The methods for constructing parallel lines using various tools are based on the signs of parallel lines.
Constructing parallel lines with a compass and straightedge
Consider the principle of constructing a parallel line passing through given point , using a compass and ruler.
Let a line be given, and some point A, which does not belong to the given line.
It is necessary to construct a line passing through the given point $A$ parallel to the given line.
In practice, it is often required to construct two or more parallel lines without a given line and point. In this case, it is necessary to draw a line arbitrarily and mark any point that will not lie on this line.
Consider steps for constructing a parallel line:
In practice, the method of constructing parallel lines using a drawing square and a ruler is also used.
Construction of parallel lines using a square and a ruler
For constructing a line that will pass through the point M parallel to the given line a, necessary:
- Attach the square to the straight line $a$ with a diagonal (see the 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 desired 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 the lines $а$ and $b$ is seen from the equality corresponding angles, which are marked in the figure by the letters $\alpha$ and $\beta$.
Construction of a parallel line at a given 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 line $MN$ and a distance $a$ be given.
- We mark on the given line $MN$ arbitrary point and call it $B$.
- Through the point $B$ we draw a line perpendicular to the line $MN$ and call it $AB$.
- On the line $AB$ from the point $B$ we plot the segment $BC=a$.
- With the help of a square and a straightedge, draw the line $CD$ through the point $C$, which will be parallel to the given line $AB$.
If we postpone on the line $AB$ from the point $B$ the segment $BC=a$ to the other side, then we get one more line parallel to the given one, spaced from it by set distance$a$.
Other ways to draw parallel lines
Another way to build parallel lines is to build with a T-square. Most often this way used in drawing practice.
When performing carpentry work for marking and building parallel lines, a special drawing tool- Malka - two wooden planks that are fastened with a hinge.
Lessons on the KOMPAS program.
Lesson number 4. Auxiliary straight lines in Compass 3D.
When developing drawings on a drawing board, designers always use thin lines, their counterpart in Compass 3D are auxiliary straight lines. They are necessary for preliminary constructions and for setting projection relationships between views. Auxiliary straight lines when printing Auxiliary, it cannot be changed.
There are several ways to construct auxiliary lines. In this lesson, we will look at some of these methods.
1. Arbitrary straight line on two points.
In the main menu of the program, sequentially press the commands Tools-Geometry-Auxiliary lines-Auxiliary line.
Or in the compact panel, press the buttons Geometry-Auxiliary line.
Click the left mouse button to specify the first base point (for example, the origin). Now we specify 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 the angle through the properties panel. For example, enter an angle of 45º and press the key Enter.
To complete the build, click on the icon "Abort Command" in the properties panel. This command can be implemented through the context menu, which is called by clicking the right mouse button.
In a similar way through the base point, you can build any number of arbitrary 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: "Don't put intersection points"(active by default) and "Set intersection points". If you need to mark the intersection points of the line with other objects, activate the switch "Set intersection points", now the system will automatically put intersection points with all graphic objects in the current view.
Dot style will be- Auxiliary. To remove all auxiliary elements, use the commands of the main menu 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.
Commands are called to construct a horizontal line Tools-Geometry-Auxiliary lines-Horizontal line.
Or via the compact panel by pressing the buttons: Geometry-Horizontal line. The toolbar for constructing auxiliary lines is not visible on the screen. To see it, press the auxiliary lines button, which is active at the time of construction, and hold for a few seconds.
Now it is enough, by clicking the left mouse button, to specify the point through which the horizontal line will pass. You can build as many lines as you want at the same time. To complete the build, click the button "Abort Command" on the properties panel.
It must be remembered that the horizontal line is parallel to the x-axis of the current coordinate system. Horizontal, built 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 will figure it out yourself.
It must be remembered that the vertical line is parallel to the y-axis of the current coordinate system. Vertical, built 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 build a parallel line, we need an object parallel to which it will pass. Such objects can be: auxiliary 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 commands Tools-Geometry-Auxiliary lines-Parallel line.
In any design training course, they learn to use thin auxiliary lines when creating drawings. Previously, they were applied on a drawing board, and then wiped off from the finished document. Currently using electronic programs for the 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 the right 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.
1. Create a straight line using two points.
One of the most popular ways. To activate, you need to 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 setting the first point, then specifying the end point of the line. At the same time, the program itself will generate the desired angle of inclination for the created straight line. However, you can change the angle by entering your own values in the box below, then just press Enter.
The auxiliary line is formed, now you need to click on the familiar icon Abort command, located in the properties panel. However, you can activate this command by completing the work with the line by simply right-clicking the mouse, and then selecting the appropriate item in the drop-down menu.
Using base point you can create infinite number straight lines going at any angle. By the way, if you have coordinates or it is more convenient to work with a coordinate grid, then you can always set desired values in the menu below. You will place a straight line, without any adjustments on the sheet. It is worth paying attention to the group Modes, it has two important switches. The first one is active on standard startup - Do not place intersection points, and the second you can choose yourself - Set intersection points. Using this setting, you can automatically put points at any intersection, without additional options and manual setting.
However, here you need to specify the style Auxiliary. By the way, to remove all auxiliary elements from the finished drawing, it is enough to activate the item in the main menu Editor-Delete-Auxiliary curves and points. The work with points on curves was considered in detail in lesson number 3.
2. Draw a horizontal line
Construction lines can be drawn using horizontal lines. Let's open the already familiar menu Tools-Geometry-Auxiliary lines-Horizontal line.
A faster option, using the compact panel, select Geometry - Horizontal line. However, the base panel will not be visible on the screen, to correct the position, press the auxiliary lines button and hold it for a while.
It remains to specify by clicking the left button desired point, through which we pass our straight line. You can create any number of horizontal lines. To complete your work, just press Abort command in the properties panel or in the drop-down menu, by right-clicking.
You also need to remember that the 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 already on the sheet.
3. Draw a vertical straight line.
The general mechanism for calling the line drawing mechanism is absolutely identical to the one described above, with the exception of choosing vertical straight.
However, there are a few important things to keep in mind. The created vertical line is always parallel only to the current coordinate axis, here the case is identical with the horizontal straight line. Therefore, if you have a modified coordinate system, the vertical straight lines will not be parallel to the sheet.
4. Create a parallel straight line.
You can draw 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 binding, from straight and auxiliary lines to the faces of polygonal objects. So, let's within the framework of the lesson, for the main take the horizontal line that goes from the origin on our sheet.
Calling a parallel straight line is identical, open Tools - Geometry - Auxiliary lines - Parallel line.
Or use the compact panel, here you need to call Geometry-Parallel line.
Now let's specify the base object, to which we will draw parallel line. As agreed, a horizontal straight line acts as an object, select it with the mouse. Then, you need to set the distance at which our parallel line will be. Below you can specify numerical value, for example 30 mm, or drag the straight line with the mouse to the desired distance.
When setting the distance by numbers, the system will suggest two phantom lines at the same distance. This can be disabled if in the properties Number of straight lines - Two straight lines remove the activation by translating it into the creation of a single straight line. To fix the created line, just select the active phantom with the mouse and click the create object button. When you need to create both lines, click create object again, and then abort the command.
When you need to draw a new parallel line, but near another object, just click on the button Specify again. Now, you can specify a new object and build a line, in the way described in this chapter of the lesson.
That's all, in the lesson we revealed the basics of creating auxiliary straight lines.
The construction of a straight line parallel to a given plane is based on
following position known from geometry: a straight line is parallel to a plane,
if this line is parallel to any line in the plane.
Through a given point in space, you can draw innumerable
set of straight lines parallel to a given plane: To obtain
the only solution requires some additional condition.
For example, through a point (Fig. 180) it is required to draw a straight line,
parallel to the plane given triangle ABC, and projection planes!
(additional condition).
Obviously, the desired line must be parallel to the line of intersection
both planes, i.e. should be parallel to the horizontal track
plane, given by a triangle ABC. To determine the direction of this
trace, you can use the horizontal plane given by the triangle
ABC. On fig. 180 a horizontal line DC is drawn and then through point M a
straight line parallel to this horizontal line.
Let us pose the inverse problem: draw a plane through a given point,
parallel to a given straight line. planes passing through some
point A parallel to some straight line BC, form a bundle of planes, the axis
which is a line passing through the point A parallel to the line BC.
To obtain a unique solution, some additional
For example, it is necessary to draw a plane parallel to the straight line CD, not through
point, but through the straight line AB (Fig. 181). Straight AB and CD - crossing. If a
through one of the two intersecting lines it is required to draw a plane,
parallel-
Rice. 180 Fig. 181
another, then the problem has a unique solution. through point B
a straight line is drawn parallel to the straight line CD; straight lines AB and BE define
a plane parallel to the straight line CD.
How can you tell if a given line is parallel to a given plane?
You can try to draw a straight line parallel to this plane
this straight line. If such a line in the plane cannot be constructed, then
the given line and plane are not parallel to each other.
You can also try to find the point of intersection of the given line with the given
plane. If such a point cannot be found, then the given line and
plane are mutually parallel.
§ 28. Construction of mutually parallel planes
Let a point K be given through which a plane must be drawn,
parallel to some plane given by intersecting lines AF and BF
Obviously, if we draw lines SK and DK through the point K, respectively
parallel to the straight lines AF and BF, then the plane defined by the straight lines CK and DK,
will be parallel given plane.
Another example of construction is given in Fig. 183 right. Through point A
held pl. parallel to sq. a. First, a straight line is drawn through point A,
obviously parallel square. . This is a horizontal with projections "" and "",
where A"N"\\h" o. So
Rice. 182 Fig. 183
since the point N is the front trace of the horizontal AN, then through this
the trace f "o% f" o will pass through the point, and the trace h "o || h" o will pass through X. planes
and are mutually parallel, since their similarly intersecting traces are mutually
are parallel.
On fig. 184 shows two planes parallel to each other - one
one of them is given by the triangle LBC, the other by the parallel lines DE and FG.
What establishes the parallelism of these planes? The one in the plane
given by the straight lines DE and FG, it turned out to be possible to draw two intersecting
lines KN and KM, respectively, parallel to intersecting lines AC and
Sun another plane.
Of course, one could try to find the intersection point at least
line DE with plane of triangle ABC. Failure would confirm
plane parallelism.
QUESTIONS To §§ 27-28
1. What is the basis for the construction of a straight line, which should be
parallel to some plane?
2. How to draw a plane through a line parallel to a given line?
3. What determines the mutual parallelism of two planes?
4. How to draw a plane parallel to a given plane through a point?
5. How to check in the drawing whether the given ones are parallel to each other
