Point, line, straight line, ray, segment, broken line. Constructing parallel lines How to draw a line parallel to a given one

Good day, dear readers of my blog. It would seem, what does it cost to draw a straight line in Photoshop? Hold down Shift and there you go. Nevertheless, this can be done in as many as three ways. Everyone's result will be different.

In this article you will learn three ways to draw a straight line in Photoshop. Which filter to use to create a wave. How to do this using another interesting tool. I'll show you how to achieve a dotted line and draw at a certain angle.

A lot of information awaits you. Shall we get started?

Line Tool

First, I'll show you how to use a tool that is designed to create straight lines. In this place you can have a rectangle, oval, ellipse or polygon. Just hold down the left mouse button for a few seconds to open a menu with additional tools.

First things first. One of the most important parameters is thickness. Thanks to the line, you can even draw rectangles. You just need to make it fatter.

Next comes “Fill” and “Stroke”. Click on the color block to the left of the inscriptions and select a shade. If you want to make a stroke, enter its width. Now, my screenshot shows the option without it. The missing color icon looks like this. Gray line crossed out in red.

You can see the settings and the result in this screenshot. It's not very visible, but the thickness here is 30 pixels. In a large picture, 30 pixels may look like a modest stripe. Everything needs to be adjusted to your own dimensions.

This is what the line will look like if you select red for the stroke color.

The next button will allow you to create a dotted stroke.

If you reduce the thickness and remove the fill, you will just get a dotted line.

Here you can align the stroke to the inner edge, outer edge, or center of your outline.

And round the corners. True, it will not be so noticeable.

If you press Shift while you're drawing a line, Photoshop will automatically create a straight line. Horizontal or vertical. Depending on where you are taking her.

If you need a line at a certain angle, then the easiest way is to look at what the information window shows and adjust it manually, pointing it in a certain direction.

Well, now I’ll show you another one.

Brush Tool

I drew these rectangles using lines drawn with a brush.

Choose the type and size that suits your brush line.

Place a dot at the expected beginning of the line, hold Shift and left-click where the strip should end.

There are two lines in front of you. The yellow one was painted using the Line tool, and the purple one was painted with a brush.

How to make a wave

No matter what tool you use, the easiest way to create a wavy line is to use a filter. Go to this category, find “Distortion” and select “Wave”.

Based on the preview picture, you will quickly understand what's what and how to set it up. The amplitude should be approximately the same. If it doesn’t work, you can just click on “Randomize” until a suitable one appears.

The last applied filter is always quickly accessible. I apply it to the layer with the yellow stripe drawn with the tool.

This is the result I got. As you can see, it is different.

Pen tool

To be honest, I still can’t use a pen professionally. I know that you can draw anything with it: smoothly, quickly, fun and cool, but it takes me a lot of time and the result is not always at the level I expected. And yet I can even draw straight lines with a pen. It's worse with curves, but I'll try. I choose “Feather”.

I put a dot, then a second one. While I haven't released the mouse button, I adjust the smoothness.

I do the same thing with each new point.

After all the manipulations are completed, right-click and select “Stroke outline” from the menu that appears.

You can choose several tools: pencil, brush, stamp, pattern, and so on. Now let this one be a brush.

I press the right mouse button again and select “Delete outline”.

This is the result I got.

Well, don’t forget that you can always use your collage making skills. Read the article about how to take a line from any picture and insert it into your image.

If you want to learn how to professionally use the pen and other tools found in Photoshop. I can offer you a course " Photoshop for beginners in video format ».

Lessons created by professionals will teach you everything you need to know about this program. You will save a lot of time searching for answers to this or that question. Ideas will spontaneously appear in your head on how to complete the task.

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Given a circle with center ABOUT and period A outside the circle. A) The diameter of the circle is drawn. Using only a ruler*, lower the perpendicular from point A to this diameter. b) Through the point A a straight line is drawn that has no common points with the circle. Using only a ruler, lower the perpendicular from point ABOUT to this straight line.

*Note. In construction tasks, a “ruler” always means not a measuring tool, but a geometric one - with its help you can only draw straight lines (through two existing points), but not measure the distance between points. In addition, a geometric ruler is considered one-sided - it cannot be used to draw a parallel line by simply applying one side of the ruler to two points and drawing a line along the other side.

Hint 1

Use the ends of the diameter rather than the center of the circle.

Hint 2

An angle with a vertex on a circle based on its diameter is a right angle. Knowing this, you can construct two altitudes in a triangle formed by the ends of the diameter and the point A.

Hint 3

Try to solve first a simpler case than the one given in paragraph b), - when a given line intersects a circle.

Solution

A) Let Sun- given diameter (Fig. 1). To solve the problem, just remember the first two tips: if you draw straight lines AB And AC, and then connect the points of their intersection with the circle with the desired vertices of the triangle ABC, then you get two heights of this triangle. And since the altitudes of the triangle intersect at one point, then the straight line CH will be the third height, that is, the desired perpendicular from A to diameter Sun.

b) The solution to this point, however, even in the case given in the third hint, does not seem simpler: yes, we can draw the diameters, connect their ends and get a rectangle ABCD(Fig. 2, in which, for simplicity, the point A marked on the circle), but how does this bring us closer to constructing a perpendicular from the center of the circle?

Here's how: since the triangle AOB isosceles, then perpendicular (height) OK will go through the middle K sides AB. This means that the task has been reduced to finding the middle of this side. Surprisingly, we no longer need a circle at all, and period D also, in general, “superfluous.” And here is the segment CD- not superfluous, but on it we will need not some specific point, but a completely arbitrary point E! If we designate as L intersection point BE And A.C.(Fig. 3) and then extend A.E. until the intersection with the continuation B.C. at the point M, then straight L.M.- this is the solution to all our worries and problems!

Is it true, is very similar, What L.M. crosses AB in the middle? This is true. Try to prove it. We will postpone the proof until the end of the problem.

So, we have learned to find the midpoint of a segment AB, which means we have learned to lower the perpendicular to AB from the center of the circle. But what to do with the original problem in which the given line does not intersect the circle, as in Fig. 4?

Let's try to reduce the problem to something already solved. This can be done, for example, like this.

First, we construct a straight line symmetrical to the given one relative to the center of the circle. The construction is clear from Fig. 5, on which this straight line is horizontal under the circle, and the one constructed symmetrical to it is highlighted in red (the two blue points can be taken on the circle completely arbitrarily). At the same time we’ll take you through the center ABOUT another straight line perpendicular to one of the sides of the resulting rectangle in a circle in order to obtain on this straight line two segments of equal length.

Having two parallel lines, on one of which two ends and the middle of the segment are already marked, let’s take an arbitrary point T(for example, on a circle) and construct such a point S, which is straight T.S. will be parallel to the existing two straight lines. This construction is shown in Fig. 6.

Thus, we have obtained a chord of the circle parallel to the given line, that is, we have reduced the problem to the previously solved version, because we already know how to draw a perpendicular to such a chord from the center of the circle.

It remains to provide a proof of the fact that we used above.

Quadrangle ABCE in Fig. 3 - trapezoid, L is the point of intersection of its diagonals, and M- the point of intersection of the extensions of its sides. According to the well-known property of a trapezoid (it is also called remarkable property of the trapezoid; you can see how it is proven) direct M.L. passes through the middle of the bases of the trapezoid.

Actually, once again we actually relied on the same theorem already in the last subtask, when we drew the third parallel line.

Afterword

The theory of geometric constructions using a single ruler, when an auxiliary circle with a center is given, was developed by the remarkable German geometer of the 19th century Jacob Steiner (it is more correct to pronounce his surname Steiner as “Steiner”, but in Russian literature the spelling with two “e” has long been established). We have already talked about his mathematical achievements once in the problem “In short, Sklifosovsky”. In the book “Geometric Constructions Performed with a Straight Line and a Fixed Circle,” Steiner proved the theorem according to which any construction that can be performed with a compass and ruler can be performed without a compass if only one circle is given and its center is marked. . Steiner's proof boils down to demonstrating the possibility of carrying out basic constructions usually performed using a compass - in particular, drawing parallel and perpendicular lines. Our task, as is easy to see, is a special case of this demonstration.

However, Steiner’s solution to some problems was not the only one. We will also present the second method.

Take two arbitrary points on this line A And B(Fig. 7). First we construct a perpendicular from A to the (blue) straight line B.O.- this is actually the solution to our first problem, because this straight line contains the diameter of the circle; all corresponding constructions in Fig. 7 are in blue. Then we construct a perpendicular from B to the (green) straight line A.O.- this is exactly the same solution to exactly the same problem, the constructions are made in green. Thus we got two heights of the triangle AOB. The third altitude of this triangle passes through the center O and the point of intersection of the other two heights. It is the desired perpendicular to the line AB.

But that's not all. Despite the (relative) simplicity of the second method, it is “excessively long”. This means that there is another construction method that requires fewer operations (in construction problems, each line drawn with a compass or ruler is counted as one operation). Constructions that require the minimum number of operations among the known ones were called by the French mathematician Emile Lemoine (1840–1912) geometric(see: Geometrography).

So, we bring to your attention a geometric solution to the point b). It only requires 10 steps, with the first six being “natural” and the next three being “amazing”. The very last step, drawing a perpendicular, should perhaps also be called natural.

We want to draw a red dotted perpendicular (Fig. 8), for this we need to find some point on it other than ABOUT. Go.

1) Let A is an arbitrary point on a line, and C- an arbitrary point on a circle. We carry out a direct A.C..

2)–3) We draw the diameter O.C.(secondarily intersecting the circle at the point D) and straight line AD. Mark the second points of intersection of the lines A.C. And AD with a circle - B And E, respectively.

4)–6) We carry out BE, BD And C.E.. Direct CD And BE crossed at a point H, A BD And C.E.- at the point G(Fig. 9).

By the way, could it happen that BE would be parallel CD? Yes, definitely. In case the diameter CD perpendicular A.O., then this is exactly what happens: BE And CD are parallel and the points A, O And G lie on the same straight line. But the opportunity to take the point C arbitrarily assumes our ability to choose it so that CO And A.O. were not perpendicular!

And now the promised amazing construction steps:

7) Conduct G.H. until it intersects a given line at a point I.
8) Conduct C.I. until it intersects the circle at the point J.
9) Conduct B.J., which intersects with G.H.... Where? That's right, at the red point, which is located on the vertical diameter of the circle (Fig. 10).

10) Draw the vertical diameter.

Instead of step 8, you could draw a straight line D.I., and then in step 9 connect the second point of its intersection with the circle with the point E. The result would be the same red dot. Isn't this surprising? Moreover, it is not even clear what is more surprising - the fact that the red dot turns out to be the same for the two construction methods, or the fact that it lies on the desired perpendicular. However, geometry is not the “art of fact”, but the “art of proof”. So try to prove it.

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:

1. Apply the square to the straight line $a$ diagonally (see figure), and attach a ruler to its larger leg.
2. Move the square along the ruler until the given point $M$ is on the diagonal of the square.
3. 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 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 straight line $MN$ and a distance $a$ be given.

1. Let us mark an arbitrary point on the given line $MN$ and call it $B$.
2. Through the point $B$ we draw a line perpendicular to the line $MN$ and call it $AB$.
3. On the line $AB$ from the point $B$ we plot the segment $BC=a$.
4. 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 obtain another parallel line to the given one, spaced from it at a given distance $a$.

Other ways to construct parallel lines

Another way to construct parallel lines is to construct using a crossbar. Most often this method is 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.

A point is an abstract object that has no measuring characteristics: no height, no length, no radius. Within the scope of the task, only its location is important

The point is indicated by a number or a capital (capital) Latin letter. Several dots - with different numbers or different letters so that they can be distinguished

point A, point B, point C

point 1, point 2, point 3

You can draw three dots “A” on a piece of paper and invite the child to draw a line through the two dots “A”. But how to understand through which ones? A A A

A line is a set of points. Only the length is measured. It has no width or thickness

Indicated by lowercase (small) Latin letters

line a, line b, line c

The line may be

1. closed if its beginning and end are at the same point,
2. open if its beginning and end are not connected

closed lines

open lines

1. self-intersecting
2. without self-intersections

self-intersecting lines

lines without self-intersections

1. straight
2. broken
3. crooked

straight lines

broken lines

curved lines

A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions

Even when a small section of a straight line is visible, it is assumed that it continues indefinitely in both directions

Indicated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters - points lying on a straight line

straight line a

straight line AB

Direct may be

1. intersecting if they have a common point. Two lines can intersect only at one point.
   • perpendicular if they intersect at right angles (90°).
2. Parallel, if they do not intersect, do not have a common point.

parallel lines

intersecting lines

perpendicular lines

A ray is a part of a straight line that has a beginning but no end; it can be continued indefinitely in only one direction

The ray of light in the picture has its starting point as the sun.

Sun

A point divides a straight line into two parts - two rays A A

The beam is designated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters, where the first is the point from which the ray begins, and the second is the point lying on the ray

ray a

beam AB

The rays coincide if

1. located on the same straight line
2. start at one point
3. directed in one direction

rays AB and AC coincide

rays CB and CA coincide

A segment is a part of a line that is limited by two points, that is, it has both a beginning and an end, which means its length can be measured. The length of a segment is the distance between its starting and ending points

Through one point you can draw any number of lines, including straight lines

Through two points - an unlimited number of curves, but only one straight line

curved lines passing through two points

straight line AB

A piece was “cut off” from the straight line and a segment remained. From the example above you can see that its length is the shortest distance between two points. ✂ B A ✂

A segment is denoted by two capital (capital) Latin letters, where the first is the point at which the segment begins, and the second is the point at which the segment ends

segment AB

Problem: where is the line, ray, segment, curve?

A broken line is a line consisting of consecutively connected segments not at an angle of 180°

A long segment was “broken” into several short ones

The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line.

The vertices of a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, and the point at which the broken line ends.

A broken line is designated by listing all its vertices.

broken line ABCDE

vertex of polyline A, vertex of polyline B, vertex of polyline C, vertex of polyline D, vertex of polyline E

broken link AB, broken link BC, broken link CD, broken link DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

The length of a broken line is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305

Task: which broken line is longer, A which has more vertices? The first line has all the links of the same length, namely 13 cm. The second line has all links of the same length, namely 49 cm. The third line has all links of the same length, namely 41 cm.

A polygon is a closed polyline

The sides of the polygon (the expressions will help you remember: “go in all four directions”, “run towards the house”, “which side of the table will you sit on?”) are the links of a broken line. Adjacent sides of a polygon are adjacent links of a broken line.

The vertices of a polygon are the vertices of a broken line. Adjacent vertices are the endpoints of one side of the polygon.

A polygon is denoted by listing all its vertices.

closed polyline without self-intersection, ABCDEF

polygon ABCDEF

polygon vertex A, polygon vertex B, polygon vertex C, polygon vertex D, polygon vertex E, polygon vertex F

vertex A and vertex B are adjacent

vertex B and vertex C are adjacent

vertex C and vertex D are adjacent

vertex D and vertex E are adjacent

vertex E and vertex F are adjacent

vertex F and vertex A are adjacent

polygon side AB, polygon side BC, polygon side CD, polygon side DE, polygon side EF

side AB and side BC are adjacent

side BC and side CD are adjacent

CD side and DE side are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

The perimeter of a polygon is the length of the broken line: P = AB + BC + CD + DE + EF + FA = 120 + 60 + 58 + 122 + 98 + 141 = 599

A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, etc.