Parallel projection

The transformation P is the orthogonal projection onto the line m.

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P² = P. Projections map the whole vector space to a subspace and leave the points in that subspace unchanged.^[1]

Parallel Projection Media

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".
Error creating thumbnail: About to transcode 1 SVG file(s)
Converting Various_projections_of_cube_above_plane.svg to /var/www/html/w/images/temp/transform_9384139deb28.png ... org.apache.batik.transcoder.TranscoderException: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_e764c1566d4a1382355a6f99/Various_projections_of_cube_above_plane.svg:-1 Cannot find the referenced element: "file:/var/www/html/w/images/temp/svg_e764c1566d4a1382355a6f99/Various_projections_of_cube_above_plane.svg#clip1" specified on the element <g> - may be a problem of 'id' at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:228) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) Caused by: org.apache.batik.bridge.BridgeException: file:/var/www/html/w/images/temp/svg_e764c1566d4a1382355a6f99/Various_projections_of_cube_above_plane.svg:-1 Cannot find the referenced element: "file:/var/www/html/w/images/temp/svg_e764c1566d4a1382355a6f99/Various_projections_of_cube_above_plane.svg#clip1" specified on the element <g> - may be a problem of 'id' at org.apache.batik.bridge.BridgeContext.getReferencedNode(BridgeContext.java:762) at org.apache.batik.bridge.BridgeContext.getReferencedElement(BridgeContext.java:804) at org.apache.batik.bridge.CSSUtilities.convertClipPath(CSSUtilities.java:718) at org.apache.batik.bridge.AbstractGraphicsNodeBridge.buildGraphicsNode(AbstractGraphicsNodeBridge.java:146) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:224) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) ... 7 more ... error (SVGConverter.error.while.rasterizing.file)

Various projections and how they are produced
Comparison of several types of graphical projection. The presence of one or more 90° principal angles is usually a good indication that the perspective is oblique.
Paul Kuniholm Mural 1924-1st-Ave-Created-2019-July-6
Optical-grinding engine model (1822), drawn in 30° isometric perspective
Example of a dimetric perspective drawing from a US Patent (1874)
Example of a trimetric projection showing the shape of the Bank of China Tower in Hong Kong.

Notes

↑ Meyer, pp 386+387

References

N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience, 1958.
Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics, 2000. ISBN 978-0-89871-454-8.

Other websites

MIT Linear Algebra Lecture on Projection Matrices Archived 2008-12-20 at the Wayback Machine at Google Video, from MIT OpenCourseWare

[1] Meyer, pp 386+387

[1]