Axonometric prospect

In a certain number of situations, and in particular in Technical design, the drawing is the representation of real objects. The reader of the drawing must be able to represent the part in volume starting from his representation on paper, into two Dimension S. poses the problem of the passage then three dimensions → two dimensions, which is in particular the field of the Perspective.

In space, one can choose a point of reference and three scales having distinct directions (they in general are chosen Perpendiculaire S) and noncoplanar (i.e. the three rules are not in the same plan). One can then locate a point of the object to be drawn (for example a top) by three numbers, called “coordinated”, which are the distance to be traversed according to the three directions to go from the point of reference at the point concerned.

See the article Location in the plan and space.

A axonometric prospect is a drawing on which:

• the three reference axes are represented by three lines;
• the lengths measured on the scales (coordinates) are deferred with a constant factor for each right-hand side (but this factor can be different from one line to another).

Thus, the drawing is particularly simple to carry out, that it is with the hand or by data-processing calculation (Infographie, Dessin computer-assisted, Synthèse of image 3D).

Axonometric prospect and real vision

The real vision is better returned with a conical Perspective. With the axonometric prospects, the distance compared to the observer results only in one displacement in the plan. There is in particular no reduction in size of the objects with the distance. On the other hand, if the object represented is not very deep, the effect of reducing is not very important, an axonometric prospect can thus give a good illusion of what the eye would see.

All the problem consists in choosing directions and reports/ratios which make a drawing easily interpretable by the reader - one thinks well that by taking axes and reports/ratios randomly, one would obtain a “realistic” drawing little.

Axonometric prospect and visual arts

The Chinese Peinture used much the drawing without reducing with the distance. It is not a question itself of “construction to the rule”, the concept is similar.

Defects of the axonometric prospects

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.

Formalism

Let us consider a Repère orthonormé direct $(O,\; \backslash \; vec\; \{e\_1\},\; \backslash \; vec\; \{e\_2\},\; \backslash \; vec\; \{e\_3\})$, the vectors respectively defining the axes of the X , the there and Z . The three axes are represented by three lines on the plan (drawing), of unit directing vectors $\backslash \; vec\; \{e\text{'}\_1\}$, $\backslash \; vec\; \{e\text{'}\_2\}$ and $\backslash \; vec\; \{e\text{'}\_3\}$ such as:

• the representation of $\backslash \; vec\; \{e\_1\}$ is $\backslash \; vec\; \{E$ _1} = k_1 \ cdot \ vec {e'_1} ;
• the representation of $\backslash \; vec\; \{e\_2\}$ is $\backslash \; vec\; \{E$ _2} = k_2 \ cdot \ vec {e'_2} ;
• the representation of $\backslash \; vec\; \{e\_3\}$ is $\backslash \; vec\; \{E$ _3} = k_3 \ cdot \ vec {e'_3} .

If one knows the coordinates ( X , there , Z ) of a point in space, then the placement of this point within projection is particularly simple: it is enough to defer these coordinates on the axes projected by applying the coefficients K ₁, K ₂ and K ₃.

Orthogonal projections

The orthogonal Projection is a mathematical operation. In the case which interests us, it is a question of projecting a point of space on a plan, perpendicular to this plan.

For example, the Ombre created by the Sun, when this one is with the vertical of the place where one is, is an orthogonal projection of the object.

Orthogonal projections are linear applications, which means enter others that two Vecteur S proportional remain proportional once projettés; it is thus many axonometric prospects.

If projection can be managed simply in computer graphics, the determination of the directions of the projected axes and the proportionality factors for the manual layout is not very simple in the general case. One frequently uses in fact dimetric prospects for which two of the coefficients are equal.

Determination of the directions of the axes and the reports/ratios

One can describe the plan of projection by rotations transforming a given plan, for example the plan ( Oxz ). If one asserts oneself that the projection of $\backslash \; vec\; \{e\_3\}$ remains vertical, then it is seen that the plan of projection can be obtained by two rotations, for example:

1. a rotation around the axis ( OX );
2. then a rotation around the projection of ( OZ ) on the plan.

One can also proceed in “the inverse order”:

1. a rotation around ( OZ );
2. then a rotation around the trace of the plan ( Oxy ) within projection.

See also the article Angles of Euler.

It is this second manner of making that we will retain. Let us notice that one obtains the same result by considering that the plan of projection remains fixed, but that it is the reference mark which turns (with opposite angles). Let us consider that the plan of projection is ( Oxz ). If one operates a rotation around ( OZ ) of an angle ω, the vectors of the base change into:

$\backslash \; vec\; \{e\_1\}\; \backslash \; rightarrow\; \backslash \; vec\; \{e\text{'}\_1\}\; =\; \backslash \; cos\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{e\_1\}\; +\; \backslash \; sin\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{e\_2\}$

$\backslash \; vec\; \{e\_2\}\; \backslash \; rightarrow\; \backslash \; vec\; \{e\text{'}\_2\}\; =\; -\; \backslash \; sin\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{e\_1\}\; +\; \backslash \; cos\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{e\_2\}$

If one applies then a rotation of angle α around the axis initial OX (which is well the trace of ( Oxy ) within projection) then that one makes projection on the plan, one sees that, $(0,\; \backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$ being the direct orthonormé reference mark of the plan of projection (transformed $(O,\; \backslash \; vec\; \{e\_1\},\; \backslash \; vec\; \{e\_3\})$ by rotations if it is the plan which turns, or $(O,\; \backslash \; vec\; \{e\_1\},\; \backslash \; vec\; \{e\_3\})$ original if it is the reference mark which turns):

• the vector $\backslash \; vec\; \{e\_3\}$ is projected according to $\backslash \; cos\; \backslash \; alpha\; \backslash \; vec\; \{J\}$;
• the vector $\backslash \; vec\; \{e\_1\}$ is projected like itself;
• the projection of the vector $\backslash \; vec\; \{e\_2\}$ is $-\; \backslash \; sin\; \backslash \; alpha\; \backslash \; cdot\; \backslash \; vec\; \{J\}$.

Projections axes are thus given by the following vectors, whose standard is the coefficient of carryforward:

• OX : $\backslash \; vec\; \{E$ _1} = \ cos \ Omega \ cdot \ vec {I} - \ sin \ Omega \ sin \ alpha \ cdot \ vec {J} ; $k\_1\; =\; \backslash \; sqrt\; \{\backslash \; cos^2\; \backslash \; Omega\; +\; \backslash \; sin^2\; \backslash \; Omega\; \backslash \; sin^2\; \backslash \; alpha\}$
• OY : $\backslash \; vec\; \{E$ _2} = - \ sin \ Omega \ cdot \ vec {I} - \ cos \ Omega \ sin \ alpha \ cdot \ vec {J} ; $k\_2\; =\; \backslash \; sqrt\; \{\backslash \; sin^2\; \backslash \; Omega\; +\; \backslash \; cos^2\; \backslash \; Omega\; \backslash \; sin^2\; \backslash \; alpha\}$
• OZ : $\backslash \; vec\; \{E$ _3} = \ cos \ alpha \ cdot \ vec {J} ; $k\_3\; =\; |\; \backslash \; cos\; \backslash \; alpha\; |$

The angles of the projected axes $\backslash \; vec\; \{E$ _1} and $\backslash \; vec\; \{E$ _2} compared to horizontal the $\backslash \; vec\; \{I\}$ can be calculated using trigonometry, for example:

• $(\backslash \; widehat\; \{\backslash \; vec\; \{I\},\; \backslash \; vec\; \{E$ _1}}) = \ arctan \ left (\ frac {\ sin \ Omega \ cdot \ sin \ alpha} {\ cos \ Omega} \ right)
• $(\backslash \; widehat\; \{\backslash \; vec\; \{I\},\; \backslash \; vec\; \{E$ _2}}) = \ arctan \ left (\ frac {\ cos \ Omega \ cdot \ sin \ alpha} {\ sin \ Omega} \ right)

angles not being directed here.

If X , there and Z is the punctual coordinates of space in the reference mark $(O,\; \backslash \; vec\; \{e\_1\},\; \backslash \; vec\; \{e\_2\},\; \backslash \; vec\; \{e\_3\})$, and x" and y" the coordinates of sound projected in the reference mark $(O,\; \backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$, one can define the matrix P of projection such as

$\backslash \; begin\; \{pmatrix\}$

X \ \ there \end{pmatrix}

P \ cdot

\begin{pmatrix} X \ \ there \ \ Z \end{pmatrix} (see the article Produces matric ), with

$P\; =\; \backslash \; begin\; \{pmatrix\}$

\ cos \ Omega & - \ sin \ Omega & 0 \ \ - \ sin \ Omega \ cdot \ sin \ alpha & - \ cos \ Omega \ cdot \ sin \ alpha & \ cos \ alpha \ \ \end{pmatrix} and

$\backslash \; begin\; \{pmatrix\}$

X \ \ there \end{pmatrix}

\begin{pmatrix}

\ cos \ Omega \ cdot X - \ sin \ Omega \ cdot there \ \ - \ sin \ Omega \ cdot \ sin \ alpha \ cdot X - \ cos \ Omega \ cdot \ sin \ alpha \ cdot there + \ cos \ alpha \ cdot Z \end{pmatrix}

For example, for ω = 30° and α = 20°, one a:

• K ₁ ≈ 0,88;
• K ₂ ≈ 0,58;
• K ₃ ≈ 0,94;
• ( I , e" ₁ ) ≈ 11,17°
• ( I , e" ₂ ) ≈ 30,64°
• x" ≈ 0,87· X - 0,50· there ;
• y" ≈ -0,17· X - 0,30· there + 0,94· Z .

Dimetric prospects

A dimetric prospect is a prospect for which two of the reports/ratios are equal.

Descriptive geometry

The sights in descriptive Géométrie are a particular case in which two of the coefficients are equal to 1, and the third coefficient is equal to 0.

They are also orthogonal projections.

Riding prospect

It is of an oblique projection and not about a true axonometry.

In the riding prospect , two of the axes are orthogonal and have a factor of carryforward of 1. The third axis tilted, in general of 30 or 45° compared to the horizontal one, is called “angle of escape”, and has a factor of carryforward lower than 1, in general 0,7 or 0,5.

See the detailed article riding Prospect.

Dimetric orthogonal projections

Let us choose K ₁ = K ₂; projections of the axes X and are symmetrical there compared to the vertical. This situation is a particular case of orthogonal projection with ω = 45 °; there is cos ω = sin ω = √2/2, that is to say

• OX : $\backslash \; vec\; \{E$ _1} = \ frac {\ sqrt {2}} {2} \ cdot \ vec {I} - \ frac {\ sqrt {2}} {2} \ sin \ alpha \ cdot \ vec {J} ; $k\_1\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{2\}\}\; \{2\}\; \backslash \; sqrt\; \{1\; +\; \backslash \; sin^2\; \backslash \; alpha\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{1\; +\; \backslash \; sin^2\; \backslash \; alpha\}\; \{2\}\}$; ( I , e" ₁ ) = arctan (sin α);

• OY : $\backslash \; vec\; \{E$ _2} = - \ frac {\ sqrt {2}} {2} \ cdot \ vec {I} - \ frac {\ sqrt {2}} {2} \ sin \ alpha \ cdot \ vec {J} ; $k\_2\; =\; k\_1$; ( I , e" ₂ ) = ( I , e" ₁ );
• OZ : $\backslash \; vec\; \{E$ _3} = \ cos \ alpha \ cdot \ vec {J} ; $k\_3\; =\; |\; \backslash \; cos\; \backslash \; alpha\; |$.

The plan of projection turns around the second Bissectrice of the plan ( Oxy ), i.e. around the vector $\backslash \; vec\; \{e\_1\}\; +\; \backslash \; vec\; \{e\_2\}$.

There is

$P\; =\; \backslash \; begin\; \{pmatrix\}$

\ frac {\ sqrt {2}} {2} & - \ frac {\ sqrt {2}} {2} & 0 \ \ - \ frac {\ sqrt {2}} {2} \ cdot \ sin \ alpha & - \ frac {\ sqrt {2}} {2} \ cdot \ sin \ alpha & \ cos \ alpha \ \ \end{pmatrix} and

$\backslash \; begin\; \{pmatrix\}$

X \ \ there \end{pmatrix}

\begin{pmatrix}

\ frac {\ sqrt {2}} {2} \ cdot (X - there) \ \ - \ frac {\ sqrt {2}} {2} \ cdot \ sin \ alpha \ cdot (X + there) + \ cos \ alpha \ cdot Z \end{pmatrix}

For example, for α = 45 °, there is

• K ₃ ≈ 0,71;
• K ₁ = K ₂ ≈ 0,87;
• ( I , e" ₁ ) ≈ 35,26 ° (vector e" ₁ directed downwards);

and for α = -10 °, one has

• K ₃ ≈ 0,98;
• K ₁ = K ₂ ≈ 0,72;
• ( I , e" ₁ ) ≈ 9,85 (vector e" ₁ directed upwards).

Isometric prospect

The isometric prospect is the particular case where the three reports/ratios are equal. It is about an orthogonal projection.

One a:

K ₁ = K ₃

that is to say

$\backslash \; sqrt\; \{\backslash \; frac\; \{1\; +\; \backslash \; sin^2\; \backslash \; alpha\}\; \{2\}\}\; =\; \backslash \; cos\; \backslash \; alpha$

by using the fact that cos ² α + sin ² α = 1, one obtains

$\backslash \; sin\; \backslash \; alpha\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{3\}\}$

and thus also $\backslash \; cos\; \backslash \; alpha\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{2\}\; \{3\}\}$ that is to say

• OX : $\backslash \; vec\; \{E$ _1} = \ frac {\ sqrt {2}} {2} \ cdot \ vec {I} - \ frac {1} {\ sqrt {6}} \ cdot \ vec {J} ; $k\_1\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{2\}\; \{3\}\}$; ( I , e" ₁ ) = arctan (1/√3) = 30 °;
• OY : $\backslash \; vec\; \{E$ _2} = - \ frac {\ sqrt {2}} {2} \ cdot \ vec {I} - \ frac {1} {\ sqrt {6}} \ cdot \ vec {J} ; $k\_2\; =\; k\_1$; ( I , e" ₂ ) = ( I , e" ₁ );
• OZ : $\backslash \; vec\; \{E$ _3} = \ sqrt {\ frac {2} {3}} \ cdot \ vec {J} ; $k\_3\; =\; k\_1$.

It is thus about a dimetric orthogonal projection (ω = 45 °), for which one has α ≈ 35,26 ° and K ₁ = K ₂ = K ₃ ≈ 0,82.

$P\; =\; \backslash \; begin\; \{pmatrix\}$

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

$\backslash \; begin\; \{pmatrix\}$

X \ \ there \end{pmatrix}

\begin{pmatrix}

\ frac {\ sqrt {2}} {2} \ cdot (X - there) \ \ - \ frac {1} {\ sqrt {6}} \ cdot (X + there) + \ sqrt {\ frac {2} {3}} \ cdot Z \end{pmatrix} that is to say

x" ≈ 0,71·( X - there  )

y" ≈ -0,41·( X + there  ) + 0,82· Z

See the detailed article isometric Prospect.

In Synthesis of image 3D

Orthogonal projection in synthesis of image

It is seen that if one knows the coordinates X_3D, Y_3D and Z_3D of the point in space, his coordinates on the screen X_2D and Y_2D, by considering an orthogonal projection, will be form:

X_2D = X_2D_0 + facteur* (A1*X_3D + A2*Y_3D)

Y_2D = Y_2D_0 + facteur* (B2* (A2*X_3D - A1*Y_3D) + B1*Z_3D)

where X_2D_0 and Y_2D_0 are constants making it possible “to center” the image, and facteur is a constant of scale. The constants A1, A2, B1 and B2 characterize the direction of the axes and the proportion of the carryforwards on these axes; they can be defined by:

A1 = cos (Omega)

A2 = sin (Omega)

B1 = cos (alpha)

B2 = sin (alpha)

omega and alpha being constants (compared to the preceding study, the sign for sin ω changed, which corresponds to a change of the sign of the angles, therefore with the reference for the direction of rotation). One can also define them without relationship to the angles, in an “empirical” way (for example adjusted by test-error to obtain a “pleasant” result), as lying between -1 and 1 and checking:

A1^2 + A2^2 = 1

B1^2 + B2^2 = 1

one can thus define only two parameters, A1 and B1, and calculate:

A2 = sqrt (1 - A1^2) or A2 = - sqrt (1 - A1^2)

B2 = sqrt (1 - B1^2) or B2 = - sqrt (1 - B1^2)

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