Which points are called frontally competing. Point and its projections. Competing points. What will we do with the received material?
Points located in space on the same projecting line are called competing. They are projected onto the corresponding projection plane at one point in accordance with Figure 1.2.15. So, A And IN– horizontally competing points; C and D – frontally competing points; E And F– profile competing points.
To increase the clarity of the drawing, they resort to some conditional visibility. It can be determined using competing points. We will assume that the direction of the rays of vision coincides with the direction of the projecting lines. Question about visibility of points A And IN on a horizontal projection it is solved as follows: the point whose height is greater is visible.
Figure 1.2.15 – Competing points
Figure 1.2.16 – Complex drawing of competing points
In accordance with Figure 1.2.16, the frontal projection shows that the point A located higher than the point IN. A similar visibility criterion is applied to points WITH And D, and to the points E And F. Yes, dots WITH And D are compared in depth, and the points E And F- by latitude.
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“Engineering Graphics” Descriptive geometry is the science of graphic images.
Various engineering structures, their individual structures, architectural
Basic designations
- Points in space are designated by capital letters of the Latin alphabet A, B, C, D... or Arabic numerals 1, 2, 3, 4, 5... - straight or curved lines in space - with
With the help of drawings, that is, with the help of images on a plane, the spatial forms of objects and the corresponding geometric patterns are studied. Development of methods for
Center projection
Let
Parallel projection
Visualization is a valuable property of centrally projected images. However, in practice, other qualities of projection drawings are also of great importance, in particular, ease of construction and reversibility.
Orthogonal projection
A parallel projection is called orthogonal (rectangular) if the projection direction s is perpendicular to the projection plane П′ (s^П’).
V o
Illustration of a straight line in a complex drawing
The projection of a straight line as a set of projections of all its points is a straight line. Consequently, the spatial line is determined in a two-picture complex drawing by a pair of its projections.
Direct private provisions
As already noted, straight lines of particular position include straight lines of level, i.e. parallel projection planes (in accordance with Figure 1.3.1 these are straight lines h, f, p), and projecting
Traces of a straight line
The points of intersection of a straight line with projection planes are called traces of a straight line.
The point of intersection of a straight line with a horizontal projection plane is called the horizontal line.
Frontal trace
The horizontal projection of the frontal trace F1 is the point of intersection of the horizontal projection of the straight line with the x12 axis.
Frontal projection of frontal s
Determining the natural size of a straight line segment
Determination of the natural size of a straight line segment in general position and its angles of inclination to the projection planes is carried out using the right-angled triangle method.
As can be seen from p
Mutual position of two straight lines
Two lines in space can intersect, be parallel, or cross.
If lines a and b intersect at some point K, then based on
Right Angle Projection Theorem
Mutual belonging (incidence) of a point and a plane
If a point belongs to a plane in space, then the projections of this point belong to the corresponding projections of any straight line lying in this plane (in accordance with Figure 1.3.16 straight
Plane traces
The trace of a plane is the line of its intersection with the projection plane. In Figure 1.3.17, the plane W is defined by the traces l and m: l=W ∩П2 and
Partial planes
It was noted above that the planes of particular position include level planes (parallel to the projection planes) and projecting planes (perpendicular to the projection planes). In the first case
Parallelism of a line and a plane
A line is parallel to a plane if it is parallel to any line lying in this plane. Thus, straight line l is parallel to straight line b located in the Q plane
Parallelism of planes
Planes are parallel if two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane. So, intersecting lines c and d plane
Perpendicularity of a line and a plane
From elementary geometry it is known that a line f2 is perpendicular to a plane if it is perpendicular to two lines lying in this plane.
On a given plane in quality
Intersection of a straight line with a plane
This is a positional task, because it defines a common data element of geometric objects, i.e. their point of intersection, which corresponds to Figure 1.3.24.
Algorithm for solving the problem
Intersection of two planes
In this positional problem, the common element of these geometric objects is a straight line. It can be constructed in two ways: using intermediary planes of a particular position, at the same time
Curved lines
A curved line can be seen as the trace of a moving point. This point can be a single point or a point belonging to a line or surface moving in space.
Curved lines mo
Projection properties of plane curves
Let us assume that this curve l lies in a certain plane W. Let us project the curve l onto the projection plane П¢ in the direction s in accordance with Figure 1.2.27.
A ruled surface is a surface that can be formed by the movement of a straight line in space. Depending on the nature of the movement of the generatrix
Surfaces of revolution
A surface of revolution is a surface that is described by some generatrix as it rotates around a fixed axis.
The generatrix can be either flat or
Surfaces of revolution of the second order
When a second-order curve rotates around its axis, a second-order surface of rotation is formed.
The following types of second-order surfaces are considered:
Intersection of a surface with a plane
This is a positional task to determine for given geometric objects their common element, which is a curved line.
To construct it, auxiliary planes are used
Conic sections
The lines that are obtained by intersecting the surface of a second-order cone with a plane are called conic sections.
These lines include the following: ell
General algorithm for solving the problem
Let two arbitrary surfaces Ф and Q be given. It is necessary to construct a line of their intersection, i.e. construct the points that belong to this line (Figure 1.3.52).
Thu
Special cases of intersection of second-order surfaces
Since second-order surfaces are algebraic, the line of their intersection is an algebraic curve. Since the order of the intersection line is equal to the product of the orders of n
Convert a complex drawing
The solution of many spatial problems (positional and metric) in a complex drawing is often complicated due to the fact that the given geometric objects are located arbitrarily relatively flat
Method for replacing projection planes
A distinctive feature of the method of replacing projection planes is the transition from a given system of planes, in which the projections of an object are specified, to a new system of two mutually perpendicular planes
This method is a special case of the plane-parallel movement method.
Indeed, if in the method of plane-parallel movement the point of the figure described some plane curve
Method of rotation around the projecting axis
When solving problems using the rotation method, the position of given geometric elements is changed by rotating them around a certain axis.
If the axis of rotation is placed perpendicular to the plane
The main problems solved by the rotation method
Task No. 1. Convert the general position straight line into the frontal level straight line (Figure 1.4.14).
Let's consider solving the problem by rotating the straight line AB around the horizontally projecting straight line
Construction of sweeps
A surface development is a flat figure formed by consistent alignment of a surface with a plane without breaks or folds. When unfolding the surface, consider
Prism surface development
There are two ways to develop a prism: the “normal section” method and the “rolling” method.
The “normal section” method is used to develop a surface
Development of the surface of the pyramid
The side faces of the pyramid are triangles, each of which can be built on three sides. Therefore, to obtain the development of the pyramid, it is enough to determine the natural values of its lateral edges and
Development of a cylindrical surface
Cylindrical surfaces are deployed in the same ways as prismatic ones. An n-gonal prism is first inscribed into a given cylinder, and then the scan is determined
Development of a conical surface
The development of a conical surface is performed similarly to the development of a pyramid in the following order. First, an n-gonal pyramid is inscribed into a given cone (number n from mass
Axonometric projections
The method of obtaining a single-projection reversible drawing is called axonometric. It gives a more visual image of the object.
The axonometric drawing consists only of
Two points whose horizontal projections coincide will be called horizontally competing. The frontal projections of such points (see points A and B in Fig. 41) do not cover each other, but the horizontal ones compete, i.e. It is not clear which point is visible and which is closed.
Of two horizontally competing points in space, the one that is higher is visible; its frontal projection is higher on the diagram. This means that from two points A and B in Fig. 41 point A on the horizontal projection plane is visible, and point B is closed (not visible).
Two points whose frontal projections coincide will be called frontally competing (see points C and D in Fig. 41). Of the two frontally competing points, the one that is closer is visible, its horizontal projection on the diagram is lower.
We have similar pairs of competing points 1, 2 and 3, 4 in Fig. 42 on intersecting lines m and n. Points 3 and 4 are frontally competing, of which point 3 is not visible as the more distant one. This point belongs to line n (this can be seen on the horizontal projection), which means that in the vicinity of points 3 and 4 on the frontal projection, line n is behind line m.
Points 1 and 2 are horizontally competing. Based on their frontal projections, we establish that point 1 is located above point 2 and belongs to straight line m. This means that on the horizontal projection in the vicinity of points 1 and 2, line n is below it, i.e. not visible.
In this way, the visibility of the planes of polyhedra and linear surfaces is determined, because Competing points on intersecting lines: edges and forming bodies are easily identified.
Rice. 42
Right angle projections
If the plane of the right angle is parallel to any projection plane, for example P 1 (Fig. 43, Fig. 44), then the right angle is projected onto this plane without distortion. In this case, both sides of the angle are parallel to plane P1. If both sides of a right angle are not parallel to any of the planes, then the right angle is projected with distortion onto all projection planes.
If one side of a right angle is parallel to any projection plane, then the right angle is projected in full size onto this projection plane (Fig. 45, Fig. 46).
Let us prove this position.
Let side BC of angle ABC be parallel to plane P1. B 1 C 1 – its horizontal projection; B 1 C 1 ║BC. A 1 – horizontal projection of point A. Plane A 1 AB, projecting straight line AB onto plane P 1, is perpendicular to BC (since BC AB and BC BB 1). And because BC║B 1 C 1, which means plane AB B 1 C 1. In this case, A 1 B 1 B 1 C 1. So A 1 B 1 C 1 is a right angle. Consider what the diagram of a straight ABC looks like, the side BC of which is parallel to the plane P 1.
Rice. 43 Fig. 44
Rice. 45 Fig. 46
Similar reasoning can be carried out regarding the projection of a right angle, one side of which is parallel to the plane P2. In Fig. 47 shows a visual image and diagrams of a right angle.
Intersecting lines. If the lines intersect, then the point of their intersection on the diagram will be on the same connection line
Parallel lines. Projections of parallel lines on a plane are parallel.
-Crossing straight lines. If the lines do not intersect or are parallel, then they intersect. The intersection points of their projections do not lie on the same projection connection line 
-Mutually perpendicular lines 
In order for a right angle to be projected in full size, it is necessary and sufficient that one of its sides be parallel and the other not perpendicular to the projection plane.
Sometimes, points in space can be located in such a way that their projections onto the plane coincide. These points are called competing points. 
Figure a – horizontally competing points. The one that is higher on the frontal projection is visible.
Figure b – frontally competing points. The one below on the horizontal plane is visible.
Figure c – profile competing points. The one that is further from the Oy axis is visible
The point can be in any of the eight octants. A point can also be located on any projection plane (belong to it) or on any coordinate axis. In Fig. Figure 15 shows points located in different quarters of space. Dot IN is in the first octant. It is removed from the projection plane P 1 , at a distance equal to the distance from its frontal projection IN to the projection axis, and from the plane P 2 to a distance equal to the distance from its horizontal projection to the axis of projections. When transforming a spatial layout, the horizontal plane of projections P 1 unfolds in the direction indicated by the arrow, and the horizontal projection of the point unfolds along with it IN , the frontal projection remains in place.
Dot A is in the second octant. When the projection planes are rotated, both projections of this point (horizontal and frontal) on the diagram will be located on the same connection line above the projection axis X . From the projections it can be determined that the point A located somewhat closer to the projection plane P 2 than to the plane P 1 , since its frontal projection is located above the horizontal one.
Dot WITH is in the fourth octant. Here the horizontal and frontal projections of the point WITH located below the projection axis. Since the horizontal projection of a point WITH closer to the projection axis than the frontal one, then the point WITH is located closer to the frontal plane of projections, similar to the projections of a point A on the frontal plane of projections.
Thus, by the location of the projections of points relative to the axis of the projections, one can judge the position of the points in space, that is, one can establish in which corners of space they are located and at what distances they are separated from the projection planes, etc.
In Fig. 16 also shows points occupying some particular (special positions). Dot E belongs to the horizontal plane P 1 ; frontal projection E 2 of this point is on the projection axis, and the horizontal projection E 1 coincides with the point itself.
Dot F belongs to the frontal plane P 2 ; horizontal projection F 1 this point is on the projection axis, and the frontal projection F 2 matches her. Dot G belongs to the projection axis. Both projections of this point are on the coordinate axis.
If a point belongs to the projection plane, then one of its projections is on the axis, and the other coincides with the point.
The distance of a point from the frontal plane of projections is called depth points, from the profile - width and from the horizontal projection plane – height. These parameters can be determined by segments of communication lines on the diagram. For example, in Fig. 13 point depth A equal to the segment A X A 1, width 0A x or A 2 A z , height – to segments A X A 2 or A at A 3. Also, the depth of a point can be determined by the size of the segment A z A 3, since it is always equal to the segment A X A 1.
In Fig. 17 shows some points. As you can see from this figure, one of the projections of the point WITH , in this case frontal, belongs, i.e. is located, on the axis X . If you write down the coordinates of a point WITH , then they will look like this: WITH (x, y, 0). From this we conclude, since the coordinate of the point WITH along the axis Z (height) is zero, then the point itself is on the horizontal projection plane at the location of its horizontal projection.
Recording the coordinates of a point A as follows: A (0, 0, z). Point coordinate A along the axis x equals zero, which means a point A cannot be located on the frontal or horizontal projection planes. Point coordinate A and along the axis y is also equal to zero, therefore, the point cannot be on the profile plane of projections. From this we conclude that the point A located on the axis z , which is the line of intersection of the frontal and profile projection planes.
Frontal projection of the point TO in Fig. 17 is located below the axis x , therefore the point itself is located below the horizontal projection plane. Below the horizontal plane are octants III and IV (see Fig. 12). And since the projection K 1 located on the diagram below the axis y , then we conclude that the point itself TO located in the fourth octant of space.
Dot IN located in the first octant of space, and from the location of the projections we can judge that the point IN belongs neither to projection planes nor to coordinate axes.
A special place in descriptive geometry is given to competing points. Competing are called points whose projections coincide on any projection plane. The competing point method is used to solve various problems, in particular to determine the visibility of objects. In Fig. 18 shows two pairs of competing points: B–T And A–E . Points B–T are horizontally competing, since their projections coincide on the horizontal projection plane, and the points A–E – frontally competing, since their projections coincide on the frontal plane of projections.
According to Fig. 18, it can be determined that a point will be visible on the horizontal projection plane IN , since in space it is located above the point T . On the diagram, the visibility of two horizontally competing points on the horizontal plane of projections is determined by comparing the height of the frontal projections of these points: height of the point IN greater than the height of the point T , therefore, on the horizontal plane of projections the point will be visible IN , since on the frontal plane of projections its projection is located above the projection of the point T .
The visibility of two frontally competing points is determined in a similar way, only in this case the location of the projections of the two points on the horizontal projection plane is compared. In Fig. 18 it is clear that the point A is located in space closer to the observer than the point E , at the point A axial distance y more than a point E . On the diagram, the projection of a point A – A 1 is located lower than the projection of the point E – E 1 , therefore, on the frontal plane of projections the point will be visible A .
The visibility of profile-competing points is determined by comparing the location of projections along the axis X . The point whose axis coordinate X more, will be visible on the profile plane of projections.
Using a diagram on a complex drawing, having certain knowledge and skills, it is easy to determine the location of a point in space relative to projection planes, coordinate axes or any other objects. Being able to recognize the position of a point from a diagram, you can also determine the position of any other object in space, since any geometric object can be represented as a set of points located in a certain way.
a B C
In Fig. 19, A it is clear that the point A located further than the point IN from the observer in space and both of them are located at the same height. In the complex drawing (Fig. 19, b) frontal projections of both points are located at equal distances from the axis X ,horizontal projection of a point A located closer to the axis X than the projection of the point IN . Since the position of a straight line in space is given by two points, connecting the points A And IN straight line, we get an image of the line in the drawing. If the frontal projections of two points of a straight line are located at the same distance from the horizontal plane of projections, therefore, the straight line is located parallel to this plane (Fig. 19, V).