Parallel Projection Beispiel Essay

Our recently completed Devil’s Slide Trail project has already been recognized with several awards, including the Project of the Year Award by the Silicon Valley Chapter of the American Public Works Association in the under $2 million category; the Bikeways & Trails Project of the Year Award from the American Society of Civil Engineers Region 9, which encompasses all of California; and an Excellence in Motion Merit Award from the Metropolitan Transportation Commission.

PlaceWorks was hired by the County of San Mateo to provide environmental review, community facilitation, and landscape architectural services for project. The new trail is located along the former State Highway 1, which was abandoned after the new Tom Lantos tunnels were completed through the adjacent mountain. With dramatic ocean views, this multi-use trail is expected to receive over 60,000 visitors a year and will provide access for pedestrians, bicyclists, and horseback riders along the former highway. The project required extensive coordination with wildlife agencies to carefully plan, schedule, and monitor construction in order to avoid disrupting a pair of nesting Peregrine Falcons on adjacent cliffs. PlaceWorks also worked with the County and the project civil engineers, BKF, to develop custom, aesthetic solutions for barrier fencing; to identify native plantings; and to design three overlooks to take in the spectacular views.

A parallel projection is a projection of an object in three-dimensional space onto a fixed plane, known as the projection plane or image plane, where the rays, known as lines of sight or projection lines, are parallel to each other. It is a basic tool in descriptive geometry. The projection is called orthographic if the rays are perpendicular (orthogonal) to the image plane, and oblique or skew if they are not.

Overview[edit]

A parallel projection is a particular case of projection in mathematics and graphical projection in technical drawing. Parallel projections can be seen as the limit of a central or perspective projection, in which the rays pass through a fixed point called the center or viewpoint, as this point is moved towards infinity. Put differently, a parallel projection corresponds to a perspective projection with an infinite focal length (the distance between the lens and the focal point in photography) or "zoom". In parallel projections, lines that are parallel in three-dimensional space remain parallel in the two-dimensional projected image.

A perspective projection of an object is often considered more realistic than a parallel projection, since it more closely resembles human vision and photography. However, parallel projections are popular in technical applications, since the parallelism of an object's lines and faces is preserved, and direct measurements can be taken from the image. Among parallel projections, orthographic projections are the most realistic, and are commonly used by engineers. On the other hand, certain types of oblique projections (for example cavalier projection, military projection) are very simple to implement, and are used to create quick and informal pictorials of objects.

The term parallel projection is used in the literature to describe both the procedure itself (a mathematical mapping function) as well as the resulting image produced by the procedure.

Properties[edit]

Every parallel projection has the following properties.

Orthographic projection[edit]

Main articles: Orthographic projection, Multiview projection, and Axonometric projection

Orthographic projection is derived from the principles of descriptive geometry, and is a type of parallel projection where the projection rays are perpendicular to the projection plane. It is the projection type of choice for working drawings.

The term orthographic is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are parallel with the projection plane (or the paper on which the orthographic or parallel projection is drawn). However, the term multiview projection is also used. In multiview projections, up to six pictures of an object are produced, with each projection plane perpendicular to one of the coordinate axes. Sub-types of multiview orthographic projections include plans, elevations and sections.

When the principal planes or axes of an object are not parallel with the projection plane, but are rather tilted to some degree to reveal multiple sides of the object, it is called an axonometric projection.[1]Axonometric projection (not to be confused with the closely related principle of axonometry, as described in Pohlke's theorem) is further subdivided into three groups: isometric, dimetric and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal.[2][3] A typical characteristic of axonometric pictorials is that one axis of space is usually displayed as vertical.

Oblique projection[edit]

Main article: Oblique projection

In an oblique projection, the parallel projection rays are not perpendicular to the viewing plane, but strike the projection plane at an angle other than ninety degrees.[2] In both orthographic and oblique projection, parallel lines in space appear parallel on the projected image. Because of its simplicity, oblique projection is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial drawing, the displayed angles among the axes as well as the foreshortening factors (scale) are arbitrary. The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel with the plane of projection thereby creating a true shape, full-size image of the chosen plane. Special types of oblique projections include military, cavalier and cabinet projection.[4]

Analytic representation of a parallel projection[edit]

If the image plane is given by equation and the direction of projection by , then the projection line through the point is parametrized by

with .

The image of is the intersection of line with plane ; it is given by the equation

In several cases, these formulas can be simplified.

(S1) If one can choose the vectors and such that , the formula for the image simplifies to

(S2) In an orthographic projection, the vectors and are parallel. In this case, one can choose and one gets

(S3) If the image plane contains the origin one has and the parallel projection is a linear mapping:

(Here is the identity matrix and the outer product.)

From this analytic representation of a parallel projection one can deduce most of the properties stated in the previous sections.

See also[edit]

References[edit]

  • Schaum's Outline: Descriptive Geometry, McGraw-Hill, (June 1, 1962),ISBN 978-0070272903
  • Joseph Malkevitch (April 2003), "Mathematics and Art", Feature Column Archive, American Mathematical Society 
  • Ingrid Carlbom, Joseph Paciorek (December 1978), "Planar Geometric Projections and Viewing Transformations", ACM Computing Surveys, 10 (4): 465–502, doi:10.1145/356744.356750 
Parallel projection terminology and notations. The two blue parallel line segments to the right remain parallel when projected onto the image plane to the left.
Two parallel projections of a cube. In an orthographic projection (at left), the projection lines are perpendicular to the image plane (pink). In an oblique projection (at right), the projection lines are at a skew angle to the image plane.
A parallel projection corresponds to a perspective projection with a hypothetical viewpoint; i.e. one where the camera lies an infinite distance away from the object and has an infinite focal length, or "zoom".
Various projections and how they are produced
Comparison of several types of graphical projection. The presence of one or more 90° angles is usually a good indication that the perspective is oblique.

0 thoughts on “Parallel Projection Beispiel Essay

Leave a Reply

Your email address will not be published. Required fields are marked *