^{-1}={\begin{bmatrix}{\dfrac {{\text{right}}-{\text{left}}}{2}}&0&0&{\dfrac {{\text{left}}+{\text{right}}}{2}}\\0&{\dfrac {{\text{top}}-{\text{bottom}}}{2}}&0&{\dfrac {{\text{top}}+{\text{bottom}}}{2}}\\0&0&{\dfrac {{\text{far}}-{\text{near}}}{-2}}&{\dfrac {{\text{far}}+{\text{near}}}{2}}\\0&0&0&1\end{bmatrix}}

Multiview orthographic projections

Main article: Multiview orthographic projection

With multiview orthographic projections, 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 6-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a 3D object. These views are known as front view, top view and end view.

Pictorials

Main article: Axonometric projection

Within orthographic projection there is the subcategory known as pictorials. Axonometric pictorials show an image of an object as viewed from a skew direction in order to reveal all three directions (axes) of space in a single picture.^[3] Orthographic pictorial instrument drawings are often used to approximate graphical perspective projections, but there is attendant distortion in the approximation. Because pictorial projections inherently have this distortion, in the instrument drawing of pictorials, great liberties may then be taken for economy of effort and best effect. Orthographic pictorials rely on the technique of axonometric projection ("to measure along axes").

[3]

Computer Graphics

3D Orthogonal projection

Orthogonal projection

Geometry

Multiview orthographic projections

Pictorials

Post a Comment