Tangent plane. Tangent plane to a sphere

DEFINITION. Tangent plane to the surface at a point
is called a plane containing all the tangents to the curves drawn on the surface through this point. Normal is called a straight line perpendicular to the tangent plane and passing through the point of tangency.

Let's show that
directed normal to the surface
at the point
­.

Consider the curve , lying on the surface and passing through the point
(Fig. 15). Let it be given by parametric equations

.

If
– radius vector of a point
, moving when changing along , then, and
– radius vector of a point
.

Because lies on the surface, then. Let us differentiate this identity with respect to :

. (6.6)

A-priory
, A. Therefore (6.6) means that the scalar product
at all points of the curve .

The equality of the scalar product of vectors to zero is a necessary and sufficient condition for their perpendicularity. So, at the point

. But the vector
– velocity vector – directed tangentially to the trajectory of the point

, that is, tangent to the curve (Fig. 15). Because chosen randomly, then
perpendicular to all possible tangents drawn to lines lying on
and passing through the point
. And this by definition means that
perpendicular to the tangent plane, that is, it is its normal.

Hence the equation of the tangent plane to a given surface has the form (see Chapter 3):

Normal equation (see Chapter 3):

. (6.8)

In particular, if the surface is given by the explicit equation
, we get: – tangent equation

plane, and
– normal equation.

EXAMPLE. Write equations for the tangent plane and normal to the sphere
at the point
.

Obviously

Tangent plane equation (6.7):

Normal equations (6.8):

.

Note that this line passes through the origin, that is, the center of the sphere.

EXAMPLE. Write the equation of the tangent plane to an elliptic paraboloid
at the point
.

This surface is given by an explicit equation and
.

Therefore, the equation of the tangent plane at a given point has the form: or.

Extrema of a function of two variables

Let the function
defined at all points of a certain region
.

DEFINITION. Dot
called the maximum (minimum) point of the function
, if its neighborhood exists
, everywhere within which.

From the definition it follows that if
is the maximum point, then

;
If

is the minimum point, then THEOREM
(a necessary condition for the extremum of a differentiable function of two variables). Let the function
has at the point

extremum. If first order derivatives exist at this point, then PROOF
. Let's fix the value
. Then . It has an extremum at
and by necessary condition extremum of a differentiable function of one variable (see Chapter 5)
.

Similarly, fixing the value
, we get that
.

Q.E.D.

DEFINITION. Stationary point functions
called a point
, in which both first-order partial derivatives are equal to zero:

.

NOTE 1. The formulated necessary condition is not a sufficient condition for an extremum.

Let
. Means,
is the stationary point of this function. Consider an arbitrary - neighborhood of the origin.

Within this vicinity it obviously has different signs(Fig. 16). And this means that the point
by definition is not an extremum point.

Thus, not every stationary point is an extremum point.

NOTE 2. A continuous function can have an extremum but not have a stationary point.

Consider the function
. Its graph is the top
half a cone, and obviously
– minimum point (Fig. 17).

DEFINITION. Points at which the first order partial derivatives of the function
are equal to zero or do not exist are called its critical dots.

is the minimum point, then(sufficient condition for the extremum of the function
). Let the function
has second order partial derivatives in some neighborhood stationary points
. Let, in addition,

.

Then if

1)
, That
– extremum point, namely: maximum point, if
, or minimum point, if
;

2)
, then the extremum at the point
No;

3)
, then additional research is required to clarify the nature of the point
.

(No proof).

EXAMPLE. Examine the extremum function
.

Let's find stationary points:
. There are no stationary points, which means the function has no extremum.

EXAMPLE. Examine the extremum function.

To find stationary points, you need to solve the system of equations:

That is this function has four stationary points.

Let us check the sufficient condition for the extremum for each of them:

.

Because
, then at points
there is no extreme.

And
, Means,
– minimum point and
;
And
, Means,
– maximum point and
.

Lesson 10. Tangent plane to a sphere.

Purpose of the lesson: to consider theorems about the tangent plane to a sphere, to teach how to solve problems on this topic.

During the classes

Updating basic knowledge.

Repetition of information from planimetry.

Definition of tangent.

Property of the radius drawn to a tangent point.

If from one point lying outside the circle we draw two tangents to it, then:

a) the lengths of the segments from a given point to the points of contact are equal:

b) the angles between each tangent and secant passing through the center of the circle are equal.

If from one point lying outside the circle we draw a tangent and a secant to it, then the square of the tangent is equal to the product of the secant and its outer part.

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

The relative position of the sphere and the plane.

Explanation new topic. (Slide 26 – 32)

So, a sphere and a plane can intersect along a circle, not intersect, and have one common point.

Let's consider the last case in more detail.

A plane that has only one common point with a sphere is called a tangent plane to the sphere, and their common point called the touch point.

TO
a tangent plane has a property similar to that of a tangent to a circle.

Given: sphere with center ABOUT and radius R, α - tangent to the sphere at a point A plane.

Prove: O.A. A.

Proof: Let O.A. not perpendicular to the plane A, Then O.A. is inclined to the plane, which means the distance from the center to the plane d R. Those. the sphere must intersect the plane along the circle, but this does not satisfy the conditions of the theorem. Means, O.A. A.

Let's prove the converse theorem.

Given: sphere with center ABOUT and radius O.A., A, O.A. A.

Prove: A– tangent plane.

Proof: Because O.A. A, then the distance from the center of the sphere to the plane is equal to the radius. This means that the sphere and the plane have one common point. By definition, a plane is tangent to a sphere.

Formation of skills and abilities of students.

How far can he see? earth man standing on the plain? (Not taking into account the refraction of light).

Solution: CN 2 = h(h + 2 R) (see paragraph I of lesson above)

Let the height of a person (up to the eyes) 1.6 m, R land 6400 km.

We will return to this problem later to find out what the viewing area is.

Work according to table 33.

AK OK(Why?). According to the Pythagorean theorem AK = = 15 . A.M.- closest distance from the point A to the sphere (if you have time, you can let students think about the obvious question - why?)

A.M.= AO-OM=9.

Lesson summary.

Homework: paragraph 61, No. 591, 592.

A surface is defined as a set of points whose coordinates satisfy a certain type of equation:

If the function F (x , y , z) (\displaystyle F(x,\,y,\,z)) is continuous at some point and has continuous partial derivatives at it, at least one of which does not vanish, then in the neighborhood of this point the surface given by equation (1) will be the right surface.

In addition to the above implicit way of specifying, the surface can be defined obviously, if one of the variables, for example, z, can be expressed in terms of the others:

More strictly simple surface is called the image of a homeomorphic mapping (that is, a one-to-one and mutually continuous mapping) of the interior of a unit square. This definition can be given an analytical expression.

Let a square be given on a plane with a rectangular coordinate system u and v, the coordinates of the internal points of which satisfy the inequalities 0< u < 1, 0 < v < 1. Гомеоморфный образ квадрата в пространстве с прямоугольной системой координат х, у, z задаётся при помощи формул х = x(u, v), у = y(u, v), z = z(u, v) (параметрическое задание поверхности). При этом от функций x(u, v), y(u, v) и z(u, v) требуется, чтобы они были непрерывными и чтобы для различных точек (u, v) и (u", v") были различными соответствующие точки (x, у, z) и (x", у", z").

Example simple surface is a hemisphere. The entire sphere is not simple surface. This necessitates further generalization of the concept of surface.

A subset of space, each point of which has a neighborhood that is simple surface, called the right surface .

Surface in differential geometry

Helicoid

Catenoid

The metric does not uniquely determine the shape of the surface. For example, the metrics of a helicoid and a catenoid, parameterized accordingly, coincide, that is, there is a correspondence between their regions that preserves all lengths (isometry). Properties that are preserved under isometric transformations are called internal geometry surfaces. The internal geometry does not depend on the position of the surface in space and does not change when it is bent without tension or compression (for example, when a cylinder is bent into a cone).

Metric coefficients E , F , G (\displaystyle E,\ F,\ G) determine not only the lengths of all curves, but also in general the results of all measurements inside the surface (angles, areas, curvature, etc.). Therefore, everything that depends only on the metric refers to internal geometry.

Normal and normal section

Normal vectors at surface points

One of the main characteristics of a surface is its normal- unit vector perpendicular to the tangent plane at a given point:

[ r u ′ , r v ′ ] |

(\displaystyle \mathbf (m) =(\frac ([\mathbf (r"_(u)) ,\mathbf (r"_(v)) ])(|[\mathbf (r"_(u)) ,\mathbf (r"_(v)) ]|))) The sign of the normal depends on the choice of coordinates. A section of a surface by a plane containing the surface normal at a given point forms a certain curve called

normal section surfaces. The main normal for a normal section coincides with the normal to the surface (up to sign). If the curve on the surface is not a normal section, then its main normal forms a certain angle with the normal of the surface θ (\displaystyle \theta ). Then the curvature k (\displaystyle k) normal section (with the same tangent) by Meunier's formula:

Coordinates of the normal unit vector for different ways surface assignments are given in the table:

Here D (y , z) D (u , v) = |.

y u ′ y v ′ z u ′ z v ′ | , D (z , x) D (u , v) = |.

z u ′ z v ′ x u ′ x v ′ |

, D (x, y) D (u, v) = | x u ′ x v ′ y u ′ y v ′ |(\displaystyle (\frac (D(y,z))(D(u,v)))=(\begin(vmatrix)y"_(u)&y"_(v)\\z"_(u) &z"_(v)\end(vmatrix)),\quad (\frac (D(z,x))(D(u,v)))=(\begin(vmatrix)z"_(u)&z" _(v)\\x"_(u)&x"_(v)\end(vmatrix)),\quad (\frac (D(x,y))(D(u,v)))=(\ begin(vmatrix)x"_(u)&x"_(v)\\y"_(u)&y"_(v)\end(vmatrix))) All derivatives are taken at the point(x 0 , y 0 , z 0) (\displaystyle (x_(0),y_(0),z_(0)))

Curvature For different directions e 2 (\displaystyle e_(2)), in which the normal curvature takes minimum and maximum values; these directions are called main. The exception is the case when the normal curvature in all directions is the same (for example, near a sphere or at the end of an ellipsoid of revolution), then all directions at a point are principal.

Surfaces with negative (left), zero (center) and positive (right) curvature.

Normal curvatures in the principal directions are called main curvatures; let's designate them κ 1 (\displaystyle \kappa _(1)) different directions κ 2 (\displaystyle \kappa _(2)). Size:

called Gaussian curvature, total curvature, or simply surface curvature. There is also the term curvature scalar, which implies the result of convolution of the curvature tensor; in this case, the curvature scalar is twice as large as the Gaussian curvature.

Gaussian curvature can be calculated via a metric, and is therefore an object of the intrinsic geometry of surfaces (note that the principal curvatures do not belong to the intrinsic geometry). You can classify surface points based on the sign of curvature (see figure). The curvature of the plane is zero. The curvature of a sphere of radius R is equal everywhere 1 R 2 (\displaystyle (\frac (1)(R^(2)))). There is also a surface of constant negative curvature -

Date: 02/02/2016

Topic: Tangent to a sphere (ball) of a plane.

Purpose of the lesson: To develop students’ knowledge and skills on the topic, to consider theorems

o, teach how to solve problems on this topic.
Cultivate attentiveness, conscientious attitude to learning, accuracy

Develop memory, thinking, spatial imagination, speech

Lesson structure

Organizing time

Setting a lesson goal

Checking homework

Protecting presentations by students

Individual independent work

Solving problems in pairs

Solving problems in a group

Mindfulness game

Issuing homework

Lesson summary
During the classes

At the beginning of the lesson, oral work is carried out. Repetition of basic concepts related to ball and sphere.

Homework No. 26 (page 61), No. 34

Those on duty at the blackboard (during recess) complete drawings for homework. During class, the teacher calls two students to the board to check their homework. After answering at the board, students give themselves grades on evaluation sheets.

Presentation protection:

Group I: History of the ball

Group II: Mutual arrangement of sphere and plane

Group III: Ball and sphere in living nature

Independent work

1. Find the coordinates of the center and radius of the sphere given by the equation:

1 option

(x-2) 2 + (y+3) 2 + z 2 = 25

Option 2

(x+3) 2 + y 2 + ( z-1) 2 = 16

2. Write the equation of a sphere with radiusRwith the center of the circle at point A, if:

1 option

A (2; 0; -1), R = 7

Option 2

A (-2; 1; 0) , R = 6

3. Check whether point A lies on the sphere given by the equation:

1 option

(x + 2) 2 + (y – 1) 2 + ( z– 3) 2 = 1 if A (-2; 1; 4)

Option 2

(x - 3) 2 + (y + 1) 2 + ( z- 4) 2 = 4 if A (5; - 1; 4)

4. Prove that given equation is the equation of the sphere:

1 option

x 2 + y 2 + z 2 + 2 z- 2у= 2

Working in pairs

Option 2

x 2 + y 2 + z 2 – 2x + 2 z = 7

The radius of the sphere is 112 cm. A point lying on a plane tangent to the sphere is 15 cm away from the point of tangency. Find the distance from this point to the point of the sphere closest to it.

Group work

All sides of triangle ABC touch a sphere of radius 5 cm. Find the distance from the center of the sphere to the plane of the triangle if AB = 13 cm, BC = 14 cm, CA = 15 cm

Game of attention

The basic formulas for the surface areas of polyhedra and bodies of revolution are written on colored papers. These cards are attached to a magnetic board. The teacher asks you to look carefully at the formulas and remember them. Naturally, students begin to memorize the formulas themselves. Having closed the board, the teacher asks the following questions: “What color is the card on which the formula for the area of ​​the lateral surface of the pyramid is written?” etc. Naturally, the students did not expect such a question. The teacher gives another opportunity, but this time the students try to remember the color of the card.

Lesson summary.

Grading scale

"5" for 8-9 points

“4” - for 6-7 points

“3” - for 4-5 points

Homework: No. 28 (page 61), No. 29 (page 62)

The Tale of the Origin of the Ball

One day, left alone at home, the handsome Semicircle spent a long time dressing up and simulating in front of a small mirror in tin frames and could not stop admiring himself.

“Why do people want to proclaim that I’m good?” he said. “People lie, I’m not good at all.” Why did the girls proclaim that best guy and has never been and never will be in the village of Khatanga?

The semicircle knew and heard everything that was said about him, and was capricious, like a handsome man. He could admire himself in front of the mirror all day, examining himself from all sides. And suddenly a miracle happened when the Semicircle turned around in front of the mirror, he saw his own reflection in the mirror in the shape of a Ball.

From the history of occurrence

A ball is usually called a body limited by a sphere, that is, a ball and a sphere are different geometric bodies. However, both the words "ball" and "sphere" come from the same Greek word "sphaira" - ball. Moreover, the word “ball” was formed from the transition of consonants sf V w.

In Book XI of the Elements, Euclid defines a ball as a figure described by a semicircle rotating around a fixed diameter. In ancient times, the sphere was held in high esteem. Astronomical observations of the firmament invariably evoked the image of a sphere.

The sphere has always been widely used in various fields of science and technology.

Definition

• A sphere is a surface consisting of all points in space located at a given distance from a given point.
• A body bounded by a sphere is called a ball.

General concepts

• This point is called the center of the sphere, and this distance is called the radius of the sphere.
• A segment connecting two points of a sphere and passing through its center is called the diameter of the sphere.
• The center, radius, diameter of a sphere is also called the center, radius and diameter of a ball.

Tangent plane to a sphere

A plane that has only one common point with a sphere is called a tangent plane to the sphere, and their common point is called the point of tangency between the plane and the sphere.

Section of a sphere by a plane

• Any section of a ball by a plane is a circle. The center of this circle is the base of the perpendicular dropped from the center of the ball onto the cutting plane.
• The section passing through the center of the ball is a large circle. (diametrical section).

Problem on the theme ball (d/z)

There are three points on the surface of the ball. The straight-line distances between them are 6 cm, 8 cm, 10 cm. The radius of the ball is 13 cm. Find the distance from the center to the plane passing through these points. (1.7 cm, 2.15 cm, 3.12 cm, 4.20 cm)