Axonometric Projection

Axonometric perspective. It is a graphic representation system, consisting of representing geometric elements or volumes in a plane, by means of an orthogonal projection , referred to three orthogonal axes, in such a way that they preserve their proportions in the three directions of space: height, width and length.

Summary

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  • 1 Definition of Axonometric Projection
  • 2 Elements of the projection system
  • 3 Properties
  • 4 Coordinates and scales
  • 5 Characteristics of the axonometric projection
  • 6 Construction methods in axonometric projection
  • 7 Foundations of the axonometric system
  • 8 Types of axonometric perspective
  • 9 Types of lines of the drawings of the plane figures
  • 10 External link
  • 11 Source

Definition of Axonometric Projection

The axonometric projection is a projection on a plane (Axonometric) that has an arbitrary position in space. If the rays are perpendicular to the axonometric plane, this is an orthogonal axonometric projection. This projection system is very similar to the way of observing objects in space, yet retains all the properties of the cylindrical projection (parallelism, perpendicularity).

The projections of the axonometric plane in the horizontal XY plane determine the XY line whose projection is perpendicular to the Z axis. In effect: Both lines (Z and XY axis) are orthogonal, the XY line is contained in the axonometric plane and the axonometric projection is an orthogonal projection.

Projection system elements

Axonometric projection system.

The elements of a projection system is:

  • Three perpendicular planes (called trirectangular trihedron).
  • The lines where the three coordinate planes (called axes) intersect.
  • Cut of the three axes (called vertex).

Properties

The axonometric perspective fulfills two important properties that distinguish it from the conical perspective

  • The scale of the represented object does not depend on its distance from the observer
  • Two parallel lines in reality are also parallel in their axonometric representation.

Coordinates and scales

The coordinates of the points on the axes can be measured, taking into account their corresponding deformation. (From there the axonometric name is derived, which in Greek means measure on the axes).

Each axis has its predetermined scale according to the axonometric plane and its respective direction of the projection rays. All lines parallel to the axonometric plane are preserved in this true-size projection. To determine the scales on the axes, we refute these on the axonometric plane where they must be projected in true size. To define the axonometric projection, it is enough to fix the angles under the X, Y, Z axes, whose sum must be 360º and none can be 90º. It can also be defined using the axonometric triangle.

  • Trimetry: the three angles are different, the three scales are different.
  • Bimetry: two angles are equal and two scales are also equal (the different scale is on the axis opposite the different angle).
  • Isometry(Monometría): the three angles equal to 120 °, the three scales are also equal.

Characteristics of the axonometric projection

Axonometry projection.

The axonometric projection is a cylindrical, orthogonal projection where: Properties:

  • a) The parallelism and proportionality, as well as the conjugate diameters of a conic.
  • b) The axonometric plane is projected at its true size.
  • c) The line perpendicular to a line parallel to the axonometric plane is projected at a right angle onto it.
  • d) A sphere projects as a circle .

Use. The axonometric projection is advantageously used to represent diagrams of installations, mechanical parts, buildings, etc. It gives an illusion more similar to the object than the oblique projection since it is closer to the way of looking (but sometimes it is more laborious to carry it out.

Representation and visibility. It is customary to review only the projection (perspective) although the horizontal projection is equally essential.

Construction methods in axonometric projection

  • Indirect: refuting the horizontal projection of the object and then fixing the points according to the respective heights.
  • a) To determine the axonometric horizontal projection, the orthogonal projection is first determined (in the XR, YR axis system).
  • b) Search for homology the axonometric horizontal projection, being: XY the axis of homology; homology rays perpendicular to the XY homology axis; a conjugate couple: O – OR.
  • c) The axonometric projection is determined according to the heights of the points. These heights correspond to the scale of the Z axis.

OM = House height. ON = Ridge height.

  • Direct: constructing the object according to those lines that are parallel to the projection axes and according to their scale.
  • a) Tetrahedron: regular with a horizontal base and an edge parallel to the Y axis. The height of the tetrahedron was determined separately.
  • b) Cube with faces parallel to the projection planes, that is, edges parallel to the axes.
  • c) Cube with main section parallel to the projection plane XZ and YZ, that is, diagonals of one face are parallel to the X and Y axes.
  • d) Regular octahedron : with diagonals parallel to the coordinate axes.
  • e) Regular octahedron: with main section parallel to the XZ plane, that is, edges parallel to the X and Y axes, and a diagonal parallel to the Z axis.
  • Projective: Similar to oblique projection, only the axes are projected differently.

Foundations of the axonometric system

  • Every body with volume is structured on three fundamental axes or directions, in them the three dimensions of the objects are distributed, on the z axis the heights are placed, on the x axis the widths and on the axis and depths.
  • The axonometric system places the basic edges of the bodies on these three coordinate axes and projects them on a flat surface equivalent to the sheet of paper and called the plane of the table.
  • Changing the configuration of the coordinate axes, the axonometric coordinate axes in space form an angle of 90º just like the edges of a cube. When they are projected orthogonally on the plane of the frame, they transform and measure more than 90º, and in turn the axes are no longer three-dimensionally structured, to adopt a new two-dimensional configuration on the plane of the frame.

Types of axonometric perspective

There are several types of axonometric perspectives:

Types of axonometric perspective.

  • Isometric axonometric perspective (the 3 equal angles)
  • Dimetric axonometric perspective (2 equal angles and another unequal)
  • Trimetric axonometric perspective (the 3 equal angles)

The isometric perspective. It is in which the axes form three equal angles of 120º each.

The dimetric. The axes form two equal angles and an unequal third.

The trimetric. Their axes form angles of different degrees.

Types of lines in drawings of plane figures

  • Isometric lines: they are all those whose sides are perpendicular to each other and when passing them to isometric their sides will be parallel to the isometric axes.
  • Non-isometric lines: the sides of these figures do not maintain parallelism with the axes, because the angles they form are different from 90º. In these cases it is solved by inscribing the figure in a coordinate grid.

 

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