Isometric prospect

The isometric prospect is a method of representation in Perspective in which the three Direction S of the space are represented with the same importance, from where the term.

It is an axonometric particular case of Perspective.

Principle

In analytical Geometry, one defines a Repère orthonormé.

The isometric prospect corresponds at a sight according to the line of directing vector (1, 1,1) in this reference mark. Thus, a cube of which stop them follow the axes of the reference mark is seen according to its large diagonal, like a hexagon.

The axes are thus projected on a level perpendicular to this large diagonal. The length undergo a reduction (the Projection is a Isométrie, the factor of reduction is the same one for all the lengths on a given axis).

It is a prospect which is easy to carry out form in the case of simple. It is an approximation of the “real” sight, which is satisfactory as long as the depth remains low: in particular it does not take into account the reduction of size with the distance.

Basic rules to draw in isometric prospect

Measurements

One speaks about isometric prospect because the distances are deferred same manner on the three axes. One applies to all the lengths which are colinéraires with an axis a reduction factor of 0,82.

In the case of the representation of an object, one defines initially a face of the object which one regards as the front face, and one places a Repère there; in this plan, there are thus only two visible axes, third is perpendicular to the drawing. The origin of the reference mark is in general placed in a corner.

One carries out then two sights (at least) which are the orthogonal projections of the object on the front face and a perpendicular face (face of left, of right-hand side, top or lower part). Then, it is enough to measure the coordinates of the points in this reference mark starting from the two figures, and to defer these coordinates on the axes of the isometric prospect by applying this coefficient for 0,82.

Angles

The Angle S between the axes ( X , there and Z ) are all equal (120°).

Circles

The circles are important forms in the technical design; this is a consequence of proceeded of manufacture of the parts (Usinage): Drilling, Milling, Tournage… They are also important in civil engineering (emerged of pipes, Arc in full-clotheshanger, Giratoire S…). When one generates the isometric prospect by odinator, this one can calculate the transformation of the circle. But this becomes complicated when one draws with the hand.

Initially let us notice that a circle is always registered in a square at which it is 4 times tangent, in the middle of the sides. For face, one thus constrained the circle in a Square .

In isometric prospect, this square becomes a Parallélogramme. Tangencies remain the same ones (medium on the sides), but the circle becomes a ellipse.

Oblique projection varies the diameter of the circle between 1 (large diameter of the ellipse, therefore horizontal diameter of the full-scale projected starting circle) and 0,58 (its small diameter, seen under its more important reduction in the direction of the greatest slope).

Trace-ellipses standardized make it possible to trace ellipses respecting these proportions for several sizes of main roads.

Defects and limits of the isometric prospect

Like all projections and all the prospects, the loss of the third dimension induced of the possible errors of interpretation. This was abundantly used by the artist Mr. C. Escher to create impossible situations.

In fact, a displacement of 1 cm on the axis Z is graphically translated same manner as a displacement of 1 cm according to the axis of the X and of the there , is a displacement of √2 ≈ 1,41 according to the “diagonal” of ( X , there ).

Uses of the isometric prospect

Use in technical design

In Draftsmanship, one represents a part under various visual angles, perpendicular to axes. These axes are “natural”: a part having a mechanical function (connection and movement with other parts), it has constraints of form and Usinage which makes that it in general has an axis of symmetry or plane faces. These axes or stop them these faces allow to define a orthogonal Repère (which one chooses orthonormé).

One can thus easily carry out an isometric prospect for a part starting from the sights in descriptive Géométrie usually used.

The isometric prospect makes it possible to the reader to easily represent the shape of the part, but does not allow to transmit useful informations to the design and the realization of the part.

Use in architecture

Eugene Purple-the-Duke used it in several of its tables of castles (and their additional buildings) to avoid accentuating the importance of some of these elements and position of the observer (the rider of the riding Perspective in the observation for the fortifications).

Use in the video games

A certain number of video games implementing characters uses a seen objectifies in isometric prospect; one often speaks, in this field, of “prospect 3/4”. From a practical point of view, that makes it possible to move the graphic elements (sprite S) without changing the size it, which was essential when the computers were not very powerful, and is of always great interest for the consoles of pocket.

That poses some problems of confusion however (because of applatissement of the image, the depth is returned by a displacement in the plan).

Because of the Pixellisation, unquestionable plays, before the use of the algorithms Antialiasing, made progress the axes according to a report/ratio of 2:1, they were thus tilted of an angle of 26,6° (arctan 0,5) instead of 30°. It was thus not isometric prospect strictly speaking, but a dimetric Perspective (another type of axonometric Perspective), but the “isometric” term is however used by abuse language.

Approaches mathematical

The isometric prospect is in fact a projection on a plan according to an orthogonal axis in this plan: a orthogonal Projection. It is a Linear application. a prospect is a setting in plan in various sights in space. A isometry is perceived 120°.

Factor of carryforward on the axes

One can simply calculate the factor of proportionality on the axes thanks to the Trigonométrie:

• let us consider stops it cube which goes from the origin at the item (0,0,1); it forms an angle α with the plan of projection, projected thus has a length of cos α;
• α is also the angle between the normal in the plan of projection passing by the origin and the point (1,1,1), and the bisectrix of the axes X and there which passes by (1,1,0);
• in the triangle formed by the items (0,0,0), (1,1,0) and (1,1,1) is a right-angled triangle; the segment (1,1,0) has as a length √2 (diagonal square), the segment (1,1,1) has as a length 1, and the hypotenuse (1,1,1) has as a length √3

There is thus

$\backslash \; cos\; \backslash \; alpha\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{2\}\; \{3\}\}\; \backslash \; simeq\; 0,82$.

One from of deduced that α ≈ 35,26 °.

One can also use the scalar product:

• the unit vector carried by the large diagonal is (1/√3, 1/√3, 1/√3);
• stops it (0,0,1) is projected on the large diagonal in a segment length K ₁, and on the normal level with this large diagonal in a segment length K ₂
• K ₁ is the scalar product of $\backslash \; vec\; \{has\}$ and of $\backslash \; vec\; \{B\}$, and can be calculated with the coordinates: $k\_1\; =\; \backslash \; vec\; \{has\}\; \backslash \; cdot\; \backslash \; vec\; \{B\}\; =\; 1\; \backslash \; times\; 1\; \backslash \; sqrt\; \{3\}\; +\; 0\; \backslash \; times\; 1\; \backslash \; sqrt\; \{3\}\; +\; 0\; \backslash \; times\; 1\; \backslash \; sqrt\; \{3\}\; =\; 1\; \backslash \; sqrt\; \{3\}$
• the Théorème of Pythagore indicates to us that K ₁ + K ₂ = 1 (length of stops cube).

One thus has:

$k\_2\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{2\}\; \{3\}\}\; \backslash \; simeq\; 0,82$.

The lengths of the segments on the axes of the reference mark are thus projected with a factor of 0,82.

One also arrives at this conclusion by using the general formula of orthogonal projections, to see axonometric Perspective > isometric Perspective .

In addition, if one considers the Cercle unit plan ( X , there ), the ray being projected according to the line of greater slope is the first bisectrix of the plan, with a factor of projection being worth sin α = K ₁ = 1/√3 ≈ 0,58, which corresponds to the small axis of the ellipse.

Transformation of the coordinates

The Cartesian transformation of the coordinates is used to calculate the sights starting from the coordinates of the points, for example in the case of video games or of software of chart 3D.

Let us suppose the space provided with a direct orthonormée Base $(\backslash \; vec\; \{e\_1\},\; \backslash \; vec\; \{e\_2\},\; \backslash \; vec\; \{e\_3\})$. Projection P is done according to the vector $\backslash \; vec\; \{U\}$ of components (1,1,1), i.e. the vector $\backslash \; vec\; \{U\}\; =\; \backslash \; vec\; \{e\_1\}\; +\; \backslash \; vec\; \{e\_2\}\; +\; \backslash \; vec\; \{e\_3\}$, according to the plan represented by this same vector.

Like any linear application, it can be represented by the transformation of the vectors of the base, since a vector $\backslash \; vec\; \{v\}\; =\; v\_1\; \backslash \; cdot\; \backslash \; vec\; \{e\_1\}\; +\; v\_2\; \backslash \; cdot\; \backslash \; vec\; \{e\_2\}\; +\; v\_3\; \backslash \; cdot\; \backslash \; vec\; \{e\_3\}$ unspecified changes according to

$P\; (\backslash \; vec\; \{v\})\; =\; P\; (v\_1\; \backslash \; cdot\; \backslash \; vec\; \{e\_1\}\; +\; v\_2\; \backslash \; cdot\; \backslash \; vec\; \{e\_2\}\; +\; v\_3\; \backslash \; cdot\; \backslash \; vec\; \{e\_3\})$

$P\; (\backslash \; vec\; \{v\})\; =\; v\_1\; \backslash \; cdot\; P\; (\backslash \; vec\; \{e\_1\})\; +\; v\_2\; \backslash \; cdot\; P\; (\backslash \; vec\; \{e\_2\})\; +\; v\_3\; \backslash \; cdot\; P\; (\backslash \; vec\; \{e\_3\})$

That is to say $\backslash \; vec\; \{e\text{'}\_n\}\; =\; P\; (\backslash \; vec\; \{e\_n\})$. Let us call $(\backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$ the direct base orthonormée in the plan of projection. One chooses arbitrarily that $\backslash \; vec\; \{e\text{'}\_1\}$ forms an angle of - π/6 with $\backslash \; vec\; \{I\}$.

The application of calculations for orthogonal projections to the particular case of the isometric prospect gives us (see axonometric Perspective > isometric Perspective ):

• $\backslash \; vec\; \{e\text{'}\_1\}\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{2\}\}\; \{2\}\; \backslash \; cdot\; \backslash \; vec\; \{I\}\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{6\}\}\; \backslash \; cdot\; \backslash \; vec\; \{J\}$; $k\_1\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{2\}\; \{3\}\}$

• $\backslash \; vec\; \{e\text{'}\_2\}\; =\; -\; \backslash \; frac\; \{\backslash \; sqrt\; \{2\}\}\; \{2\}\; \backslash \; cdot\; \backslash \; vec\; \{I\}\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{6\}\}\; \backslash \; cdot\; \backslash \; vec\; \{J\}$; $k\_2\; =\; k\_1$
• $\backslash \; vec\; \{e\text{'}\_3\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{2\}\; \{3\}\}\; \backslash \; cdot\; \backslash \; vec\; \{J\}$; $k\_3\; =\; k\_1$

The matrix of projection M_P is thus

$M\_P\; =\; \backslash \; begin\; \{pmatrix\}$

\ frac {\ sqrt {2}} {2} & - \ frac {\ sqrt {2}} {2} & 0 \ \ - \ frac {1} {\ sqrt {6}} & - \ frac {1} {\ sqrt {6}} & \ sqrt {\ frac {2} {3}} \ \ \end{pmatrix}

Let us consider a point ( X , there , Z ) of the space which is projected in ( X ', there '). Its projection will be thus:

$\backslash \; begin\; \{pmatrix\}\; x\text{'}\; \backslash \; \backslash \; y\text{'}\; \backslash \; \backslash \; \backslash \; end\; \{pmatrix\}\; =\; M\_P\; \backslash \; cdot\; \backslash \; begin\; \{pmatrix\}\; X\; \backslash \; \backslash \; there\; \backslash \; \backslash \; Z\; \backslash \; \backslash \; \backslash \; end\; \{pmatrix\}$

\ begin {pmatrix} \ frac {\ sqrt {2}} {2} (X there) \ \ \ sqrt {\ frac {2} {3}} Z - \ frac {1} {\ sqrt {6}} (X + there) \ \ \ end {pmatrix}

See also Projection (geometry) > Projection on a plan parallel to a line in analytical geometry.

Transformation of a circle of a plan containing two axes

Let us consider the trigonometrical circle of the plan $(\backslash \; vec\; \{e\_1\},\; \backslash \; vec\; \{e\_3\})$. The parametric coordinates of its points are:

$\backslash \; begin\; \{pmatrix\}\; X\; \backslash \; \backslash \; there\; \backslash \; \backslash \; Z\; \backslash \; \backslash \; \backslash \; end\; \{pmatrix\}\; =$

\ begin {pmatrix} \ cos \ theta \ \ \ sin \ theta \ \ 0 \ \ \ end {pmatrix} The coordinates of the points projected in the base $(\backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$ are thus

$\backslash \; begin\; \{pmatrix\}\; x\text{'}\; \backslash \; \backslash \; y\text{'}\; \backslash \; \backslash \; \backslash \; end\; \{pmatrix\}\; =$

\ begin {pmatrix} \ frac {\ sqrt {2}} {2} (\ cos \ theta - \ sin \ theta) \ \ - \ frac {1} {\ sqrt {6}} (\ cos \ theta + \ sin \ theta) \ \ \ end {pmatrix} The distance to the origin is $r\; =\; \backslash \; sqrt\; \{x\text{'}^2\; +\; y\text{'}^2\}$, that is to say

$r^2\; =\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash \; left\; (1\; -\; \backslash \; sin\; \backslash \; theta\; \backslash \; cdot\; \backslash \; cos\; \backslash \; theta\; \backslash \; right)$

\ frac {2} {3} \ left (1 - \ frac {1} {2} \ cdot \ sin 2 \ theta \ right)

(Formula of Moivre); this provides in the passing a parametric equation of the ellipse. This distance thus varies between 1 and $\backslash \; sqrt\; \{1/3\}\; \backslash \; simeq\; 0,58$. One finds the reports/ratios of the main roads and the small axis of the ellipse.

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