AXONOMETRY OF PLANE FIGURES

METHODS OF MODELING THE FIGURES AXONOMETRIC PROJECTIONS

AKSONOMETRICAL OBLIQUE-ANGLED PROJECTIONS STATE STANDARD 2.317-69

 

 

Figure 3.5 - Frontal izometry

Figure 3.6 - Horizontal izometry

Figure 3.7 - Frontal dimetry

 

 

For modeling of a point in axonometry a coordinate broken line is drawn taking into account the coefficients of distortion on the axes of х, y, and z depending on the type of axonometry.

Let’s consider modeling of an axonometry of point, line and plane figure in rectangular izometry.

In izometry coefficients of distortion on the axes of х, y, and z are equal and equal 1.

a) b)

Figure 3.8 – Modeling of a point in axonometry

The segment ОАх (figure 3.8, a) put from a point 00 for the axes 0х of the axonometrical system of coordinates (figure 3.8, b). Through the obtained point 0Ах draw a straight line, parallel to 0у, on which a segment is drown, equal a segment Ах 1А. . Point 01А is obtained from which draw a straight line, parallel 0z . On this line draw the segment 01А02А, equal the segment of Ах2А. The obtained point 0А is the isometric projection of point A. The axonometrical segments make an axonometrical coordinate plane.

Making the considered model for every point of axonometrical figure, we can draw the model of this figure in axonometrical projections. Figure 3.9 b shows the construction of rectangular izometry for the segment AB, and figure 3.10 b shows a construction of rectangular izometry of a plane figure ABC.

а) b) а) b)

Figure 3.9 – Figure 3.10 –

Construction of rectangular izometry onstruction of rectangular izometry for the segment AB of a plane figure ABC

 

 

Let’s consider the modeling of rectangular izometry of a plane figurewhich is in a plane of projections (or in a level plane). As flat figures have two measurings that is why for their modeling in axonometry two axes are used, which are chosen depending on to which of the planes of projections the figure is parallel.

а) b) c)

Figure 3.11 – The modeling of correct hexagon

 

Figure 3.11 shows a correct hexagon which is placed: a) parallel to the horizontal plane; b) – parallel to the frontal plane of projections; c) parallel to the profile plane of projections. The modeling of every point in a axonometrical projection generally is carried out after the description as shown in fig.3.8, we will use the well-known rule: the axonometrical projections of parallel lines are parallel between themselves. Through auxiliary points 1, 2, 3, 4, which are on axonometrical axes, draw lines parallel the proper axonometrical axes (see figure 3.12, a) and on their crossing mark the points of 0B, 0C, 0E, 0F. Points 0A, 0D are also on the proper axonometrical axes while connecting all of axonometrical projections of points, we’ll get an axonometrical projection of a figure (see fig. 3.12 b).

 

а) b)

Figure 3.12 – The modeling of correct hexagon

 

Axonometry of a free circle can be built as an aggregate of axonometrical projections of certain number of points of this circle. In any type of axonometry a circle will be projected in an ellipse. For the modeling of a rectangular іzometry of a circle (see fig. 3.13, a) which is in a coordinate plane (or in a level plane ), at first it is necessary to build axonometry of its center (points of 0X, 0Y, 0Z (figure 3.8)) to draw through the obtained point lines, parallel to the proper two axonometrical axes (segment: [1, 2] = D (there is a diameter of a circle inplane), and [3, 4] = D). Then draw the minor axis of an ellipse (segment [CD] = 0,71 ´ D (dash-dotted line)) parallel the axonometrical axis out of this plane, and a major axis (segment [AB] = 1,22 ´ D) which will be perpendicular (see figure 7.13, b). An ellipse is drawn on the obtained eight points by the curve (figure 3.13, c).

а) b) c)

Figure 3.13 – Axonometry of a free circle

 

Let’s consider the modeling of a rectangular dimetry of a plane figurewhich lies in a plane of projections (or in a plane level). Let’s remind, that it is a axonometrical projection with the identical indexes of distortion on two axes – X and Z.

а) b) c)

Figure 3.14 – Axonometry of a free circle

 

Figure 3.14 shows a square which is placed: a) parallel to the horizontal plane; b) – parallel to the frontal plane of projections; c) parallel to the profile plane of projections. Taken together these squares are the projection of a cube on the ortogonal planes of projections. In a rectangular dimetry this cube is shown in figure 3.4 d. It is evident, that the length of a verge to the direction of axis Y is twice less. The modeling of every point in an axonometrical projection is generally carried out after the description in figure 3.8, that is why we’ll use the well-known rule: the axonometrical projections of parallel lines are parallel between themselves. Through auxiliary points 1, 2, 3, 4, which are on axonometrical axes, draw lines parallel the proper axonometrical axes (see fig. 3.15, a) and on their crossing mark the points of 0B, 0C, 0E, 0F. Points of 0A, 0D are also found on the proper axonometrical axes. Connecting all the axonometrical projections of points, we’ll obtained an axonometrical projection of a figure (see fig. 3.15, b).

а) b)

Figure 3.15 – Axonometrical projection of a figure

 

For the modeling of a rectangular dimetry circle (fig 3.16, a) which is in a co-ordinate plane (or in a plane level), first it is necessary to draw an axonometry of its center (points of 0X, 0Y, 0Z (figure 3.8)), to draw through the obtained point lines, parallel to the proper two axonometrical axes (segment: [1, 2] = D; [3, 4] = D; [5, 6] = 0.5´D).

а) b) c)

Figure 3.16 – The modeling of a rectangular dimetry circle

 

Then we draw the minor axis of an ellipse (segment [CD] = 0,95 ´ D (dash-dotted line) or [EF] = 0,35 ´ D) placed parallell to the axonometrical axis which doesn’t exist in this plane, and also major axis (segment [AB] = 1,06 ´ D) which will be perpendicular to it (figure 3.16, b). The ellipse is traced on the eight points obtained by curve (figure 3.16, c).