How to find the correlation function of a random process example. Correlation function of a stationary process

We see circle shapes and circles everywhere: this is the wheel of a car, the horizon line, and the disk of the Moon. Mathematicians began to study geometric figures - a circle on a plane - a very long time ago.

A circle with a center and radius is a set of points on a plane located at a distance not greater than . A circle is bounded by a circle consisting of points located exactly at a distance from the center. The segments connecting the center with the points of the circle have a length and are also called radii (of a circle, circle). The parts of a circle into which it is divided by two radii are called circular sectors(Fig. 1). A chord - a segment connecting two points on a circle - divides the circle into two segments, and the circle into two arcs (Fig. 2). A perpendicular drawn from the center to the chord divides it and the arcs subtended by it in half. The chord is longer, the closer it is located to the center; the longest chords - the chords passing through the center - are called diameters (of a circle, circle).

If a straight line is removed from the center of the circle by a distance , then at it does not intersect with the circle, at it intersects the circle along a chord and is called a secant, at at it has a single common point with the circle and the circle and is called a tangent. A tangent is characterized by the fact that it is perpendicular to the radius drawn to the point of tangency. Two tangents can be drawn to a circle from a point outside it, and their segments from a given point to the points of tangency are equal.

Arcs of a circle, like angles, can be measured in degrees and fractions. Part of the entire circle is taken as a degree. The central angle (Fig. 3) is measured in the same number of degrees as the arc on which it rests; an inscribed angle is measured by half an arc. If the vertex of an angle lies inside the circle, then this angle in degrees is equal to half the sum of the arcs and (Fig. 4,a). An angle with a vertex outside the circle (Fig. 4,b), cutting out arcs and on the circle, is measured by the half-difference of arcs and. Finally, the angle between the tangent and the chord equal to half the arc of a circle enclosed between them (Fig. 4,c).

Circle and circumference have infinite set axes of symmetry.

From the theorems on the measurement of angles and the similarity of triangles follow two theorems on proportional segments in a circle. The chord theorem says that if a point lies inside a circle, then the product of the lengths of the segments of chords passing through it is constant. In Fig. 5,a. The theorem about secant and tangent (meaning the lengths of segments of parts of these lines) states that if a point lies outside the circle, then the product of the secant and its external part is also unchanged and equal to the square of the tangent (Fig. 5,b).

Even in ancient times, people tried to solve problems related to the circle - to measure the length of a circle or its arc, the area of a circle or sector, segment. The first of them has a purely “practical” solution: you can lay a thread along a circle, and then unroll it and apply it to a ruler, or mark a point on the circle and “roll” it along the ruler (you can, on the contrary, “roll” a circle with a ruler). One way or another, measurements showed that the ratio of the circumference to its diameter is the same for all circles. This relationship is usually denoted Greek letter(“pi” is the initial letter Greek word perimetron, which means “circle”).

However, the ancient Greek mathematicians were not satisfied with such an empirical, experimental approach to determining the circumference of a circle: a circle is a line, i.e., according to Euclid, “length without width,” and such threads do not exist. If we roll a circle along a ruler, then the question arises: why do we get the circumference and not some other value? In addition, this approach did not allow us to determine the area of the circle.

The solution was found as follows: if we consider regular -gons inscribed in a circle, then as , tending to infinity, in the limit they tend to . Therefore, it is natural to introduce the following, already strict, definitions: the length of a circle is the limit of the sequence of perimeters of regular triangles inscribed in a circle, and the area of a circle is the limit of the sequence of their areas. This approach is also accepted in modern mathematics, and in relation not only to the circle and circle, but also to other curved areas or areas limited by curvilinear contours: instead of regular polygons, sequences of broken lines with vertices on curves or contours of areas are considered, and the limit is taken when the length tends to the greatest links of the broken line to zero.

The length of the arc of a circle is determined in a similar way: the arc is divided into equal parts, the division points are connected by a broken line and the length of the arc is given equal to the limit the perimeters of such broken lines as , tending to infinity. (Like the ancient Greeks, we do not clarify the concept of limit itself - it no longer refers to geometry and was quite strictly introduced only in the 19th century.)

From the definition of the number itself, the formula for the circumference follows:

For the length of an arc, we can write a similar formula: since for two arcs and with a common central angle, considerations of similarity imply the proportion , and from it the proportion , after passing to the limit we obtain the independence (of the radius of the arc) of the ratio . This ratio is determined only by the central angle and is called the radian measure of this angle and all corresponding arcs with center at. This gives the formula for the arc length:

where is the radian measure of the arc.

The written formulas for and are just rewritten definitions or notations, but with their help we obtain formulas for the areas of a circle and a sector that are far from just notations:

To derive the first formula, it is enough to go to the limit in the formula for the area of a regular triangle inscribed in a circle:

By definition left side tends to the area of the circle, and the right one tends to the number

and , bases of its medians and , midpoints and line segments from the point of intersection of its heights to its vertices.

This circle, found in the 18th century. by the great scientist L. Euler (which is why it is often also called Euler’s circle), was rediscovered in the next century by a teacher at a provincial gymnasium in Germany. This teacher's name was Karl Feuerbach (he was the brother famous philosopher Ludwig Feuerbach). Additionally, K. Feuerbach found out that a circle of nine points has four more points that are closely related to the geometry of any given triangle. These are the points of contact with the four circles special type(Fig. 2). One of these circles is inscribed, the other three are excircles. They are inscribed in the corners of the triangle and touch externally its sides. The points of contact of these circles with a circle of nine points are called Feuerbach points. Thus, the circle of nine points is actually the circle of thirteen points.

This circle is very easy to construct if you know its two properties. Firstly, the center of the circle of nine points lies in the middle of the segment connecting the center of the circle circumscribed about the triangle with a point - its orthocenter (the point of intersection of its altitudes). Secondly, its radius for a given triangle is equal to half the radius of the circle circumscribed around it.

Circle is a flat closed line, all points of which are at the same distance from a certain point (point O), which is called the center of the circle.
(Circumference - geometric figure, consisting of all points located at a given distance from a given point.)

Circle is a part of the plane limited by a circle. Point O is also called the center of the circle.

The distance from a point on a circle to its center, as well as the segment connecting the center of the circle to its point, is called the radius circle/circle.
See how circle and circumference are used in our life, art, design.

Chord - Greek - a string that binds something together
Diameter - "measurement through"

ROUND SHAPE

Angles can occur in ever-increasing quantities and, accordingly, acquire an ever-increasing turn - until they completely disappear and the plane becomes a circle.It's very simple and at the same time very difficult case, which I would like to talk about in detail. It should be noted here that both simplicity and complexity are due to the absence of angles. The circle is simple because the pressure of its boundaries, in comparison with rectangular shapes, is leveled - the differences here are not so great. It is complex because the top imperceptibly flows into the left and right, and the left and right into the bottom.

V. Kandinsky

IN Ancient Greece the circle and circumference were considered the crown of perfection. Indeed, at each point the circle is arranged in the same way, which allows it to move on its own. This property of the circle made possible occurrence wheels, since the axle and wheel hub must be in contact at all times.

A lot is studied at school useful properties circles. One of the most beautiful theorems is the following: let us draw through given point the straight line intersecting given circle, then the product of the distances from this point to the intersection points of a circle with a straight line does not depend on exactly how the straight line was drawn. This theorem is about two thousand years old.

In Fig. Figure 2 shows two circles and a chain of circles, each of which touches these two circles and two neighbors in the chain. The Swiss geometer Jacob Steiner proved about 150 years ago next statement: if for some choice of the third circle the chain is closed, then it will be closed for any other choice of the third circle. It follows from this that if the chain is not closed once, then it will not be closed for any choice of the third circle. To the artist who painteddepicted chain, one would have to work hard to make it work, or turn to a mathematician to calculate the location of the first two circles, at which the chain is closed.

We mentioned the wheel first, but even before the wheel, people used round logs- rollers for transporting heavy loads.

Is it possible to use rollers of some other shape than round? Germanengineer Franz Relo discovered that rollers, the shape of which is shown in Fig. 1, have the same property. 3. This figure is obtained by drawing arcs of circles with centers at the vertices equilateral triangle connecting two other vertices. If we draw two parallel tangents to this figure, then the distance betweenthey will be equal to the length of the side of the original equilateral triangle, so such rollers are no worse than round ones. Later, other figures were invented that could serve as rollers.

Enz. "I explore the world. Mathematics", 2006

Each triangle has, and moreover, only one, nine point circle. Thisa circle passing through the following three triplets of points, the positions of which are determined for the triangle: the bases of its altitudes D1 D2 and D3, the bases of its medians D4, D5 and D6the midpoints of D7, D8 and D9 of straight segments from the point of intersection of its heights H to its vertices.

This circle, found in the 18th century. by the great scientist L. Euler (which is why it is often also called Euler’s circle), was rediscovered in the next century by a teacher at a provincial gymnasium in Germany. This teacher's name was Karl Feuerbach (he was the brother of the famous philosopher Ludwig Feuerbach).Additionally, K. Feuerbach found that a circle of nine points has four more points that are closely related to the geometry of any given triangle. These are the points of its contact with four circles of a special type. One of these circles is inscribed, the other three are excircles. They are inscribed in the corners of the triangle and externally touch its sides. The points of contact of these circles with the circle of nine points D10, D11, D12 and D13 are called Feuerbach points. Thus, the circle of nine points is actually the circle of thirteen points.

This circle is very easy to construct if you know its two properties. Firstly, the center of the circle of nine points lies in the middle of the segment connecting the center of the circumscribed circle of the triangle with point H - its orthocenter (the point of intersection of its altitudes). Secondly, its radius for a given triangle is equal to half the radius of the circle circumscribed around it.

Enz. reference book for young mathematicians, 1989

Circle- a geometric figure consisting of all points of the plane located at a given distance from a given point.

This point (O) is called center of the circle.
Circle radius- this is a segment connecting the center with any point on the circle. All radii have the same length (by definition).
Chord- a segment connecting two points on a circle. A chord passing through the center of a circle is called diameter. The center of a circle is the midpoint of any diameter.
Any two points on a circle divide it into two parts. Each of these parts is called arc of a circle. The arc is called semicircle, if the segment connecting its ends is a diameter.
The length of a unit semicircle is denoted by π .
The sum of the degree measures of two arcs of a circle with common ends is equal to 360º.
The part of the plane bounded by a circle is called all around.
Circular sector- a part of a circle bounded by an arc and two radii connecting the ends of the arc to the center of the circle. The arc that limits the sector is called arc of the sector.
Two circles having general center, are called concentric.
Two circles intersecting at right angles are called orthogonal.

The relative position of a straight line and a circle

If the distance from the center of the circle to the straight line less than radius circle ( d), then a straight line and a circle have two common points. In this case the line is called secant in relation to the circle.
If the distance from the center of the circle to the straight line is equal to the radius of the circle, then the straight line and the circle have only one common point. This line is called tangent to the circle, and their common point is called point of tangency between a line and a circle.
If the distance from the center of the circle to the straight line greater than radius circles, then a straight line and a circle have no common points

Central and inscribed angles

Central angle is an angle with its vertex at the center of the circle.
Inscribed angle- an angle whose vertex lies on a circle and whose sides intersect the circle.

Inscribed angle theorem

An inscribed angle is measured by the half of the arc on which it subtends.

Corollary 1.
Inscribed angles subtending the same arc are equal.

Corollary 2.
An inscribed angle subtended by a semicircle is a right angle.

Theorem on the product of segments of intersecting chords.

If two chords of a circle intersect, then the product of the segments of one chord is equal to the product of the segments of the other chord.

Basic formulas

Circumference:

C = 2∙π∙R

Circular arc length:

R = С/(2∙π) = D/2

Diameter:

D = C/π = 2∙R

Circular arc length:

l = (π∙R) / 180∙α,
Where α - degree measure of the length of a circular arc)

Circle area:

S = π∙R 2

Area of the circular sector:

S = ((π∙R 2) / 360)∙α

Equation of a circle

IN rectangular system coordinate equation of a circle radius r centered at a point C(x o;y o) has the form:

(x - x o) 2 + (y - y o) 2 = r 2

The equation of a circle of radius r with center at the origin has the form:

x 2 + y 2 = r 2

First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on the plane located on equal distance from a single central point. But, if the circle consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that limits it (circle(r)), and an innumerable number of points that are inside the circle.

For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).

A segment that connects two points on a circle is its chord.

A chord passing directly through the center of a circle is diameter this circle (D). The diameter can be calculated using the formula: D=2R

Circumference calculated by the formula: C=2\pi R

Area of a circle: S=\pi R^(2)

Arc of a circle is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.

Central angle An angle that lies between two radii is called.

Arc length can be found using the formula:

Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
Using radian measure: CD = \alpha R

The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it in half.

If chords AB and CD of a circle intersect at point N, then the products of segments of chords separated by point N are equal to each other.

AN\cdot NB = CN\cdot ND

Tangent to a circle

Tangent to a circle It is customary to call a straight line that has one common point with a circle.

If a line has two common points, it is called secant.

If you draw the radius to the tangent point, it will be perpendicular to the tangent to the circle.

Let's draw two tangents from this point to our circle. It turns out that the tangent segments will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.

AC = CB

Now let’s draw a tangent and a secant to the circle from our point. We find that the square of the length of the tangent segment will be equal to the product the entire segment secant to its outer part.

AC^(2) = CD \cdot BC

We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external part.

AC\cdot BC = EC\cdot DC

Angles in a circle

Degree measures central angle and the arc on which it rests are equal.

\angle COD = \cup CD = \alpha ^(\circ)

Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.

You can calculate it by knowing the size of the arc, since it is equal to half of this arc.

\angle AOB = 2 \angle ADB

Based on a diameter, inscribed angle, right angle.

\angle CBD = \angle CED = \angle CAD = 90^ (\circ)

Inscribed angles that subtend the same arc are identical.

Inscribed angles resting on one chord are identical or their sum is equal to 180^ (\circ) .

\angle ADB + \angle AKB = 180^ (\circ)

\angle ADB = \angle AEB = \angle AFB

On the same circle are the vertices of triangles with identical angles and a given base.

An angle with a vertex inside a circle and located between two chords is identical to half the sum angular values arcs of a circle that are contained within a given and vertical angle.

\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)

An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular values of the arcs of the circle that are contained inside the angle.

\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)

Inscribed circle

Inscribed circle is a circle tangent to the sides of a polygon.

At the point where the bisectors of the corners of a polygon intersect, its center is located.

A circle may not be inscribed in every polygon.

The area of a polygon with an inscribed circle is found by the formula:

S = pr,

p is the semi-perimeter of the polygon,

r is the radius of the inscribed circle.

It follows that the radius of the inscribed circle is equal to:

r = \frac(S)(p)

Sums of lengths opposite sides will be identical if the circle is inscribed in a convex quadrilateral. And vice versa: a circle fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.

AB + DC = AD + BC

It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors intersect internal corners figure, the center of this inscribed circle will lie.

The radius of the inscribed circle is calculated by the formula:

r = \frac(S)(p) ,

where p = \frac(a + b + c)(2)

Circumcircle

If a circle passes through each vertex of a polygon, then such a circle is usually called described about a polygon.

At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumcircle.

The radius can be found by calculating it as the radius of the circle that is circumscribed about the triangle defined by any 3 vertices of the polygon.

Eat next condition: a circle can be described around a quadrilateral only if its sum opposite corners is equal to 180^( \circ) .

\angle A + \angle C = \angle B + \angle D = 180^ (\circ)

Around any triangle you can describe a circle, and only one. The center of such a circle will be located at the point where they intersect perpendicular bisectors sides of the triangle.