Parallel projection

Principle of parallel projection

A parallel projection is a mapping of points in three-dimensional space onto points of a given plane , the projection rays being parallel to one another . If the projection rays hit the projection plane at a right angle , it is an orthogonal projection . A parallel projection can be viewed as a borderline case of a central projection in which the projection center is at infinity . Parallel projections are often used to create oblique images of geometric bodies.

description

The image point of any point in space is obtained in a parallel projection by bringing the line parallel to the projection direction through this point to intersect with the projection plane. Straight lines are generally mapped back onto straight lines by means of parallel projection. However, this does not apply to parallels to the projection direction, as these merge into points. The image lines of parallel lines are - if defined - also parallel to one another. The length of a line is retained if it runs parallel to the projection plane. The size of a projected angle usually does not match the size of the original angle. For this reason, a rectangle is generally mapped onto a parallelogram , but only in exceptional cases onto a rectangle. The same applies to circles , which generally turn into ellipses .

Parallel projection of a cube: a) orthogonal, b) oblique

In general, the projection rays strike the projection surface at an angle. One then speaks of an oblique or oblique parallel projection. Examples of this are the cavalier projection and bird's eye view .

Most frequently is orthogonal (also called orthogonal or orthographic parallel projection) applied. Here the projection rays hit the projection plane at a right angle. The technical drawings of the engineers and architects are based on this projection , whereby the special case dominates that one of the three main levels of the often cube-shaped technical objects is parallel to the projection surface ( three-panel projection ). In order to create drawings with a spatial impression, this parallelism is removed. The objects are tilted. Depending on the angle of inclination (s), isometrics or dimensions are created, for example . The images obtained in this way are often mistakenly viewed as images from a gentlemanly perspective. The orthogonal projection corresponds to a photograph with a telecentric lens or approximately a photograph from a great distance, advantageously taken with a telephoto lens .

Calculation of pixels

If a point is to be mapped onto a plane (given in normal form ) by means of a parallel projection with the projection direction , the image point is from the intersection of the straight line through with the direction vector : ${\displaystyle P}$ ${\displaystyle E:~{\vec {n}}\cdot {\vec {x}}-d=0}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle {\vec {v}}}$

${\displaystyle g:~{\vec {x}}={\overrightarrow {OP}}+\lambda {\vec {v}},~\lambda \in \mathbb {R} }$

If you let the plane and the straight line intersect, the following results for the parameter : ${\displaystyle \lambda }$

${\displaystyle \lambda ={\frac {d-{\overrightarrow {OP}}\cdot {\vec {n}}}{{\vec {n}}\cdot {\vec {v}}}}}$

If you insert this into the straight line , you get the intersection of this with and thus the image point : ${\displaystyle g}$ ${\displaystyle E}$ ${\displaystyle P'}$

${\displaystyle {\overrightarrow {OP'}}={\overrightarrow {OP}}+{\frac {d-{\overrightarrow {OP}}\cdot {\vec {n}}}{{\vec {n}}\cdot {\vec {v}}}}\cdot {\vec {v}}}$

If the direction of projection is the same as the normal direction of the plane , the orthogonal projection of the point onto the plane is obtained as a special case . ${\displaystyle ({\vec {v}}={\vec {n}})}$

Synthetic geometry

In synthetic geometry , the parallel projection of a straight line in an affine plane onto another straight line in the same plane plays a fundamental role. The definition here is: Be an affine plane and be and different lines of the level regarded as sets of points lying on it. A bijective mapping is called a parallel projection of on if: ${\displaystyle A}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle \pi :g\rightarrow h}$ ${\displaystyle g}$ ${\displaystyle h}$

1. If they intersect and at one point , then applies ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle S}$ ${\displaystyle \pi (S)=S}$
2. For two different points that do not belong, always applies ${\displaystyle P,Q\in g}$ ${\displaystyle h}$

${\displaystyle P\pi (P)\parallel Q\pi (Q)}$.

In addition, for formal reasons, the following is defined: For the identical image is the only parallel projection. ${\displaystyle g=h}$

Properties and meaning

The most important formal properties of the parallel projections defined in this way between straight lines of any, but here firmly selected affine plane:

• Every parallel projection of the plane is reversible and its inverted image is a parallel projection.
• There is always a parallel projection for any two straight lines on the plane . ${\displaystyle g,h}$ ${\displaystyle \pi :g\rightarrow h}$

• This parallel projection is the identity, if is. ${\displaystyle g=h}$
• For such a parallel projection is through a single point-pixel pair uniquely determined, if not the intersection of the line is. ${\displaystyle g\neq h}$ ${\displaystyle (P,\pi (P))}$ ${\displaystyle P}$
• If you select two points , neither of intersections of the lines and , then there exists a parallel projection of on , the on maps. ${\displaystyle P\in g,P'\in h}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle P}$ ${\displaystyle P'}$

• The composition of two parallel projections of the plane , is always a bijective map, but it is generally not parallel projection. ${\displaystyle \pi _{12}:g_{1}\rightarrow g_{2},\,\pi _{23}:g_{2}\rightarrow g_{3}}$ ${\displaystyle \pi _{23}\circ \pi _{12}:g_{1}\rightarrow g_{3}}$

The concept of parallel projection allows the concept of affinity to be generalized to non-desargue affine levels. In general it is defined:

A collineation on an affine plane is called affinity if the restriction can be represented for every straight line by a finite composition of parallel projections. ${\displaystyle \alpha :A\rightarrow A}$ ${\displaystyle g\subset A}$ ${\displaystyle \left.\alpha \right|_{g}:g\rightarrow \alpha (g)}$

Through this definition and the formal properties of the parallel projections, the generalized affinities form a subgroup of the group of all collineations on the affine plane. The supplementary definition for parallel projections, with which the identical mapping of the plane becomes an affinity, ensures the existence of at least one affinity. It is not known whether there are affine planes where the identical map is the only affinity.

Through their definition and the formal properties of the parallel projections, affinities inherit all invariance properties of the parallel projections :

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

In an affine translation plane applies

• If three collinear points are commensurable , then so are their images under every parallel projection and every affinity. ${\displaystyle (T,P,Q)}$
• The stretching factor and the division ratio of three different collinear and commensurable points are invariant under parallel projections and affinities.

Conversely, since every partial ratio collineation on a Desargue level fulfills the generalized definition of an affinity, precisely the partial ratio collineations are affinities for Desargue levels. A Desargue plane is always isomorphic to a coordinate plane over an inclined body and an affine translation plane with the additional property that collinear points are always commensurable.

The generalized term “affinity” for Desargue levels thus coincides with that which is familiar from analytic 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 level ). However, there are also affine incidence levels which, apart from identity, do not allow any further translation.

See also

• Cauchy's theorem , on the mean surface of convex bodies when projected in parallel
• Axonometry , orthogonal axonometry

literature

Descriptive Geometry:

• Fucke, Kirch, Nickel: Descriptive Geometry. Fachbuch-Verlag, Leipzig, 1998, ISBN 3-446-00778-4 .
• Cornelie Leopold: Geometric Basics of Architectural Representation. Verlag W. Kohlhammer, Stuttgart, 2005, ISBN 3-17-018489-X .
• Kurt Peter Müller: spatial geometry: spatial phenomena - construction - calculation . Mathematics ABC for the teaching profession. 2nd revised and expanded edition. Teubner, Stuttgart / Leipzig / Wiesbaden 2004, Chapter 2, 2.2.3, p. 38 ff . (Inclined parallel projection).
• Eduard Stiefel : Textbook of the descriptive geometry . In: Textbooks and monographs from the field of the exact sciences . Mathematical series. 2nd revised edition. tape 6 . Birkhäuser, Basel / Stuttgart 1960 (detailed and application-oriented representation of the vertical parallel and especially three-panel projection).

Synthetic geometry:

• Wendelin Degen and Lothar Profke: Fundamentals of affine and Euclidean geometry . In: Mathematics for teaching at high schools . 1st edition. Teubner, Stuttgart 1976, ISBN 3-519-02751-8 .

The history of the term:

• Jeremy Gray : Worlds out of nothing . A course in the history of geometry of the 19th century. 1st edition. Springer, Berlin / Heidelberg / New York 2007, ISBN 978-0-85729-059-5 , 1st chapter.
• Gaspard Monge : Géométrie descriptive . 7th edition. Paris 1847 (French, first systematic treatment of three-panel projections and parallel projection in general, first edition 1811).
• Guido Schreiber: Textbook of descriptive geometry . after Monge's Géométrie descriptive. 1st completely revised edition. Herder, Karlsruhe and Freiburg 1928 (Heavily revised German translation of G. Monge's textbook).