Lessons on the COMPASS program.
Lesson #4. Auxiliary lines in Compass 3D.
When developing drawings on a drawing board, designers always use thin lines; their analogue in Compass 3D is auxiliary straight lines. They are necessary for preliminary constructions and for specifying projection connections between views. When printing, auxiliary lines Auxiliary, it is impossible to change it.
There are several ways to construct auxiliary lines. In this lesson we will look at some of these methods.
1. An arbitrary straight line based on two points.
In the main menu of the program, press the commands sequentially Tools-Geometry-Auxiliary Lines-Auxiliary Line.
Or press the buttons in the compact panel Geometry-Auxiliary line.
By clicking the left mouse button we indicate the first base point (for example, the origin of coordinates). Now we indicate the second point through which the line will pass. The angle of inclination between the straight line and the abscissa axis of the current coordinate system will be determined automatically. You can enter an angle through the properties panel. For example, enter an angle of 45º and press the key Enter.
To complete the construction, click on the icon "Abort command" in the properties panel. This command can be carried out through the context menu, which is called up by right-clicking the mouse.
In a similar way through the base point, you can construct any number of arbitrary straight lines at any angle. You have probably already noticed that the coordinates of points can be entered from the keyboard using the properties panel. In addition, in the properties panel there is a group Modes, which has two switches: “Do not put intersection points”(active by default) and "Place intersection points". If you need to mark the intersection points of a line with other objects, activate the switch "Place intersection points", now the system will automatically set the intersection points with all graphic objects in its current form.
The dot style will be - Auxiliary. To remove all auxiliary elements, use the main menu commands Editor-Delete-Auxiliary curves and points. How to mark intersection points not with all, but only with some objects is described in lesson No. 3.
2.Horizontal straight line.
To construct a horizontal line, use the commands Tools-Geometry-Auxiliary Lines-Horizontal Line.
Or through the compact panel by pressing the buttons: Geometry-Horizontal line. The toolbar for constructing auxiliary lines is not entirely visible on the screen. To see it, click on the auxiliary lines button, active at the time of construction, and hold for several seconds.
Now it is enough to click the left mouse button to indicate the point through which the horizontal line will pass. You can build as many straight lines as you like at the same time. To complete the construction, click the button "Abort command" in the properties panel.
It must be remembered that the horizontal line is parallel to the x-axis of the current coordinate system. Horizontal ones constructed in a coordinate system rotated relative to the absolute system will not be parallel to the horizontal sides of the sheet.
3. Vertical straight line.
The construction is similar to the construction of horizontal lines, so you can figure it out on your own.
It must be remembered that the vertical line is parallel to the ordinate axis of the current coordinate system. Vertical ones constructed in a coordinate system rotated relative to the absolute system will not be parallel to the vertical sides of the sheet.
4. Parallel line.
To construct a parallel line, we need an object parallel to which it will pass. Such objects can be: auxiliary straight lines, segments, polyline links, sides of polygons, dimension lines, etc. Let's construct a parallel line for the horizontal line passing through the origin.
Calling the teams Tools-Geometry-Auxiliary Lines-Parallel Line.
In any design training course, they teach you to use thin auxiliary lines when creating drawings. Previously, they were applied on a drawing board and then erased from the finished document. Currently in use electronic programs for a drawing, but the need for auxiliary lines is not even discussed. Although in Compass 3D it is even easier to work with them than on a classic drawing board. Auxiliary lines are used to form necessary connections, marking the drawing, creating certain boundaries.
The program allows you to create auxiliary lines in several ways, again, this is very convenient, since sometimes one is used, and in another situation a different method of drawing auxiliary lines is used.
1. Create a straight line using two points.
One of the most popular methods. To activate, you must open the main menu Tools – Geometry - Auxiliary lines - Auxiliary line.
Or you can click in the panel Geometry-Auxiliary line.
Let's set our line by left-clicking on the sheet, so defining the first point, then specify end point lines. At the same time, the program itself will generate the required angle of inclination for the created straight line. However, you can change the angle by entering your values in the box below, then just click Enter.
The auxiliary line has been formed, now you need to click on the familiar icon Abort command, located in the properties panel. However, you can activate this command after finishing working with the line by simply right-clicking the mouse and then selecting the appropriate item in the drop-down menu.
Using a base point you can create infinite number straight lines going at any angles. By the way, if you have the coordinates or with coordinate grid work more conveniently, then you can always ask required values in the menu below. You will place a straight line, without any adjustments, on the sheet. Worth paying attention to the group Modes, it has two important switches. The first one is active during standard startup - Don't put intersection points, and you can choose the second one yourself - Set intersection points. Using this setting, you can automatically place points at any intersections, without additional options or manual placement.
However, here you need to specify the style Auxiliary. By the way, to remove all auxiliary elements, with finished drawing just activate the item in the main menu Editor-Delete-Auxiliary curves and points. We discussed working with points on curves in detail in lesson #3.
2.Draw a horizontal line
You can build auxiliary lines using horizontal lines. Let's open the already familiar menu Tools-Geometry-Auxiliary Lines-Horizontal Line.
A faster option, using a compact panel, select Geometry - Horizontal straight line. However, the basic panel will not be visible on the screen; to correct the situation, press the auxiliary lines button and hold it for a while.
All that remains is to use a left-click to indicate the desired point through which we will pass our straight line. You can create any number horizontal lines. To finish the job, just click Abort command in the properties panel or in the drop-down menu, right-click.
You also need to remember that a horizontal straight line is always parallel to the current x-axis. However, when setting horizontal lines using a rotated coordinate system, they will not be horizontal on the sheet.
3. Draw a vertical straight line.
The general mechanism for calling the line drawing mechanism is absolutely identical to that described above, with the exception of the choice Vertical straight.
However, there are a few important things to remember here. The created vertical straight line is always parallel only to the actual coordinate axis; here the case is identical to the horizontal straight line. Therefore, if you have a modified coordinate system, vertical straight lines will not be parallel to the sheet.
4. Create a parallel straight line.
You can build a parallel straight line only if there is any object on the sheet. It is to these lines that we will create a parallel. Moreover, absolutely any object can act as objects for snapping, from straight and auxiliary lines to the faces of polygonal objects. So, as part of the lesson, let’s take as the main one the horizontal line that goes from the origin of coordinates on our sheet.
Calling a parallel straight line is identical, open Tools – Geometry - Auxiliary lines - Parallel line.
Or use a compact panel, here you need to call Geometry-Parallel Line.
Now we will indicate the base object to which we will draw a parallel line. As agreed, the object is a horizontal straight line, select it with the mouse. Then, we need to set the distance at which our parallel line will be located. Below you can specify numeric value, for example 30 mm, or pull it straight with the mouse to the desired distance.
When specifying the distance in numbers, the system will offer two phantom lines at the same distance. This can be disabled if in the properties Number of lines - Two lines remove the activation, transforming it into the creation of one straight line. To fix the created line, just select the active phantom using the mouse and click on the create object button. When you need to create both lines, click Create Object again and then abort the command.
When you need to build a new parallel line, but near another object, just press the button Specify again. Now you can specify new object and build a line in the manner described in this chapter of the lesson.
That's all, in this lesson we covered the basics of creating auxiliary straight lines.
The methods for constructing parallel lines using various tools are based on the signs of parallel lines.
Constructing parallel lines using a compass and ruler
Let's consider the principle of constructing a parallel line passing through a given point, using a compass and ruler.
Let a line be given and some point A that does not belong to the given line.
It is necessary to construct a line passing through a given point $A$ parallel to the given line.
In practice, it is often necessary to construct two or more parallel lines without a given line and point. In this case, it is necessary to draw a straight line arbitrarily and mark any point that will not lie on this straight line.
Let's consider stages of constructing a parallel line:
In practice, they also use the method of constructing parallel lines using a drawing square and a ruler.
Constructing parallel lines using a square and ruler
For constructing a line that will pass through point M parallel to a given line a, necessary:
- Apply the square to the straight line $a$ diagonally (see figure), and attach a ruler to its larger leg.
- Move the square along the ruler until given point$M$ will not be on the diagonal of the square.
- Draw the required straight line $b$ through the point $M$.
We have obtained a line passing through a given point $M$, parallel to a given line $a$:
$a \parallel b$, i.e. $M \in b$.
The parallelism of straight lines $a$ and $b$ is evident from the equality of the corresponding angles, which are marked in the figure with the letters $\alpha$ and $\beta$.
Construction of a parallel line spaced at a specified distance from a given line
If it is necessary to construct a straight line parallel to a given straight line and spaced from it at a given distance, you can use a ruler and a square.
Let a straight line $MN$ and a distance $a$ be given.
- On the given straight line $MN$ we mark arbitrary point and let's call it $B$.
- Through the point $B$ we draw a line perpendicular to the line $MN$ and call it $AB$.
- On the straight line $AB$ from the point $B$ we plot the segment $BC=a$.
- Using a square and a ruler, we draw a straight line $CD$ through the point $C$, which will be parallel to the given straight line $AB$.
If we plot the segment $BC=a$ on the straight line $AB$ from point $B$ in the other direction, we get another parallel line to the given one, spaced from it by specified distance$a$.
Other ways to construct parallel lines
Another way to construct parallel lines is to construct using a crossbar. More often this method used in drawing practice.
When performing carpentry work for marking and constructing parallel lines, a special drawing tool is used - a clapper - two wooden planks that are fastened with a hinge.
Parallel lines. Definition
Two lines in a plane are called parallel if they do not intersect.
The parallelism of lines a and b is denoted as follows: a||b. Figure 1 shows lines a and b perpendicular to line c. Such lines a and b do not intersect, i.e. they are parallel.
Along with parallel lines, they are often considered parallel lines. Two segments are called parallel if they lie on parallel lines. In the figure (Fig. 2, a) the segments AB and CD are parallel (AB||CO) and the segments MN and CD are not parallel. The parallelism of a segment and a straight line (Fig. 2, b), a ray and a straight line, a segment and a ray, and two rays (Fig. 2, c) is determined similarly.
Signs of parallelism of two lines
Line c is called a secant to lines a and b if it intersects them at two points (Fig. 3). When lines a and b intersect with transversal c, eight angles are formed, which are indicated by numbers in Figure 3.
Some pairs of these angles have special names:
crosswise angles: 3 and 5, 4 and 6;
one-sided angles: 4 and 5, 3 and 6;
corresponding angles: 1 and 5, 4 and 8, 2 and 6, 3 and 7.
Let's consider three signs of parallelism of two straight lines associated with these pairs of angles.
Theorem. If, when two lines intersect crosswise, the angles involved are equal, then the lines are parallel.
Proof. Let the intersecting lines a and b crosswise the angles AB be equal: ∠1=∠2 (Fig. 4, a).
Let us show that a||b. If angles 1 and 2 are right (Fig. 4, b), then lines a and b are perpendicular to line AB and, therefore, parallel. Let's consider the case when angles 1 and 2 are not right. From the middle O of segment AB we draw a perpendicular OH to straight line a (Fig. 4, c). On straight line b from point B we plot the segment ВН1 equal to the segment AH, as shown in Figure 4, c, and draw the segment OH1. Triangles OHA and OH1B are equal on both sides and the angle between them (AO=VO. AN=BN1 ∠1=∠2), therefore ∠3=∠4 and ∠15=∠16. From the equality ∠3=∠4 it follows that point H1 lies on the continuation of the ray OH, i.e. points H, O and H1 lie on the same straight line, and from the equality ∠5=∠6 it follows that angle 6 is a straight line (so as angle 5 is a right angle). This means that lines a and b are perpendicular to line HH1, so they are parallel. The theorem is proven.
Theorem. If, when two lines intersect with a transversal, the corresponding angles are equal, then the lines are parallel.
Proof. Suppose that when lines a and b intersect with transversal c, the corresponding angles are equal, for example ∠1=∠2 (Fig. 5). Since angles 2 and 3 are vertical, then ∠2=∠3. From these two equalities it follows that ∠1=∠3. But angles 1 and 3 are crosswise, so lines a and b are parallel. The theorem is proven.
Theorem. If, when two lines intersect with a transversal, the sum of the one-sided angles is 180°, then the lines are parallel.
Proof. Let the intersection of straight lines a and b with transversal c sum the one-sided angles equal to 180°, for example ∠1+∠4=180° (see Fig. 5). Since angles 3 and 4 are adjacent, then ∠3+∠4=180°. From these two equalities it follows that the crosswise angles 1 and 3 are equal, therefore lines a and b are parallel. The theorem is proven.
Practical ways to construct parallel lines
Signs of parallel lines underlie the methods of constructing parallel lines using various tools used in practice. Consider, for example, the method of constructing parallel lines using a drawing square and a ruler. To construct a straight line passing through the point M and parallel to the given straight line a, we will apply a drawing square to the straight line a, and a ruler to it as shown in Figure 103. Then, moving the square along the ruler, we will ensure that the point M is on the side square, and draw straight line b. Straight lines a and b are parallel, since the corresponding angles, indicated in Figure 103 by the letters alpha and beta, are equal.
There is also a way to construct parallel lines using a crossbar. This method is used in drawing practice.
A similar method is used when performing carpentry work, where a block (two wooden planks fastened with a hinge) is used to mark parallel lines.
Occupies a special place in the history of mathematics Euclid's fifth postulate (axiom of parallel lines). For a long time mathematicians tried unsuccessfully to deduce the fifth postulate from the remaining postulates of Euclid and only in the middle of the 19th century thanks to research N. I. Lobachevsky, B. Riman And Y. Bolyai it became clear that the fifth postulate cannot be deduced from the others, and the system of axioms proposed by Euclid is not the only possible one.
Axiom of parallel lines
Even the ancient Greeks came up with a simple way: how to draw a compass and ruler through a point A lying outside a given line l, another line m that does not intersect the line l. But is there the only solution to this problem? Or can you draw several different lines through point A that do not intersect the original line m?
Euclid, apparently, was the first among the Hellenes to understand that the answer to this question cannot be obtained based on other properties of lines and points - those that he formulated in the form of axioms and postulates. It is necessary to introduce an additional postulate about the uniqueness of the desired line m - and call this line parallel!
Are other formulations of the postulate about parallel lines possible - incompatible with Euclid’s postulate? For example, we can assume the existence of several different lines that do not intersect a given line l and pass through common point A. Will such an assumption lead to a logical contradiction or not? If not, then geometries other than Euclidean are possible!
The first non-Euclidean geometry was invented in the 1820s by three talented mathematics: German Carl Gauss, Russian Nikolai Lobachevsky and Hungarian Janos Bolyai. The Russian mathematician turned out to be the most courageous and persistent of the three discoverers. He was the first to publish his book with a prediction remarkable properties non-Euclidean figures. For example, on the Lobachevsky plane the sum internal corners a triangle is always less than 180 degrees. She accepts different meanings for different triangles; with two similar to a triangle necessarily equal!
At the end of the 19th century, geometers Klein and Poincaré invented quite simple models surfaces on which Lobachevsky's geometry is embodied. Even earlier, Riemann noticed that the ordinary sphere embodies the third possible geometry (projective): there are no “parallel” lines in it at all, and the sum of the internal angles of a triangle is always greater than 180 degrees.
Until the early 20th century, it was believed that non-Euclidean geometries could only be useful internally mathematical science. But in the 1910s, Einstein created General Theory Relativity: it turned out to be a four-dimensional embodiment of Lobachevsky’s non-Euclidean geometry. Since then, physicists have believed that every consistent mathematical construct is embodied somewhere in Nature. This may be true.
Historical reference
In ancient times, literally 2500 years ago, in famous school Pythagoras Greek word“parallelos” began to be used as a geometric term, although the definition of parallel lines was not yet known at that time. But historical facts they say that the ancient Greek scientist Euclid in the third century BC, in his books, nevertheless revealed the meaning of such a concept as parallel lines.
As you already know, from the material covered in previous classes, the term “parallelos” translated from Greek language means walking next to or held near each other.
In mathematics, there is a special sign to indicate parallel lines. True, the parallelism sign did not always have its current form. For example, the ancient Greek mathematician Pappus in the third century AD used the equal sign “=” to indicate parallelism. And only in the eighteenth century, thanks to William Oughtred, they began to use the sign “//” to denote parallel lines. If there are, for example, parallel a and b, then they should be written in writing as a//b
But the “=” sign was introduced into general circulation by Record and began to be used as an equals sign.
Parallel lines in everyday life
We often encounter parallel lines in the life around us, although, as a rule, we rarely focus our attention on it. During music lessons, when we open a music book, we immediately see the lines of the staff with the naked eye. But parallel lines you can see not only in music books and songbooks, but also if you look closely at musical instruments. After all, the strings of a guitar, harp or organ are also parallel.
Looking up on the street, you see electrical wires running parallel. Finding yourself on the subway or railway, it is also not difficult to notice that the rails are located parallel to each other.
Parallel lines can be found everywhere. We constantly encounter them in everyday life and painting. Architecture cannot do without them, since the concept of parallelism is strictly taken into account in the construction of buildings.
If you look closely at the image, you will immediately notice the presence of parallel lines in these architectural structures. Perhaps they last so long and remain beautiful because the architects and engineers used parallel lines when creating these iconic buildings.
Have you ever wondered why the wires in power lines are arranged in parallel? And imagine what would happen if they were not parallel and intersected or touched each other. And this would lead to bad consequences, in which a short circuit could occur, interruptions and lack of electricity. What could happen to the train if the rails were not parallel? It's scary to even think about it.
You all know well that parallel lines never intersect. But if you look into the distance for a long time, into infinity, you may eventually see how parallel lines intersect. In this case, we are faced with an illusion of vision. Perhaps it was only thanks to such illusions and visual distortions that painting appeared.
Homework
1. Give your examples of where you are Everyday life, in everyday life or in nature, you come across moments or facts of parallelism.
2. What methods do you know by which you can draw parallel lines? Name these methods.
3. Draw parallel lines in your notebook using methods that you know.
4. Under what conditions can straight lines be called parallel?
Questions:
1. Which lines are called parallel?
2. What practical ways of constructing parallel lines exist?