A widely used variety of isometric perspective is isometric drawing. In the isometric the dimension reduction coefficient. Since the reduction is identical in the three axes, the isometric drawing is made without reduction, with the dimensions parallel to the axes at 1:1 scale or natural scale, without changing the appearance of the drawing except in its size. This allows both to draw these dimensions directly on the paper, which facilitates drawing by Cartesian coordinates, and to measure those of an object directly in the drawing. The appearance of the drawing is identical although larger, and the dimensions that in the correct perspective would be equal to the real ones (those parallel to the projection plane) are larger.

The scale on which the isometric drawing is larger compared to the isometric perspective is approximately 1.22.

In design and technical drawing

In industrial design, a piece is represented from different points of view, perpendicular to the natural coordinate axes. A piece with mechanical movement generally has shapes with axes of symmetry or flat faces. Such axes, or the edges of the faces, allow us to define an orthogonal projection.

These views are generally referred to as: plan, elevation and profile.

The view from above being plant, (bird's eye view); elevation, front view and profile, side view. In other words, if we refer to the 3D Cartesian plane, X, Y and Z, the views would be: Plan - Z axis; Elevation - Y axis; and Profile - X axis.

An isometric perspective of the part can easily be drawn from such views, allowing for improved understanding of the object's shape.

In Architecture

The use of isometric projection is useful to easily visualize groups of relatively small buildings, producing images that are reminiscent of oblique photographs taken from a bird's eye view, in which the great distance between the observer and the represented model tends to attenuate the effect of convergence of parallel lines typical of real perspective.

From a more practical concept, section drawings are also common, which allow you to get a much better idea of ​​the distribution of the volume of the rooms in a house than a simple plan of the floor plan of the house.

in video games

A certain number of video games put their characters in action using an isometric perspective point of view or rather, in the usual jargon, in "3/4 perspective". From a practical angle, this allows graphic elements to be moved without modifying the size, an inevitable limitation for computers with low graphic capacity.

In order to avoid pixelation, in some cases the projection was taken to a 2:1 system, that is, to an inclination of 26.6° (arctan 0.5) instead of 30°, which does not correspond to an isometric projection per se, but "dimetric".

The progressive increase in the graphic capabilities of computers has enabled the increasingly widespread use of more realistic projection systems, based on the perspective naturally perceived by the human eye: the conical perspective.

Contenido

Siendo la perspectiva isométrica una proyección geométrica") sobre un plano según un eje perpendicular al mismo, sus características y relaciones pueden ser calculadas analíticamente mediante la trigonometría.

Reduction factor on the axes

Considering the edge of a cube that goes from the origin to the point (0,0,1), if its intersection with the projection plane defines an angle α, the projection will have a length equivalent to the cosine of α.

Consequently:.

It can be deduced that α ≈ 35.26°.

It is also possible to use the dot product:.

Consequently:.

The length of the segments on the representation axes are projected with a factor of 0.82.

This conclusion is also reached using the general formula for orthogonal projections.

On the other hand, if the unit circle of the plane (x,y) is considered, the ray is projected along the line of greatest slope, which is the first bisector of the plane, with a projection factor equivalent to sin α = k = 1/√3 ≈ 0.58, which corresponds to the minor axis of the ellipse.

Coordinate transformation

The Cartesian coordinate transformation is used to calculate views from the coordinates of points, for example in the case of a video game, or 3D simulation.

Assuming a space provided with a direct orthonormal base. The projection P is made according to the component vector (1,1,1), that is, the vector , according to the plane represented by that same vector.

Like any linear application, it can be represented by the transformation of the base vectors, plus a vector.

that transforms according to.

Be . We call the base direct orthonormal on the projection plane.

We arbitrarily choose what makes an angle of -π/6 with .

The particular application of calculus to orthogonal projections in isometric perspective results:

The projection matrix M is consequently:.

Considering a point (x, y, z) of space that is projected onto (x', y'), its projection will be:.

If we consider the trigonometric circle of the plane, the parametric coordinates of its points will be:

The coordinates of the points projected on the base will be:.

The distance to the origin is , being.

This distance varies accordingly between 1 and.

Isometry determines a viewing direction in which the projection of the coordinate axes x, y, z make up the same angle, that is, 120° relative to each other. Objects are displayed with a 30° viewpoint rotation in the three main directions (x, y, z).

This perspective can be visualized by considering the point of view located at the upper vertex of a cubic room, looking towards the opposite vertex. The x and y axes are the lines where the walls meet the floor, and the z axis, the vertical line, is the meeting of the walls. In the drawing, the axes (and their parallel lines) maintain 120° between them.

Within the set of axonometric or cylindrical projections, there are other types of perspective, which differ by the position of the main axes, and the use of different reduction coefficients to compensate for visual distortions.

A widely used variety of isometric perspective is isometric drawing. In the isometric the dimension reduction coefficient. Since the reduction is identical in the three axes, the isometric drawing is made without reduction, with the dimensions parallel to the axes at 1:1 scale or natural scale, without changing the appearance of the drawing except in its size. This allows both to draw these dimensions directly on the paper, which facilitates drawing by Cartesian coordinates, and to measure those of an object directly in the drawing. The appearance of the drawing is identical although larger, and the dimensions that in the correct perspective would be equal to the real ones (those parallel to the projection plane) are larger.

The scale on which the isometric drawing is larger compared to the isometric perspective is approximately 1.22.

In design and technical drawing

In industrial design, a piece is represented from different points of view, perpendicular to the natural coordinate axes. A piece with mechanical movement generally has shapes with axes of symmetry or flat faces. Such axes, or the edges of the faces, allow us to define an orthogonal projection.

These views are generally referred to as: plan, elevation and profile.

The view from above being plant, (bird's eye view); elevation, front view and profile, side view. In other words, if we refer to the 3D Cartesian plane, X, Y and Z, the views would be: Plan - Z axis; Elevation - Y axis; and Profile - X axis.

An isometric perspective of the part can easily be drawn from such views, allowing for improved understanding of the object's shape.

In Architecture

The use of isometric projection is useful to easily visualize groups of relatively small buildings, producing images that are reminiscent of oblique photographs taken from a bird's eye view, in which the great distance between the observer and the represented model tends to attenuate the effect of convergence of parallel lines typical of real perspective.

From a more practical concept, section drawings are also common, which allow you to get a much better idea of ​​the distribution of the volume of the rooms in a house than a simple plan of the floor plan of the house.

in video games

A certain number of video games put their characters in action using an isometric perspective point of view or rather, in the usual jargon, in "3/4 perspective". From a practical angle, this allows graphic elements to be moved without modifying the size, an inevitable limitation for computers with low graphic capacity.

In order to avoid pixelation, in some cases the projection was taken to a 2:1 system, that is, to an inclination of 26.6° (arctan 0.5) instead of 30°, which does not correspond to an isometric projection per se, but "dimetric".

The progressive increase in the graphic capabilities of computers has enabled the increasingly widespread use of more realistic projection systems, based on the perspective naturally perceived by the human eye: the conical perspective.

Contenido

Siendo la perspectiva isométrica una proyección geométrica") sobre un plano según un eje perpendicular al mismo, sus características y relaciones pueden ser calculadas analíticamente mediante la trigonometría.

Reduction factor on the axes

Considering the edge of a cube that goes from the origin to the point (0,0,1), if its intersection with the projection plane defines an angle α, the projection will have a length equivalent to the cosine of α.

Consequently:.

It can be deduced that α ≈ 35.26°.

It is also possible to use the dot product:.

Consequently:.

The length of the segments on the representation axes are projected with a factor of 0.82.

This conclusion is also reached using the general formula for orthogonal projections.

On the other hand, if the unit circle of the plane (x,y) is considered, the ray is projected along the line of greatest slope, which is the first bisector of the plane, with a projection factor equivalent to sin α = k = 1/√3 ≈ 0.58, which corresponds to the minor axis of the ellipse.

Coordinate transformation

The Cartesian coordinate transformation is used to calculate views from the coordinates of points, for example in the case of a video game, or 3D simulation.

Assuming a space provided with a direct orthonormal base. The projection P is made according to the component vector (1,1,1), that is, the vector , according to the plane represented by that same vector.

Like any linear application, it can be represented by the transformation of the base vectors, plus a vector.

that transforms according to.

Be . We call the base direct orthonormal on the projection plane.

We arbitrarily choose what makes an angle of -π/6 with .

The particular application of calculus to orthogonal projections in isometric perspective results:

The projection matrix M is consequently:.

Considering a point (x, y, z) of space that is projected onto (x', y'), its projection will be:.

If we consider the trigonometric circle of the plane, the parametric coordinates of its points will be:

The coordinates of the points projected on the base will be:.

The distance to the origin is , being.

This distance varies accordingly between 1 and.