Equation of the normal to the sphere. Theoretical material. Normal and normal section

Namely, what you see in the title. Essentially, it is a "spatial analogue" the problem of finding the tangent and normals to the graph of a function of one variable, and therefore no difficulties should arise.

Let's start with some basic questions: WHAT IS a tangent plane and WHAT IS a normal? Many are aware of these concepts at the level of intuition. The simplest model that comes to mind is a ball with a thin flat piece of cardboard resting on it. The cardboard is located as close to the sphere as possible and touches it at a single point. In addition, at the point of contact, it is fixed by a needle sticking straight up.

In theory, there is a rather ingenious definition of a tangent plane. Imagine an arbitrary surface and the point belonging to it. Obviously, a lot of spatial lines that belong to this surface. Who has what associations? =) ... I personally introduced the octopus. Suppose that each such line has spatial tangent at the point.

Definition 1: tangent plane to the surface at a point is plane containing tangents to all curves that belong to this surface and pass through the point.

Definition 2: normal to the surface at a point is straight passing through this point perpendicular to the tangent plane.

Simple and elegant. By the way, so that you do not die of boredom from the simplicity of the material, a little later I will share with you one elegant secret that allows you to forget ONCE AND FOREVER about cramming various definitions.

We will get acquainted with the working formulas and the solution algorithm directly on a specific example. In the overwhelming majority of problems, it is required to compose both the equation of the tangent plane and the equations of the normal:

Example 1

Solution: if the surface is given by the equation (i.e. implicitly), then the equation of the tangent plane to a given surface at a point can be found by the following formula:

I pay special attention to unusual partial derivatives - their not to be confused with partial derivatives of an implicitly defined function (although the surface is implicitly specified)... When finding these derivatives, you need to be guided by the differentiation rules for a function of three variables, that is, when differentiating with respect to any variable, the other two letters are considered constants:

Without leaving the checkout, we find the partial derivative at the point:

Similarly:

This was the most unpleasant moment of the decision, in which a mistake, if not allowed, then constantly appears. Nevertheless, there is an effective verification technique here, which I talked about in the lesson Directional derivative and gradient.

All the "ingredients" have been found, and now it's up to a neat substitution with further simplifications:

– general equation the desired tangent plane.

I strongly recommend that you check this stage of the solution as well. First, you need to make sure that the coordinates of the touch point really satisfy the found equation:

- true equality.

Now we "remove" the coefficients of the general equation of the plane and check them for coincidence or proportionality with the corresponding values. In this case, they are proportional. Do you remember from analytical geometry course, - this is normal vector tangent plane, and he is - direction vector normal straight line. Let's compose canonical equations normals by point and direction vector:

In principle, the denominators can be reduced by "two", but there is no special need for this

Answer:

It is not forbidden to designate the equations with some letters, however, again - why? Here, and so it is extremely clear what's what.

The next two examples are for independent decision... A little "mathematical tongue twister":

Example 2

Find the equations of the tangent plane and the normal to the surface at a point.

And a task that is interesting from a technical point of view:

Example 3

Write the equations of the tangent plane and the normal to the surface at a point

At the point.

There is every chance not only to get confused, but also to face difficulties when recording canonical equations of the line... And the equations of the normal, as you probably understood, are usually written in this form. Although, due to forgetfulness or ignorance of some of the nuances, the parametric form is more than acceptable.

Sample examples of finishing solutions at the end of the lesson.

Does a tangent plane exist at any point on the surface? In general, of course not. A classic example is tapered surface and point - the tangents at this point directly form a conical surface, and, of course, do not lie in the same plane. It is easy to be convinced of the troubles analytically:.

Another source of problems is the fact non-existence any partial derivative at a point. However, this does not mean that at a given point there is no single tangent plane.

But it was, more likely, popular science than practically significant information, and we return to our daily matters:

How to write the equations of the tangent plane and the normal at a point,
if the surface is given by an explicit function?

Let's rewrite it implicitly:

And according to the same principles, we will find the partial derivatives:

Thus, the formula for the tangent plane is transformed into the following equation:

And accordingly, the canonical normal equations:

As you might guess, - these are already "real" partial derivatives of a function of two variables at the point that we used to designate with the letter "z" and found 100,500 times.

Note that in this article it is enough to remember the very first formula, from which, if necessary, it is easy to derive everything else. (understandably, having baseline preparation)... This is the approach that should be used in the study of the exact sciences, i.e. from a minimum of information, one should strive to “pull out” a maximum of conclusions and consequences. "Soobrazhalovka" and already existing knowledge to help! This principle is also useful because with very likely will save you in a critical situation when you know very little.

Let's work out the "modified" formulas with a couple of examples:

Example 4

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

A small overlay here turned out with designations - now the letter denotes a point on the plane, but what to do - such a popular letter….

Solution: the equation of the required tangent plane is compiled by the formula:

Let's calculate the value of the function at the point:

Let's calculate 1st order partial derivatives at this point:

Thus:

carefully, not in a hurry:

We write down the canonical equations of the normal at the point:

Answer:

And a final example for a do-it-yourself solution:

Example 5

Write the equations for the tangent plane and the normal to the surface at a point.

The final one - because in fact I have explained all the technical points and there is nothing special to add. Even the functions themselves, offered in this task, are dull and monotonous - it is almost guaranteed in practice that you will come across a "polynomial", and in this sense Example # 2 with an exponent looks like a "black sheep". By the way, it is much more likely to meet the surface given by the equation, and this is another reason why the function was included in the article "second number".

And finally, the promised secret: how can you avoid cramming definitions? (I, of course, do not mean a situation where a student is frantically cramming something before the exam)

The definition of any concept / phenomenon / object, first of all, gives an answer to the following question: WHAT IS IT? (who / such / such / such). Consciously in answering this question, you should try to reflect essential signs, unequivocally identifying this or that concept / phenomenon / object. Yes, at first this turns out to be somewhat tongue-tied, inaccurate and redundant (the teacher will correct =)), but over time, a completely worthy scientific speech develops.

Practice on the most abstract objects, for example, answer the question: who is Cheburashka? It's not that simple ;-) Is this a “fairytale character with big ears, eyes and brown hair”? Far and very far from the definition - you never know there are characters with such characteristics .... But this is already much closer to the definition: "Cheburashka is a character invented by the writer Eduard Uspensky in 1966, who ... (enumeration of the main distinguishing features)"... Pay attention to how well it started

A surface is defined as a set of points whose coordinates satisfy a certain form of equations:

F (x, y, z) = 0 (1) (\ displaystyle F (x, \, y, \, z) = 0 \ qquad (1))

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 vicinity of this point the surface given by Eq. (1) will be correct surface.

Besides the above implicit way of setting, the surface can be defined clearly if one of the variables, for example, z, can be expressed in terms of the others:

z = f (x, y) (1 ′) (\ displaystyle z = f (x, y) \ qquad (1 "))

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

Let a square be given on a plane with a rectangular coordinate system u and v, the coordinates of the interior 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").

An example simple surface is a hemisphere. The whole sphere is not simple surface... This necessitates further generalization of the concept of a surface.

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

Surface in differential geometry

Helicoid

Catenoid

The metric does not uniquely define the shape of the surface. For example, the metrics of the helicoid and the catenoid, parameterized accordingly, coincide, that is, there is a correspondence between their regions that preserves all lengths (isometry). Properties preserved under isometric transformations are called internal geometry surface. 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 in general the results of all measurements inside the surface (angles, areas, curvature, etc.). Therefore, everything that depends only on the metric refers to the internal geometry.

Normal and normal section

Normal vectors at surface points

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

m = [r u ′, r v ′] | [r u ′, r v ′] | (\ displaystyle \ mathbf (m) = (\ frac ([\ mathbf (r "_ (u)), \ mathbf (r" _ (v))]) (| [\ mathbf (r "_ (u)) , \ mathbf (r "_ (v))] |))).

The normal sign depends on the choice of coordinates.

The section of a surface by a plane containing the normal of the surface at a given point forms a certain curve, which is called normal section surface. The main normal for the normal section coincides with the normal to the surface (up to a sign).

If the curve on the surface is not a normal section, then its principal normal forms a certain angle with the surface normal θ (\ displaystyle \ theta)... Then the curvature k (\ displaystyle k) curve associated with curvature k n (\ displaystyle k_ (n)) normal section (with the same tangent) by the Meunier formula:

k n = ± k cos θ (\ displaystyle k_ (n) = \ pm k \, \ cos \, \ theta)

Ort coordinates of the normal for different ways surface specifications are given in the table:

	Normal coordinates at a surface point
implicit assignment	(∂ F ∂ x; ∂ F ∂ y; ∂ F ∂ z) (∂ F ∂ x) 2 + (∂ F ∂ y) 2 + (∂ F ∂ z) 2 (\ displaystyle (\ frac (\ left (( \ frac (\ partial F) (\ partial x)); \, (\ frac (\ partial F) (\ partial y)); \, (\ frac (\ partial F) (\ partial z)) \ right) ) (\ sqrt (\ left ((\ frac (\ partial F) (\ partial x)) \ right) ^ (2) + \ left ((\ frac (\ partial F) (\ partial y)) \ right) ^ (2) + \ left ((\ frac (\ partial F) (\ partial z)) \ right) ^ (2)))))
explicit assignment	(- ∂ f ∂ x; - ∂ f ∂ y; 1) (∂ f ∂ x) 2 + (∂ f ∂ y) 2 + 1 (\ displaystyle (\ frac (\ left (- (\ frac (\ partial f ) (\ partial x)); \, - (\ frac (\ partial f) (\ partial y)); \, 1 \ right)) (\ sqrt (\ left ((\ frac (\ partial f) (\ partial x)) \ right) ^ (2) + \ left ((\ frac (\ partial f) (\ partial y)) \ right) ^ (2) +1))))
parametric assignment	(D (y, z) D (u, v); D (z, x) D (u, v); D (x, y) D (u, v)) (D (y, z) D (u , v)) 2 + (D (z, x) D (u, v)) 2 + (D (x, y) D (u, v)) 2 (\ displaystyle (\ frac (\ left ((\ frac (D (y, z)) (D (u, v))); \, (\ frac (D (z, x)) (D (u, v))); \, (\ frac (D (x , y)) (D (u, v))) \ right)) (\ sqrt (\ left ((\ frac (D (y, z)) (D (u, v))) \ right) ^ (2 ) + \ left ((\ frac (D (z, x)) (D (u, v))) \ right) ^ (2) + \ left ((\ frac (D (x, y)) (D ( u, v))) \ right) ^ (2)))))

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 at a given point on the surface, a different curvature of the normal section is obtained, which is called normal curvature; it is assigned a plus sign if the main normal of the curve goes in the same direction as the normal to the surface, or minus if the directions of the normals are opposite.

Generally speaking, at each point of the surface there are two perpendicular directions e 1 (\ displaystyle e_ (1)) and e 2 (\ displaystyle e_ (2)), in which the normal curvature takes the minimum and maximum values; these directions are called the main... An exception is the case when the normal curvature is the same in all directions (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) curvatures.

Normal curvatures in principal directions are called principal curvatures; denote them κ 1 (\ displaystyle \ kappa _ (1)) and κ 2 (\ displaystyle \ kappa _ (2))... Quantity:

K = κ 1 κ 2 (\ displaystyle K = \ kappa _ (1) \ kappa _ (2))

Tangent planes play a big role in geometry. In practical terms, the construction of tangent planes has essential, since their presence allows you to determine the direction of the normal to the surface at the point of tangency. This problem is widely used in engineering practice. Tangent planes are also used to construct sketches. geometric shapes bounded by closed surfaces. In theoretical terms, planes tangent to a surface are used in differential geometry to study the properties of a surface in the vicinity of a tangency point.

Basic concepts and definitions

The plane tangent to the surface should be considered as the limiting position of the secant plane (by analogy with the straight line tangent to the curve, which is also defined as the limiting position of the secant plane).

The plane tangent to the surface at a given point on the surface is the set of all straight lines - tangents drawn to the surface through a given point.

In differential geometry, it is proved that all tangents to a surface drawn at an ordinary point are coplanar (belong to the same plane).

Let us find out how the line tangent to the surface is drawn. The tangent t to the surface β at the point M given on the surface (Fig. 203) represents the limiting position of the secant lj intersecting the surface at two points (MM 1, MM 2, ..., MM n) when the intersection points coincide (M ≡ M n, ln ≡ l M). Obviously (M 1, M 2, ..., M n) ∈ g, since g ⊂ β. From the above, the following definition follows: tangent to a surface is a straight line tangent to any curve belonging to the surface.

Since the plane is defined by two intersecting straight lines, then to specify a plane tangent to the surface at a given point, it is enough to draw through this point two arbitrary lines belonging to the surface (preferably simple in shape), and to each of them construct tangents at the point of intersection of these lines ... The constructed tangents uniquely define the tangent plane. A visual representation of the drawing of the plane α, tangent to the surface β at a given point M, is given in Fig. 204. This figure also shows the normal n to the surface β.

The normal to the surface at a given point is a straight line perpendicular to the tangent plane and passing through the tangency point.

The line of intersection of the surface by a plane passing through the normal is called the normal section of the surface. Depending on the type of surface, the tangent plane can have, with the surface, either one or many points (line). The tangent line can be at the same time the line of intersection of the surface with the plane.

Cases are also possible when there are points on the surface at which it is impossible to draw a tangent to the surface; such points are called special. An example of singular points is the points belonging to the edge of the torso surface return, or the point of intersection of the meridian of the surface of revolution with its axis, if the meridian and the axis do not intersect at right angles.

The types of tangency depend on the nature of the curvature of the surface.

Surface curvature

The questions of surface curvature were investigated by the French mathematician F. Dupin (1784-1873), who proposed a visual way of depicting the change in the curvature of normal sections of a surface.

For this, in the plane tangent to the surface under consideration at point M (Fig. 205, 206), segments equal to the square roots of the values ​​of the corresponding radii of curvature of these sections are laid on the tangents to the normal sections on both sides of this point. The set of points - the ends of the segments define a curve called Dupin indicatrix... The algorithm for constructing the Dupin indicatrix (Fig. 205) can be written:

1. M ∈ α, M ∈ β ∧ α β;

2. = √ (R l 1), = √ (R l 2), ..., = √ (R l n)

where R is the radius of curvature.

(A 1 ∪ А 2 ∪ ... ∪ А n) is the Dupin indicatrix.

If the Dupin indicatrix of the surface is an ellipse, then the point M is called elliptic, and the surface is called a surface with elliptical points(fig. 206). In this case, the tangent plane has only one common point with the surface, and all lines belonging to the surface and intersecting at the point under consideration are located on one side of the tangent plane. Examples of surfaces with elliptical points are: a paraboloid of revolution, an ellipsoid of revolution, a sphere (in this case, the Dupin indicatrix is ​​a circle, etc.).

When drawing a tangent plane to the torso surface, the plane will touch this surface along a straight generatrix. The points of this line are called parabolic, and the surface is a surface with parabolic points... Dupin's indicatrix in this case is two parallel straight lines (Fig. 207 *).

In fig. 208 shows a surface consisting of points at which

* A curve of the second order - a parabola - under certain conditions can split into two real parallel straight lines, two imaginary parallel straight lines, two coinciding straight lines. In fig. 207 we are dealing with two real parallel lines.

The tangent plane intersects the surface. Such a surface is called hyperbolic, and the points belonging to it - hyperbolic points. Dupin's indicatrix in this case is a hyperbole.

A surface, all points of which are hyperbolic, has the shape of a saddle (oblique plane, one-sheet hyperboloid, concave surfaces of revolution, etc.).

One surface can have points different types, for example, at the torso surface (Fig. 209) the point M is elliptical; point N - parabolic; point K is hyperbolic.

In the course of differential geometry it is proved that normal sections, in which the curvature values ​​K j = 1 / R j (where R j is the radius of curvature of the section under consideration) have extreme values, are located in two mutually perpendicular planes.

Such curvatures K 1 = 1 / R max. K 2 = 1 / R min are called principal, and the values ​​H = (K 1 + K 2) / 2 and K = K 1 K 2 are, respectively, the mean curvature of the surface and the total (Gaussian) curvature of the surface at the point under consideration. For elliptic points K> 0, hyperbolic K

Specifying a plane tangent to a surface on a Monge plot

Below, using specific examples, we show the construction of a plane tangent to a surface with elliptic (example 1), parabolic (example 2), and hyperbolic (example 3) points.

EXAMPLE 1. Construct the plane α, tangent to the surface of revolution β, with elliptical points. Consider two options for solving this problem, a) a point М ∈ β and b) a point М ∉ β

Option a (Fig. 210).

The tangent plane is defined by two tangents t 1 and t 2 drawn at point M to the parallel and meridian of the surface β.

The projections of the tangent t 1 to the parallel h of the surface β will be t "1 ⊥ (S" M ") and t" 1 || the x axis. The horizontal projection of the tangent t "2 to the meridian d of the surface β passing through the point M coincides with the horizontal projection of the meridian. To find the frontal projection of the tangent t" 2, the meridional plane γ (γ ∋ М) by rotation around the axis of the surface β is translated into position γ 1 parallel to the plane π 2. In this case, point M → M 1 (M "1, M" 1). The projection of the tangent t "2 rarr; t" 2 1 is defined (M "1 S"). If we now return the plane γ 1 to its original position, then the point S "will remain in place (as belonging to the axis of rotation), and M" 1 → M "and the frontal projection of the tangent t" 2 will be determined (M "S")

Two tangents t 1 and t 2 intersecting at a point М ∈ β define the plane α tangent to the surface β.

Option b (fig. 211)

To construct a plane tangent to a surface passing through a point that does not belong to the surface, one must proceed from the following considerations: a set of planes tangent to the surface can be drawn through a point outside the surface consisting of elliptical points. The envelope of these surfaces will be some conical surface. Therefore, if there are no additional instructions, then the problem has many solutions and in this case is reduced to drawing a conical surface γ tangent to this surface β.

In fig. 211 shows the construction of a conical surface γ tangent to the sphere β. Any plane α tangent to the conical surface γ will be tangent to the surface β.

To construct projections of the surface γ from the points M "and M" we draw tangents to the circles h "and f" - the projections of the sphere. Mark the touch points 1 (1 "and 1"), 2 (2 "and 2"), 3 (3 "and 3") and 4 (4 "and 4"). Horizontal projection of a circle - the tangent line of the conical surface and the sphere will be projected in [1 "2"] To find the points of the ellipse into which this circle will be projected onto the frontal plane of projections, use the parallels of the sphere.

In fig. 211 in this way, frontal projections of points E and F (E "and F") are determined. Having a conical surface γ, we construct a tangent plane α to it. The nature and sequence of the graphic

The constructions that need to be performed for this are shown in the following example.

EXAMPLE 2 Construct the plane α tangent to the surface β with parabolic points

As in example 1, consider two options for the solution: a) point N ∈ β; b) point N ∉ β

Option a (Figure 212).

A conical surface refers to surfaces with parabolic points (see Fig. 207.) A plane tangent to a conical surface touches it along a rectilinear generatrix.

1) draw a generatrix SN (S "N" and S "N") through this point N;

2) mark the point of intersection of the generatrix (SN) with the guide d: (SN) ∩ d = A;

3) will wind also the tangent t to d at point A.

The generator (SA) and the tangent t intersecting it define the plane α tangent to the conical surface β at a given point N *.

To draw the plane α, tangent to the conical surface β and passing through the point N, does not belong

* Since the surface β consists of parabolic points (except for the vertex S), the plane α tangent to it will have in common with it not one point N, but a straight line (SN).

on a given surface, it is necessary:

1) draw a straight line a (a "and a") through a given point N and a vertex S of the conical surface β;

2) determine the horizontal trace of this straight line H a;

3) through H a draw tangents t "1 and t" 2 of the curve h 0β - the horizontal trace of the conical surface;

4) connect the points of tangency A (A "and A") and B (B "and B") to the top of the conical surface S (S "and S").

The intersecting lines t 1, (AS) and t 2, (BS) define the sought tangent planes α 1 and α 2

EXAMPLE 3. Construct the plane α tangent to the surface β with hyperbolic points.

Point K (Fig. 214) is located on the surface of the globoid (inner surface of the ring).

To determine the position of the tangent plane α, it is necessary:

1) draw through the point K a parallel to the surface β h (h ", h");

2) through the point K "draw a tangent t" 1 (t "1 ≡ h");

3) to determine the directions of the projections of the tangent to the meridional section, it is necessary to draw the plane γ through the point K and the surface axis, the horizontal projection t "2 coincides with h 0γ; to construct the frontal projection of the tangent t" 2, we first translate the plane γ by rotating it around the axis of the surface of revolution to the position γ 1 || π 2. In this case, the meridional section by the γ plane will be combined with the left outline arc of the frontal projection - the semicircle g ".

Point K (K ", K"), belonging to the curve of the meridional section, will move to position K 1 (K "1, K" 1). Through K "1 we draw a frontal projection of the tangent t" 2 1, aligned with the plane γ 1 || π 2 position and mark the point of its intersection with the frontal projection of the axis of rotation S "1. Return the plane γ 1 to its original position, point K" 1 → K "(point S" 1 ≡ S "). The frontal projection of tangent t" 2 is determined by the points K "and S".

The tangents t 1 and t 2 define the desired tangent plane α, which intersects the surface β along the curve l.

EXAMPLE 4. Construct the plane α, tangent to the surface β at point K. Point K is located on the surface of a one-sheet hyperboloid of revolution (Fig. 215).

This problem can be solved by adhering to the algorithm used in the previous example, but taking into account that the surface of a one-sheet hyperboloid of revolution is a ruled surface that has two families of rectilinear generators, and each of the generators of one family intersects all generators of the other family (see Section 32, Fig. . 138). Through each point of this surface, you can draw two intersecting straight lines - generators, which will be simultaneously tangent to the surface of a one-sheet hyperboloid of revolution.

These tangents define the tangent plane, that is, the plane tangent to the surface of a one-sheet hyperboloid of revolution intersects this surface along two straight lines g 1 and g 2. To construct the projections of these straight lines, it is enough to carry the horizontal projection of the point K to carry the tangents t "1 and t" 2 to the horizon.

the local projection of the circle d "2 - the throat of the surface of a one-sheet hyperboloid of revolution; determine points 1" and 2 at which t "1 and t" 2 intersect one it of the guiding surfaces d 1. For 1 "and 2" we find 1 "and 2", which together with K "determine the frontal projections of the sought-for straight lines.

Definition. A point lying on a second-order surface defined with respect to the GDSK by the general equation (1) is called non-singular if among the three numbers: there is at least one that is not equal to zero.

Thus, a point lying on a surface of the second order is not singular if and only if it is its center, otherwise, when the surface is conical, and the point is the vertex of this surface.

Definition. A tangent line to a second-order surface at a given non-singular point on it is a straight line passing through this point, intersecting the second-order surface at a double point, or being a rectilinear generatrix of the surface.

Theorem 3. The tangent lines to the surface of the second order at a given non-singular point on it lie in one plane, called the tangent plane to the surface at the point in question. The tangent plane equation has

Proof. Let,, be the parametric equations of the straight line passing through a non-singular point of the second-order surface, given by equation (1). Substituting into equation (1), instead of,,, we get:

Since the point lies on the surface (1), we also find from equation (3) (this value corresponds to the point). In order for the point of intersection of the straight line with the surface (1) to be double, or for the straight line to lie entirely on the surface, it is necessary and sufficient that the equality holds:

If at the same time:

The point of intersection of the straight line with the surface (1) is double. What if:

Then the whole line lies on the surface (1).

From relations (4) and,, it follows that the coordinates,, of any point lying on any tangent to the surface (1) satisfy the equation:

Conversely, if the coordinates of some point other than satisfy this equation, then the coordinates,, vectors, satisfy relation (4), which means that the line is tangent to the surface under consideration.

Since a point is a non-singular point of the surface (1), then among the numbers,, there is at least one that is not equal to zero; then equation (5) is an equation of the first degree with respect to. This is the equation of the plane tangent to the surface (1) at a given non-singular point on it.

Based on the canonical equations of second-order surfaces, it is easy to compose the equations of the tangent planes to an ellipsoid, hyperboloid, etc. at a given point on them.

1). Tangent plane to ellipsoid:

2). The tangent plane to one and two-sheet hyperboloids:

3). Tangent plane to elliptical and hyperbolic paraboloids:

§ 161. Intersection of a tangent plane with a surface of the second order.

We will take a non-singular point of the surface of the second order as the origin of coordinates of the ODSK, the axis and place it in the plane tangent to the surface at the point. Then, in the general equation of the surface (1), the free term is equal to zero:, and the equation of the plane tangent to the surface at the origin of coordinates should have the form:.

But the equation of the plane passing through the origin has the form:.

And, since this equation must be equivalent to the equation, then,,.

So, in the selected coordinate system, the surface equation (1) should have the form:

Conversely, if, then equation (6) is the equation of the surface passing through the origin, and the plane is the tangent plane to this surface at a point. The equation of the line along which the tangent plane to the surface at a point intersects the surface (6) has the form:

If . This is an invariant in the theory of invariants for second-order lines. Equation (7)

This is the second order line. By the form of this line, it is invariant, therefore:

When, here are two imaginary intersecting straight lines.

At - two real intersecting straight lines.

If, but at least one of the coefficients,, is not equal to zero, then the intersection line (7) is two coinciding straight lines.

Finally, if, then the plane

is a part of this surface, and the surface itself splits, therefore, into a pair of planes

§ 162. Elliptic, hyperbolic or parabolic points of a surface of the second order.

1. Let the tangent plane to the surface of the second order at a point intersect it along two imaginary intersecting straight lines. In this case, the point is called the elliptical point of the surface.

2. Let the tangent plane to the surface of the second order at a point intersect it along two real lines that intersect at the point of tangency. In this case, the point is called a hyperbolic point of the surface.

3. Let the tangent plane to the surface of the second order at a point intersect it along two coinciding straight lines. In this case, the point is called a parabolic point of the surface.

Theorem 4. Let the surface of the second order with respect to the ODSK be given by equation (1) and this equation (1) is the equation of a real non-decaying surface of the second order. Then if; then all points of the surface are elliptic.

Proof. Let us introduce a new coordinate system, choosing as the origin of coordinates any non-singular point of the given surface and placing the axes and in the plane tangent to the surface at the point. Equation (1) in new system coordinates is converted to the form:

Where . Let us calculate the invariant for this equation.

Since during the transition from one ODSK to another ODSK the sign does not change, then the signs are opposite, therefore, if, then; and, as follows from the classification (see § 161), the tangent plane to the surface at a point intersects the surface along two imaginary intersecting lines, i.e. is an elliptical point.

2) A one-sheet hyperboloid and a hyperbolic paraboloid consist of hyperbolic points.

3) Real cone of the second order (the vertex is excluded), elliptic (real), hyperbolic and parabolic cylinders consist of parabolic points.

Parabolic cylinder.

To determine the location of the parabolic cylinder, it is enough to know:

1) a plane of symmetry parallel to the generatrix of the cylinder;

2) the tangent plane to the cylinder, perpendicular to this plane of symmetry;

3) a vector perpendicular to this tangent plane and directed towards the concavity of the cylinder.

If the general equation defines a parabolic cylinder, it can be rewritten as:

We will select m so that the plane

would be mutually perpendicular:

With this value m plane

will be the plane of symmetry parallel to the generatrix of the cylinder.

Plane

will be the tangent plane to the cylinder, perpendicular to the specified plane of symmetry, and the vector

will be perpendicular to the found tangent plane and directed towards the concavity of the cylinder.

The graph of a 2-variable function z = f (x, y) is a surface projecting onto the XOY plane into the domain of definition of the function D.
Consider the surface σ given by the equation z = f (x, y), where f (x, y) is a differentiable function, and let M 0 (x 0, y 0, z 0) be a fixed point on the surface σ, that is, z 0 = f (x 0, y 0). Appointment. The online calculator is designed to find equations of the tangent plane and the normal to the surface... The decision is made in Word format. If you need to find the equation of the tangent to the curve (y = f (x)), then you need to use this service.

Function entry rules:

Function entry rules:

1. All variables are expressed in terms of x, y, z

The tangent plane to the surface σ at her point M 0 is the plane in which the tangents lie to all curves drawn on the surface σ through the point M 0 .
The equation of the tangent plane to the surface given by the equation z = f (x, y) at the point M 0 (x 0, y 0, z 0) has the form:

z - z 0 = f ’x (x 0, y 0) (x - x 0) + f’ y (x 0, y 0) (y - y 0)

The vector is called the normal vector to the surface σ at point М 0. The normal vector is perpendicular to the tangent plane.
Normal to surface σ at the point M 0 is called a straight line passing through this point and having the direction of the vector N.
The canonical equations of the normal to the surface given by the equation z = f (x, y) at the point M 0 (x 0, y 0, z 0), where z 0 = f (x 0, y 0), have the form:

Example # 1. The surface is given by the equation x 3 + 5y. Find the equation of the tangent plane to the surface at the point M 0 (0; 1).
Solution... Let us write the equations of the tangent line in general form: z - z 0 = f "x (x 0, y 0, z 0) (x - x 0) + f" y (x 0, y 0, z 0) (y - y 0 )
By the condition of the problem x 0 = 0, y 0 = 1, then z 0 = 5
Find the partial derivatives of the function z = x ^ 3 + 5 * y:
f "x (x, y) = (x 3 +5 y)" x = 3 x 2
f "x (x, y) = (x 3 +5 y)" y = 5
At the point М 0 (0,1), the values ​​of the partial derivatives:
f "x (0; 1) = 0
f "y (0; 1) = 5
Using the formula, we obtain the equation of the tangent plane to the surface at the point М 0: z - 5 = 0 (x - 0) + 5 (y - 1) or -5 y + z = 0

Example # 2. The surface is implicitly specified y 2 -1 / 2 * x 3 -8z. Find the equation of the tangent plane to the surface at the point M 0 (1; 0; 1).
Solution... Find the partial derivatives of the function. Since the function is set implicitly, we look for derivatives by the formula:

For our function:

Then:

At the point М 0 (1,0,1), the values ​​of the partial derivatives:
f "x (1; 0; 1) = -3 / 16
f "y (1; 0; 1) = 0
Using the formula, we obtain the equation of the tangent plane to the surface at the point М 0: z - 1 = -3 / 16 (x - 1) + 0 (y - 0) or 3/16 x + z- 19/16 = 0

An example. Surface σ given by the equation z= y / x + xy – 5x 3. Find the equation of the tangent plane and the normal to the surface σ at the point M 0 (x 0 ,y 0 ,z 0) belonging to it if x 0 = –1, y 0 = 2.
Find the partial derivatives of the function z= f(x,y) = y / x + xy – 5x 3:
f x ’( x,y) = (y / x + xy – 5x 3) 'x = - y / x 2 + y – 15x 2 ;
f y ’( x,y) = (y / x + xy – 5x 3) ’y = 1 / x + x.
Point M 0 (x 0 ,y 0 ,z 0) belongs to the surface σ , so one can calculate z 0 by substituting the given x 0 = –1 and y 0 = 2 into the surface equation:

z= y / x + xy – 5x 3

z 0 = 2/(-1) + (–1) 2 – 5 (–1) 3 = 1.
At the point M 0 (–1, 2, 1) values ​​of partial derivatives:
f x ’( M 0) = –1 / (- 1) 2 + 2 - 15 (–1) 2 = –15; f y ’( M 0) = 1/(-1) – 1 = –2.
Using formula (5), we obtain the equation of the tangent plane to the surface σ at the point M 0:
z – 1= –15(x + 1) – 2(y – 2) z – 1= –15x – 15 – 2y + 4 15x + 2y + z + 10 = 0.
Using formula (6), we obtain the canonical equations of the normal to the surface σ at the point M 0: .
Answers: tangent plane equation: 15 x + 2y + z+ 10 = 0; normal equations: .

Example # 1. Given a function z = f (x, y) and two points A (x 0, y 0) and B (x 1, y 1). Required: 1) calculate the value z 1 of the function at point B; 2) calculate the approximate value z 1 of the function at point B based on the value z 0 of the function at point A, replacing the increment of the function when passing from point A to point B by the differential; 3) compose the equation of the tangent plane to the surface z = f (x, y) at the point C (x 0, y 0, z 0).
Solution.
Let us write the equations of the tangent line in general form:
z - z 0 = f "x (x 0, y 0, z 0) (x - x 0) + f" y (x 0, y 0, z 0) (y - y 0)
By the condition of the problem x 0 = 1, y 0 = 2, then z 0 = 25
Find the partial derivatives of the function z = f (x, y) x ^ 2 + 3 * x * y * + y ^ 2:
f "x (x, y) = (x 2 +3 x y + y 2)" x = 2 x + 3 y 3
f "x (x, y) = (x 2 +3 x y + y 2)" y = 9 x y 2
At the point М 0 (1,2), the values ​​of the partial derivatives:
f "x (1; 2) = 26
f "y (1; 2) = 36
Using the formula, we obtain the equation of the tangent plane to the surface at the point М 0:
z - 25 = 26 (x - 1) + 36 (y - 2)
or
-26 x-36 y + z + 73 = 0

Example # 2. Write the equations of the tangent plane and the normal to the elliptic paraboloid z = 2x 2 + y 2 at the point (1; -1; 3).