Image of a sphere and its sections. Big encyclopedia of oil and gas

Introduction

A ball is a body that consists of all points in space that are at a distance not greater than a given distance from a given point. This point is called the center of the ball, and this distance is called the radius of the ball.

The boundary of a sphere is called a spherical surface, or sphere. Thus, the points of the sphere are all points of the ball that are at a distance from the center equal to the radius. Any line segment that connects the center of a ball to a point on the ball surface, also called a radius.

The segment connecting two points of the spherical surface passing through the center of the ball is called the diameter. The ends of any diameter are called diametrically opposite points of the ball.

A ball, like a cylinder and a cone, is a body of revolution. It is obtained by rotating a semicircle around its diameter as an axis.

Section of a sphere by a plane

Any section of a sphere by a plane is a circle. The center of this circle is the base of the perpendicular dropped from the center of the ball to the cutting plane.

Proof: Let be a cutting plane and O - the center of the ball (Fig. 1) Let's drop the perpendicular from the center of the ball to the plane and denote the base of this perpendicular by O ".

Let X be an arbitrary point of the ball belonging to the plane. According to the Pythagorean theorem, OX2 \u003d OO "2 + O" X2. Since OX is not greater than the radius R of the ball, then O "X?, i.e. any point of the section of the ball by a plane is located from the point O" at a distance not greater, therefore, it belongs to a circle with center O "and radius. Conversely: any the point X of this circle belongs to the ball, which means that the section of the ball by the plane is a circle centered at the point O". The theorem has been proven.

The area passing through the center of the sphere is called the diametrical plane. The cross section of a ball with a diametral plane is called a great circle, and the cross section of a sphere is called a great circle.

The ball to the plane is equal to the radius of the plane, then the plane touches the ball at only one point, and the cross-sectional area will be zero, that is, if b \u003d R, then S \u003d 0. If b \u003d 0, then the cutting plane passes through the center of the ball. In this case, the section will be a circle, the radius of which coincides with the radius of the ball. The area of ​​this circle will be, according to the formula, equal to S = πR^2.

These two extreme cases give the boundaries between which the desired area will always lie: 0< S < πR^2. При этом любое сечение шара плоскостью всегда является кругом. Следовательно, задача сводится к тому, чтобы найти радиус окружности сечения. Тогда площадь этого сечения вычисляется по формуле площади круга.

Since the distance from a point to a plane is defined as the length of a segment perpendicular to the plane and starting at a point, the second end of this segment will coincide with the circle of the section. Such a conclusion follows from the definition of a ball: it is obvious that all points of the circumference of the section belong to the sphere, and therefore lie at an equal distance from the center of the ball. This means that the circles of the section can be considered the vertex of a right-angled triangle, the hypotenuse of which is the radius of the ball, one of which is the perpendicular segment connecting the center of the ball with the plane, and the second leg is the radius of the circle of the section.

Of the three sides of this triangle, two are given - the radius of the ball R and the distance b, that is, the hypotenuse. According to the Pythagorean theorem, the length of the second leg must be equal to √(R^2 - b^2). This is the radius of the section circle. Substituting the found value into the formula for the area of ​​a circle, it is easy to conclude that the cross-sectional area of ​​a ball by a plane is: S = π(R^2 - b^2). found results.

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• section of a sphere by a plane

All planets in the solar system are shaped ball. In addition, many objects created by man, including details of technical devices, have a spherical or close to such shape. The ball, like any body of revolution, has an axis that coincides with the diameter. However, this is not the only important property. ball. Below are the main properties of this geometric figure and the method for finding its area.

Instruction

If you take either a circle and rotate it around its axis, you get a body called a ball. In other words, a sphere is a body bounded by a sphere. The sphere is a shell ball, and its circumference. From ball it differs in that it is hollow. Axis like ball, and the sphere coincides with the diameter and passes through the center. Radius ball called a segment from its center to any external point. As opposed to a sphere, sections ball are circles. A shape close to spherical has the majority of celestial bodies. At different points ball there are identical in shape, but unequal in size, the so-called sections - circles of different areas.

The ball and sphere are interchangeable bodies, unlike the cone, despite being also a body of revolution. Spherical surfaces always form a circle in their section, regardless of how it is - horizontally or vertically. A conical surface is obtained only by rotating the triangle along its axis perpendicular to the base. Therefore, the cone, in contrast to ball, and is not considered an interchangeable body of revolution.

The largest possible circle is obtained by cutting ball passing through the center O. All circles that pass through the center O intersect with each other in the same diameter. The radius is always half the diameter. Through two points A and B located anywhere on the surface ball, can go through an infinite number of circles or circles. It is for this reason that through

Section of the surface of the ball

Any section of the surface of the ball by a plane is a circle, which is projected without distortion only if the cutting plane is parallel to the plane of projections. In the general case, we will get an ellipse. In the event that the cutting plane is perpendicular to the plane of projections, on this plane the projection of the circle is a straight line segment that is equal to the diameter of this circle.

Figure 109 shows the intersection of the surface of the ball with a horizontal projecting plane R. The section will be projected onto the horizontal plane as a projection segment R plane R, which is enclosed between the contour of the ball and is equal to the diameter of the section circle. On the frontal plane we get an ellipse. O 1 is the center of the circle, which is obtained in the section of the ball. It is located at the same height as the center of the ball O. Horizontal projection about 1 center O 1 circle is located in the middle of the segment ab. Perpendicular that is dropped from point o to a straight line ab, hits the point about 1 , which is a horizontal projection of the center of the section circle. frontal projection about 1 of the center of the circle is the center of the ellipse we are interested in.

If we consider the ellipse as a projection of some circle, then its major axis will always be the projection of the diameter of the circle, which is parallel to the projection plane, and the minor axis of the ellipse will be the projection of the diameter perpendicular to it. As a result, the major axis of the projection ellipse is always equal to the diameter of the projected circle. Here the diameter of the circle CD perpendicular to the plane H and is projected without distortion onto the frontal plane. To find the ends of the major axis of the ellipse, it is necessary to lay down and up from the center about 1 ellipse (perpendicular to a straight line oo 1) segments about 1 With and about 1 d, which are equal to half the diameter of the circumference of the section about 1 With = about 1 d = 1/2(ab). At the same time, the diameter AB circle is parallel to the horizontal plane, and its frontal projection ab′ is the minor axis of the ellipse under consideration.

Points separating the visible part of the ellipse from the invisible. Let's start by drawing the frontal plane Q, which bisects the ball. Plane Q will intersect the surface of the ball along a circle projected onto the frontal plane in the form of a contour. Then the part of the section line located on the front of the ball will be visible if you look at the ball from the front, and the rest of it will not be visible. Plane Q will cross the plane R frontal F one . Intersecting with the contour, its frontal projection F will determine the points 1 , which separate the visible part of the curve from the invisible part. Intermediate points 2′ of the ellipse can be found using the auxiliary frontal plane R intersecting the surface of the ball along a circle of radius r 2 and the plane R- along the front F 2.

A ball is a body consisting of all points in space that are at a distance not greater than a given distance from a given point. This point is called the center of the ball, and this distance is called the radius of the ball. The boundary of a sphere is called a spherical surface or sphere. The points of the sphere are all points of the ball that are at a distance equal to the radius from the center. Any segment that connects the center of the ball with a point on the spherical surface is also called a radius. The segment passing through the center of the ball, which connects two points of the spherical surface, is called the diameter. The ends of any diameter are called diametrically opposite points of the ball.

A ball is a body of revolution, just like a cone and a cylinder. A ball is obtained by rotating a semicircle around its diameter as an axis.

The surface area of ​​a sphere can be found using the formulas:

where r is the radius of the ball, d is the diameter of the ball.

The volume of a sphere is found by the formula:

V = 4 / 3 pr 3 ,

where r is the radius of the ball.

Theorem. Any section of a sphere by a plane is a circle. The center of this circle is the base of the perpendicular dropped from the center of the ball to the cutting plane.

Based on this theorem, if a ball with center O and radius R is intersected by a plane α, then a circle of radius r with center K is obtained in the section. The radius of the section of the ball by the plane can be found by the formula

It can be seen from the formula that planes equidistant from the center intersect the ball in equal circles. The radius of the section is the greater, the closer the secant plane is to the center of the ball, that is, the smaller the distance OK. The largest radius has a section with a plane passing through the center of the ball. The radius of this circle is equal to the radius of the ball.

The plane passing through the center of the ball is called the diametrical plane. The section of the ball by the diametral plane is called the great circle, and the section of the sphere is called the great circle, and the section of the sphere is called the great circle.

Theorem. Any diametral plane of a ball is its plane of symmetry. The center of the ball is its center of symmetry.

The plane that passes through the point A of the spherical surface and is perpendicular to the radius drawn to the point A is called the tangent plane. Point A is called the touch point.

Theorem. The tangent plane has only one common point with the ball - the point of contact.

A straight line that passes through point A of the spherical surface perpendicular to the radius drawn to this point is called a tangent.

Theorem. Through any point of the spherical surface there are infinitely many tangents, and all of them lie in the tangent plane of the ball.

A spherical segment is a part of a sphere cut off from it by a plane. Circle ABC is the base of the spherical segment. The segment MN of the perpendicular drawn from the center N of the circle ABC to the intersection with the spherical surface is the height of the spherical segment. Point M is the vertex of the spherical segment.

The surface area of ​​a spherical segment can be calculated using the formula:

The volume of a spherical segment can be found by the formula:

V \u003d πh 2 (R - 1/3h),

where R is the radius of the great circle, h is the height of the spherical segment.

A spherical sector is obtained from a spherical segment and a cone, as follows. If the spherical segment is less than a hemisphere, then the spherical segment is complemented by a cone whose vertex is in the center of the ball and whose base is the base of the segment. If the segment is larger than a hemisphere, then the indicated cone is removed from it.

A spherical sector is a part of a sphere bounded by a curved surface of a spherical segment (in our figure it is AMCB) and a conical surface (in the figure it is OABC), the base of which is the base of the segment (ABC), and the apex is the center of the ball O.

The volume of the spherical sector is found by the formula:

V = 2/3 πR 2 H.

A spherical layer is a part of a sphere enclosed between two parallel planes (planes ABC and DEF in the figure) intersecting a spherical surface. The curved surface of a spherical layer is called a spherical belt (zone). Circles ABC and DEF are the bases of the spherical belt. The distance NK between the bases of the spherical belt is its height.

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