Isometric projection

نوشته شده در موضوع خرید اینترنتی در ۲۹ آبان ۱۳۹۵

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Graphical projection

Isometric projection is a process for visually representing three-dimensional objects in dual measure in technical and engineering drawings. It is an axonometric projection in that a 3 coordinate axes seem equally foreshortened and a angle between any dual of them is 120 degrees.

Overview[edit]

The tenure “isometric” comes from a Greek for “equal measure”, reflecting that a scale along any pivot of a projection is a same (unlike some other forms of graphical projection).

An isometric viewpoint of an intent can be performed by selecting a observation instruction such that a angles between a projections of a x, y, and z axes are all a same, or 120°. For example, with a cube, this is finished by initial looking true towards one face. Next, a brick is rotated ±۴۵° about a true axis, followed by a revolution of approximately ±۳۵٫۲۶۴° (precisely arcsin ۱۳ or arctan ۱۲, that is associated to a Magic angle) about a plane axis. Note that with a brick (see image) a fringe of a ensuing 2D sketch is a ideal unchanging hexagon: all a black lines have equal length and all a cube’s faces are a same area. Isometric graph paper can be placed underneath a normal square of sketch paper to assistance grasp a outcome but calculation.

In a identical way, an isometric view can be performed in a 3D scene. Starting with a camera aligned together to a building and aligned to a coordinate axes, it is initial rotated plumb (around a plane axis) by about 35.264° as above, afterwards ±۴۵° around a true axis.

Another approach isometric projection can be visualized is by deliberation a viewpoint within a cubical room starting in an top dilemma and looking towards a opposite, reduce corner. The x-axis extends diagonally down and right, a y-axis extends diagonally down and left, and a z-axis is true up. Depth is also shown by tallness on a image. Lines drawn along a axes are during 120° to one another.

The tenure “isometric” is mostly incorrectly used to impute to axonometric projections generally. (There are 3 forms of axonometric projections: isometric, dimetric and trimetric.)

Rotation angles[edit]

From a dual angles indispensable for an isometric projection, a value of a second might seem counterintuitive and deserves some serve explanation. Let’s initial suppose a brick with sides of length 2, and a core positioned during a pivot origin. We can calculate a length of a line from a core to a center of any corner as ۲ regulating Pythagoras’ postulate . By rotating a brick by 45° on a x-axis, a indicate (1, 1, 1) will therefore turn (1, 0, ۲) as decorated in a diagram. The second revolution aims to move a same indicate on a certain z-axis and so needs to perform a revolution of value equal to a arctangent of ۱۲ that is approximately 35.264°.

Mathematics[edit]

There are 8 conflicting orientations to obtain an isometric view, depending into that octant a spectator looks. The isometric renovate from a indicate ax,y,z in 3D space to a indicate bx,y in 2D space looking into a initial octant can be created mathematically with revolution matrices as:

[cxcycz]=[1000cos⁡αsin⁡α۰−sin⁡αcos⁡α][cos⁡β۰−sin⁡β۰۱۰sin⁡β۰cos⁡β][axayaz]=16[30−۳۱۲۱۲−۲۲][axayaz]{displaystyle {begin{bmatrix}mathbf {c} _{x}\mathbf {c} _{y}\mathbf {c} _{z}\end{bmatrix}}={begin{bmatrix}100\0{cos alpha }{sin alpha }\0{-sin alpha }{cos alpha }\end{bmatrix}}{begin{bmatrix}{cos beta }0{-sin beta }\010\{sin beta }0{cos beta }\end{bmatrix}}{begin{bmatrix}mathbf {a} _{x}\mathbf {a} _{y}\mathbf {a} _{z}\end{bmatrix}}={frac {1}{sqrt {6}}}{begin{bmatrix}{sqrt {3}}0-{sqrt {3}}\121\{sqrt {2}}-{sqrt {2}}{sqrt {2}}\end{bmatrix}}{begin{bmatrix}mathbf {a} _{x}\mathbf {a} _{y}\mathbf {a} _{z}\end{bmatrix}}}

where α = arcsin(tan 30°) ≈ ۳۵٫۲۶۴° and β = ۴۵°. As explained above, this is a revolution around a true (here y) pivot by β, followed by a revolution around a plane (here x) pivot by α. This is afterwards followed by an orthographic projection to a xy-plane:

[bxby0]=[100010000][cxcycz]{displaystyle {begin{bmatrix}mathbf {b} _{x}\mathbf {b} _{y}\0\end{bmatrix}}={begin{bmatrix}100\010\000\end{bmatrix}}{begin{bmatrix}mathbf {c} _{x}\mathbf {c} _{y}\mathbf {c} _{z}\end{bmatrix}}}

The other 7 possibilities are performed by possibly rotating to a conflicting sides or not, and afterwards inverting a viewpoint instruction or not.[1]

History and limitations[edit]

First formalized by Professor William Farish (1759–۱۸۳۷), a judgment of isometry had existed in a severe experimental form for centuries.[3][4] From a center of a 19th century, isometry became an “invaluable apparatus for engineers, and shortly afterward axonometry and isometry were incorporated in a curriculum of architectural training courses in Europe and a U.S.”[5] According to Jan Krikke (2000)[6] however, “axonometry originated in China. Its duty in Chinese art was identical to linear viewpoint in European art. Axonometry, and a impressive abbreviation that goes with it, has taken on a new stress with a appearance of visible computing”.[6]

As with all forms of together projection, objects drawn with isometric projection do not seem incomparable or smaller as they extend closer to or divided from a viewer. While fitting for architectural drawings where measurements need to be taken directly, a outcome is a viewed distortion, as distinct viewpoint projection, it is not how a eyes or photography routinely work. It also can simply outcome in situations where abyss and altitude are formidable to gauge, as is shown in a painting to a right. This can seem to emanate enigmatic or unfit shapes, such as a Penrose stairs.

Usage in video games and pixel art[edit]

Isometric graphics were used in early video games during a 80s and 90s, as a technique supposing a singular 3D outcome that could be achieved with a compelled resources of microcomputers of a era.

The character was also used for sprites and pixel art, achieving a evil character still used in retrogaming.

See also[edit]

  • Graphical projection
  • Axonometric projection

References[edit]

External links[edit]

Wikimedia Commons has media associated to Isometric projection.

Article source: https://en.wikipedia.org/wiki/Isometric_perspective

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