Isometric projection

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Some 3D shapes in isometric projection. Black labels denote di-mensions of the 3D object, while red labels denote dimensionsof the 2D projection (drawing).

Isometric projection is a method for visually repre-senting three-dimensional objects in two dimensions intechnical and engineering drawings. It is an axonometricprojection in which the three coordinate axes appearequally foreshortened and the angles between any two ofthem are 120 degrees.

1 OverviewThe term isometric comes from the Greek for equalmeasure, reecting that the scale along each axis ofthe projection is the same (unlike some other forms ofgraphical projection).An isometric view of an object can be obtained by choos-ing the viewing direction such that the angles between theprojections of the x, y, and z axes are all the same, or120. For example, with of a cube, this is done by rstlooking straight towards one face. Next, the cube is ro-tated 45 about the vertical axis, followed by a rotationof approximately 35.264 (precisely arcsin 1 or arc-tan 1) about the horizontal axis. Note that with thecube (see image) the perimeter of the resulting 2D draw-ing is a perfect regular hexagon: all the black lines haveequal length and all the cubes faces are the same area.Isometric graph paper can be placed under a normal pieceof drawing paper to help achieve the eect without cal-

culation.In a similar way, an isometric view can be obtained in a3D scene. Starting with the camera aligned parallel to theoor and aligned to the coordinate axes, it is rst rotatedvertically (around the horizontal axis) by about 35.264as above, then 45 around the vertical axis.Another way isometric projection can be visualized is byconsidering a view within a cubical room starting in anupper corner and looking towards the opposite, lower cor-ner. The x-axis extends diagonally down and right, they-axis extends diagonally down and left, and the z-axis isstraight up. Depth is also shown by height on the image.Lines drawn along the axes are at 120 to one another.The term isometric is often mistakenly used to referto axonometric projections generally. (There are threetypes of axonometric projections: isometric, dimetricand trimetric.)

2 Rotation anglesFrom the two angles needed for an isometric projection,the value of the second may seem counter intuitive anddeserves some further explanation. Lets rst imagine acube with sides of length 2, and its center positioned atthe axis origin. We can calculate the length of the linefrom its center to the middle of any edge as p2 usingPythagoras theorem . By rotating the cube by 45 on thex axis, the point (1, 1, 1) will therefore become (1, 0, p2) as depicted in the diagram. The second rotation aims tobring the same point on the positive z axis and so needsto perform a rotation of value equal to the arctangent of1/p2 which is approximately 35.264.

3 MathematicsThere are eight dierent orientations to obtain an isomet-ric view, depending into which octant the viewer looks.The isometric transform from a point ax;y;z in 3D spaceto a point bx;y in 2D space looking into the rst octantcan be written mathematically with rotation matrices as:

24cxcycz

35 =241 0 00 cos sin0 sin cos

3524cos 0 sin0 1 0sin 0 cos

3524axayaz

35 = 1p6

24p3 0 p31 2 1p2 p2 p2

3524axayaz

351

2 6 REFERENCES

where = arcsin(tan 30) 35:264 and = 45 .As explained above, this is a rotation around the vertical(here y) axis by , followed by a rotation around the hor-izontal (here x) axis by . This is then followed by anorthographic projection to the x-y plane:

24bxby0

35 =241 0 00 1 00 0 0

3524cxcycz

35The other 7 possibilities are obtained by either rotatingto the opposite sides or not, and then inverting the viewdirection or not.[1]

4 History and limitations

Optimal-grinding engine model (1822), drawn in 30isometric.[2]

Example of Chinese art in an illustrated edition of theRomance of the Three Kingdoms, China, c. 15th century.Main article: Axonometric projection

First formalized by Professor William Farish (17591837), the concept of isometry had existed in a roughempirical form for centuries.[3][4] From the middle of the19th century, isometry became an invaluable tool forengineers, and soon thereafter axonometry and isome-try were incorporated in the curriculum of architecturaltraining courses in Europe and the U.S.[5] According toJan Krikke (2000)[6] however, axonometry originated inChina. Its function in Chinese art was similar to linearperspective in European art. Axonometry, and the picto-rial grammar that goes with it, has taken on a new signif-icance with the advent of visual computing.[6]

An example of the limitations of isometric projection.The height dierence between the red and blue ballscannot be determined locally.

The Penrose stairs depicts a staircase which seems toascend (anticlockwise) or descend (clockwise) yet formsa continuous loop.

As with all types of parallel projection, objects drawnwith isometric projection do not appear larger or smalleras they extend closer to or away from the viewer. Whileadvantageous for architectural drawings where measure-ments need to be taken directly, the result is a perceiveddistortion, as unlike perspective projection, it is not howour eyes or photography normally work. It also can easilyresult in situations where depth and altitude are dicultto gauge, as is shown in the illustration to the right. Thiscan appear to create paradoxical or impossible shapes,such as the Penrose stairs.

5 See also Graphical projection

Isometric graphics in video games and pixel art

6 References[1] Ingrid Carlbom, Joseph Paciorek , Dan Lim (Decem-

ber 1978). Planar Geometric Projections and ViewingTransformations. ACM Computing Surveys (ACM) 10(4): 465502. doi:10.1145/356744.356750.

[2] William Farish (1822) On Isometrical Perspective. In:Cambridge Philosophical Transactions. 1 (1822).

[3] Barclay G. Jones (1986). Protecting historic architectureand museum collections from natural disasters. Universityof Michigan. ISBN 0-409-90035-4. p.243.

[4] Charles Edmund Moorhouse (1974). Visual messages:graphic communication for senior students.

3[5] J. Krikke (1996). "A Chinese perspective for cy-berspace?". In: International Institute for Asian StudiesNewsletter, 9, Summer 1996.

[6] Jan Krikke (2000). Axonometry: a matter of perspec-tive. In: Computer Graphics and Applications, IEEEJul/Aug 2000. Vol 20 (4), pp. 711.

7 External links Isometric Projection

4 8 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

8 Text and image sources, contributors, and licenses8.1 Text

Isometric projection Source: http://en.wikipedia.org/wiki/Isometric%20projection?oldid=656893801 Contributors: Tarquin, DrBob,Heron, Frecklefoot, Infrogmation, Michael Hardy, Two halves, Liftarn, Wapcaplet, TakuyaMurata, CatherineMunro, Raven in Orbit, Sb-woodside, Tedius Zanarukando, Omeomi, Furrykef, Warofdreams, AnonMoos, Cdang, Boy b, Altenmann, Wlievens, Alerante, Giftlite,Pat Kelso, Smjg, DocWatson42, Gilgamesh, Macrakis, Pne, Chowbok, Mike R, Tybruce, Antandrus, MistToys, Andux, Histrion, Allefant,Finog, Chmod007, CALR, Johan Elisson, Luvcraft, WikiPediaAid, Andrejj, Aqua008, Parklandspanaway, Helopticor, PiccoloNamek,Minghong, Mdd, Stetin, Interiot, Borisblue, Plumbago, Bngrybt, Ashley Pomeroy, Ahruman, DreamGuy, Revolt, Wtshymanski, Cbur-nett, Juhtolv, Geraldshields11, Pfahlstrom, Oleg Alexandrov, OleMaster, LOL, Quadduc, TheEvilBlueberryCouncil, JamieScuell, Gra-ham87, Keeves, Icey, Rjwilmsi, 25, SMC, Mathbot, Omega025, OrbitOne, Chobot, Whosasking, YurikBot, Hairy Dude, DT28, Muchness,ZFGokuSSJ1, Chensiyuan, Takeshi316, Jtgibson, Howcheng, TDogg310, Zagalejo, FlyingPenguins, FF2010, MagicKnight, Zero1328,Gtdp, Closedmouth, Gulliveig, HereToHelp, Lewis R, Gordmoo, Cmglee, SmackBot, Eskimbot, ZS, Bluebot, Shatner, Thumperward,Tony.rc, Gsp8181, Kotra, Frap, Zagrebo, Yaksha, Gothmog.es, Nakon, Tossrock, Algr, Ctpm, BryanEkers, Paul 012, Tktktk, Mgiganteus1,Plvekamp, Peter Horn, SohanDsouza, Dantai Amakiir, CmdrObot, CBM, Mika1h, Leevanjackson, Simeon, Mato, Gogo Dodo, Mattjball,Pipatron, Satori Son, Epbr123, Moondigger, Three Laws of Robotics, AntiVandalBot, Vendettax, Jason2gs, Geniac, Map42892, Magioladi-tis, Carlwev, DancingPenguin, STBot, Cthulhugoat, Hasanisawi, Jrsnbarn, Cocoaguy, SharkD, MezzoMezzo, Useight, Spark010, Franklinharsha, Eric Ng, Philip Trueman, TXiKiBoT, Kww, Rei-bot, Qxz, Sniperz11, Tzsche, Austriacus, SieBot, TJRC, Krawi, Brian Ammon,Oxymoron83, Hello71, Tioda, Weston.pace, Capitalismojo, Homologia, Cmac441111, ClueBot, Alnatour 2000, Madskunk, Puchiko, Elec-tron105, Conexxo, MasterOfHisOwnDomain, DumZiBoT, Ascher15, Kwjbot, Lingerie92, Kbdankbot, Addbot, DOI bot, Olli Niemitalo,Madsy, Tide rolls, Yobot, AVB, Eric-Wester, Wikieditoroftoday, Materialscientist, Citation bot, Xqbot, Martin Kraus, Feardatfro, Miym,Grandthefttoaster, FrescoBot, Pinethicket, I dream of horses, Edderso, MertyWiki, Salvidrim!, Sintau.tayua, Guerillero, EmausBot, JohnCline, Carmichael, Peter Karlsen, ClueBot NG, Peter James, Mmick66, Widr, Helpful Pixie Bot, Greifer69, Thegreatgrabber, Victor Yus,Awsomeninja, Indiana State, Monkbot, Amortias, TerryAlex and Anonymous: 219

8.2 Images File:3D_shapes_in_isometric_projection.svg Source: http://upload.wikimedia.org/wikipedia/commons/6/67/3D_shapes_in_

isometric_projection.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Cmglee File:Axonometric_projection.svg Source: http://upload.wikimedia.org/wikipedia/commons/4/48/Axonometric_projection.svg License:

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File:Optimal-grinding_engine_model.jpg Source: http://upload.wikimedia.org/wikipedia/commons/a/ac/Optimal-grinding_engine_model.jpg License: Public domain Contributors: Paper On Isometrical Perspective. In: Cambridge Philosophical Transactions. 1 (1822),Original artist: William Farish (1759 1837)

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Overview Rotation anglesMathematicsHistory and limitations See alsoReferences External linksText and image sources, contributors, and licensesTextImagesContent license