Parallel projection

A parallel projection is an image of points of the three-dimensional space of points on a given plane, wherein the projection beams are mutually parallel. Meet the projection rays at right angles to the plane of projection, it is an orthogonal projection. A parallel projection may be considered as a limiting case of a central projection, in which the center of projection is located at infinity. Parallel projections are often used to produce oblique images of geometric bodies.

Description

The image point to any point in space is obtained in a parallel projection, characterized in that one brings the parallel to the direction of projection by the point of the projection plane for cutting. Straight lines are mapped back to the line through a parallel projection in general. However, this does not apply to parallel to the projection direction, as they pass into points. The image of straight parallel lines are - if defined - also parallel to each other. The length of a line is obtained only when this is parallel to the projection plane; in all other cases appear shortened in the projection paths. Also, the size of a projected angle usually does not match the size of the original angle. For this reason, a rectangle is generally mapped to a parallelogram, but only in exceptional cases to a rectangle. The same is true for circles which pass generally in ellipses.

In general, the projection rays strike at an angle to the screen. This is called a slant or oblique parallel projection. Examples are the cavalier projection and bird's eye view.

Most commonly an orthogonal projection (also orthogonal, or orthographic parallel projection called ) applied. Here, the projection rays meet at a right angle to the plane of projection. This projection is based technical drawings of engineers and architects, with the special case that dominates one of the three principal planes of the cubic often technical objects is parallel to the projection surface ( three-panel projection). To create drawings with spatial impression, this parallelism is canceled. The objects are inclined. Depending on the inclination angle ( s) occur, for example isometrics or Dimetrien. The resulting images are often mistakenly viewed as images in cavalier perspective. The orthogonal projection advantageously corresponds taken with a telephoto lens of a photograph with a telecentric lens or approximately a photograph from a great distance.

Calculation of pixels

If a point on a plane are represented by a parallel projection of the projection direction, then the pixel of the intersection point of the straight line by using the direction vector:

Leaving flat surfaces and straight cut, the result for the parameter:

Substituting this into the straight one, we obtain the intersection of this with and therefore the pixel:

If the direction of projection is equal to the normal direction of the plane, the orthogonal projection of the point is obtained as a special case of the plain.

Synthetic geometry

In the synthetic geometry of the orthogonal projection of a straight line to another is precisely the same level plays a fundamental role in an affine plane. The definition here is: Be an affine plane and are distinct lines and the plane construed as sets of points lying on it. A bijective mapping is called parallel projection of the, if:

Addition is defined on formal grounds: For the identity map is the only parallel projection.

Features and Significance

The main formal properties of the so-defined parallel projections between straight lines an arbitrary but fixed here chosen affine plane:

• Each orthogonal projection of the plane is invertible and its inverse mapping is a parallel projection.
• Always exists a parallel projection to any two straight lines of the plane.

• This parallel projection is the identity if is.
• For such a parallel projection is uniquely determined by a single point - pixel pair, if not the intersection of the lines.
• If you select two points, both intersections of the straight and are not, then there is exactly a parallel projection of that maps to.

• The composition of two parallel projections of the plane, is always a bijective mapping, but it is generally not parallel projection.

The concept of parallel projection allows us to generalize the notion of affinity to nichtdesarguesche affine planes. It is generally defined as:

By this definition and the formal properties of parallel projections the generalized affinities form a subgroup of the group of all collineations of the affine plane. The supplementary definition for parallel projections, with the identical mapping of the plane to an affinity is, the existence of guarantees at least one affinity. It is not known whether there is an affinity levels at which the identical image is the only affinity.

Affinities inherit by their definition and the formal properties of parallel projections all invariance properties of parallel projections:

In an affine plane which satisfies the affine Fano axiom the middle of two points is invariant under parallel projections, and therefore also affinities.

Applicable in an affine translation plane

• Are three collinear points commensurable, then there are their images under any parallel projection and each affinity.
• The stretch factor and the division ratio of three collinear and commensurate points are invariant under parallel projections and affinities.

Conversely, since each part relatively faithful collineation on a desargueschen level meets the definition of a generalized affinity, some relatively faithful collineations are accurate for desarguesche levels affinities. A desarguesche level is always isomorphic to a coordinate plane over a skew field and an affine translation plane with the additional property that collinear points are always commensurable.

This fall the generalized term " affinity " for desarguesche levels together with the familiar from analytical geometry.

Example

A translation in an affine incidence level is always an affinity in the sense of the generalized definition (see the main article affine translation plane ). However, there are also affine incidence planes that do not allow more than the translational identity.