The equation is normal to the sphere. Theoretical material. Normal and Normal section

Namely, about what you see in the title. Essentially, this is a "spatial analogue" objectives of finding tangent and normal To the graph of the function of one variable, and therefore there should be no difficulties.

Let's start with the basic questions: what is a tangent plane and what is normal? Many are aware of these concepts at the level of intuition. The simplest model coming to mind is a ball on which a thin flat cardboard is lying. Cardboard is located as close as possible to the sphere and concerns it in a single point. In addition, at the touch point, it is fixed strictly up the needle.

In theory there is a rather witty determination of the tangent plane. Imagine free surface And the point belonging to it. Obviously, a lot of point passes through the point spatial lineswhich belong to this surface. Who has any associations? \u003d) ... Personally, I presented octopus. Suppose that each such line exists spatial tangent At point.

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

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

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

With working formulas and algorithm, solutions will get acquainted directly on a specific example. In the overwhelming majority of tasks, the equation of the tangent plane and the normal equation are also required:

Example 1.

Decision: if the surface is set by the equation (i.e. implicitly), The equation of the tangent plane to this surface at the point can be found according to the following formula:

Paying special attention to unusual private derivatives - their do not be confused from partial derivatives implicitly specified function (although the surface is defined). If you find these derivatives you need to be guided differentiation Rules of the Function of Three Variables, that is, when differentiated by any variable, two other letters are considered constants:

Without departing from the box office, find a private derivative at the point:

Similarly:

It was the most unpleasant moment of a solution in which an error if not allowed, it is constantly seemed. However, there is an effective receipt of the check, which I talked about in class Gradient derivative.

All "ingredients" are found and now it's a neat substitution with further simplifications:

– general equation The desired tangent plane.

I strongly recommend checking out this stage of the solution. First you need to make sure that the coordinates of the touch point are really satisfying the found equation:

- Faithful equality.

Now "remove" the coefficients of the general equation of the plane and check them for coincidence or proportionality with the corresponding values. In this case are proportional. How do you remember from course of analytical geometry, - this is vector Normal tangent plane and it's - vector guide Normal straight. Make up canonical equations Normal on the point and the guide vector:

In principle, the denominators can be reduced to the "Two", but there is no particular need for this

Answer:

The equations are not rebeling to designate some letters, however, again - why? Here and so extremely clear what's what.

The following two examples for an independent solution. Small "mathematical patter":

Example 2.

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

And the task, interesting from a technical point of view:

Example 3.

Make the equations of the tangent plane and normal to the surface at the point

At point.

There are all the chances not only to get confused, but also to face difficulties in writing. canonical equations are direct. And the equations are normal, as you probably understood, it is customary to record in this form. Although, due to the forgetfulness or ignorance of some nuances more than acceptable and parametric form.

Exemplary samples of finishing decisions at the end of the lesson.

Is there a tangent plane in any surface? In general, of course, no. Classic example is conical surface And the point - the tangents at this point directly form a conical surface, and, of course, are not lying in the same plane. In anything easy to make sure and analytically :.

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

But it was rather popularly popular than practically significant information, and we return to the urgent matters:

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

Rewrite it in an implicit form:

And on the same principles find private derivatives:

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

And, accordingly, canonical equations are normal:

How to guess, - This is already "real" private derivatives of two variables At the point that we used to identify the letter "Zet" and found 100,500 times.

Note that this article is enough to remember the very first formula from which if necessary, it is easy to remove everything else. (Clear, possessing the basic level of preparation). This approach should be used during the study of the exact sciences, i.e. From a minimum of information, it is necessary to strive to "pull out" the maximum of conclusions and consequences. "Reaching" and already existing knowledge to help! This principle is also useful in the fact that with a high probability will save in a critical situation when you know very little.

We will work out "modified" formulas for a pair of examples:

Example 4.

Make the equations of the tangent plane and normal to the surface At point.

The lining here turned out with the notation - now the letter denotes the point of the plane, but what to do is such a popular letter ....

Decision: The equation of the desired tangent plane will be according to the formula:

Calculate the value of the function at the point:

Calculate private derivatives of the 1st order At this point:

In this way:

Carefully, not in a rush:

We write canonical equations of normal at the point:

Answer:

And the final example for an independent decision:

Example 5.

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

Final - because I actually explained all the technical moments and there is nothing to add. Even the functions proposed in this task, sadness and monotonous - almost guaranteed in practice you will get "polynomial", and in this sense, Example No. 2 with an exponent looks like "White Voron". By the way, it is much likely to meet the surface specified by the equation and this is another reason why the function entered the article by the "second number".

And finally, the promised secret: so how to avoid bunches of definitions? (I certainly do not mean the situation when a student is feverishly shaved in front of the exam)

The definition of any concept / phenomenon / object, first of all, gives the answer to the next question: what is it? (who / such / such / such). Consciously Answering this question, you must try to reflect significantsigns definite Identifying this or that concept / phenomenon / object. Yes, at first it turns out somewhat obliquely, inaccurately and excessively (the teacher will correct \u003d)), but over time, a completely decent scientific speech is developing.

Repeat on the most disturbed objects, for example, answer the question: who is such Cheburashka? Not so, everything is simple ;-) This is a fabulous character with big ears, eyes and brown wool "? Far and very far from the definition - there are few characters with such characteristics .... But it is already much closer to the definition: "Cheburashka is a character invented by the writer Edward Asspensky in 1966, which ... (transfer of the main distinctive features)". Pay attention to how competent started

The surface is defined as a set of points whose coordinates satisfy a certain type of equations:

F (x, y, z) \u003d 0 (1) (\\ displaystyle f (x, \\, y, \\, z) \u003d 0 \\ qquad (1))

If the function F (x, y, z) (\\ displaystyle f (x, \\, y, \\, z)) continuous at some point and has continuous private derivatives in it, at least one of which is not accessed into zero, then in the vicinity of this point, the surface specified by the equation (1) will be right surface.

In addition to the above implicit way to task, surface can be determined obviousIf one of the variables, for example, z, can be expressed in the rest:

z \u003d f (x, y) (1 ') (\\ displaystyle z \u003d f (x, y) \\ qquad (1 "))

More strictly simple surface An image of a homeomorphic mapping is called (that is, a mutually unambiguous and mutually continuous display) of the inside of a single square. This definition can be given an analytical expression.

Suppose on the plane with the rectangular coordinate system U and V, the square is set, the coordinates of the internal points of which satisfy 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 half asphere. The whole sphere is not simple surface. This causes the need to further generalize the concept of the surface.

Subset of space, each point of which has a neighborhood, which is simple surface, called right surface .

SURFACE IN DIFFERENTIAL Geometry

Helicoid.

Kattenoid

Metric does not define the unique surface shape. For example, Helicoid and Catenoid metrics, parametrized accordingly, coincide, that is, between their regions there is a correspondence that maintains all the lengths (isometry). Properties that persist in isometric transformations are called internal geometry Surfaces. Internal geometry does not depend on the position of the surface in space and does not change when it is bent without stretching and compressing (for example, when the cylinder is bent into the 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, area, 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 the surface is its normal - single vector, perpendicular tangent plane at a specified point:

M \u003d [R U ', R V'] | [R U ', R V'] | (\\ DisplayStyle \\ MathBF (M) \u003d (\\ FRAC ([\\ MathBF (R "_ (U)), \\ MathBF (R" _ (V))]) (| [\\ MathBF (R "_ (U)) , \\ MathBF (R "_ (V))] |))).

Normal sign depends on the choice of coordinates.

The surface cross section of the plane containing the surface normal surface at a given point forms some curve called normal cross section Surfaces. The main standard for normal cross section coincides with the normal to the surface (with an accuracy of the sign).

If the curve on the surface is not a normal cross section, then its main normal is formed with a normal surface with the normal surface θ (\\ DisplayStyle \\ Theta). Then Crivale K (\\ DisplayStyle K) The curve is related to the curvature k n (\\ displaystyle k_ (n)) Normal section (with the same tangent) Menias formula:

k n \u003d ± k cos θ (\\ displaystyle k_ (n) \u003d \\ pm k \\, \\ cos \\, \\ ,ta)

The coordinates of the ort of normal for different ways to task the surface are shown in the table:

	Coordinates of normal at the point of the surface
implicit task	(∂ 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 task	(- ∂ f ∂ x; - ∂ f ∂ y; 1) (∂ F ∂ x) 2 + (∂ F ∂ Y) 2 + 1 (\\ DisplayStyle (\\ FRAC (\\ PARTI FRAC (- (\\ FRAC (\\ Partial F ) (\\ Partial x)); \\, - (\\ FRAC (\\ Partial f) (\\ Partial y)); \\, 1 \\ Right)) (\\ sqrt (\\ left ((\\ FRAC (\\ Partial F) (\\ Parametric task
	(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) \u003d | Y u 'y v' z u 'z v' | , D (z, x) d (u, v) \u003d | z u 'z v' x u 'x v' | , D (x, y) d (u, v) \u003d | x U 'x V' y u 'y v' | (\\ DisplayStyle (\\ FRAC (D (Y, Z)) (D (U, V))) \u003d (\\ begin (Vmatrix) y "_ (u) & y" _ (v) \\\\ z "_ (U) & z "_ (v) \\ end (vmatrix)), \\ quad (\\ FRAC (D (z, x)) (D (U, V))) \u003d (\\ begin (vmatrix) z" _ (u) & z " _ (v) \\\\ x "_ (u) & x" _ (v) \\ end (vmatrix)), \\ quad (\\ FRAC (D (X, Y)) (D (U, V))) \u003d (\\ All derivatives are taken at point.

(x 0, y 0, z 0) (\\ displaystyle (x_ (0), y_ (0), z_ (0))) Curvature.

For different directions at a given surface point, a different curvature of a normal cross section is obtained, which is called

Normal curvature ; She is attributed to the plus sign, if the main normal normal curve goes in the same direction as the normal to the surface, or minus, if the directions are opposite.Generally speaking, at each point of the surface there are two perpendicular directions.

E 1 (\\ DisplayStyle E_ (1)) E 2 (\\ DisplayStyle E_ (2)) and in which normal curvature takes the minimum and maximum value; These directions are calledMain . The exception is the case when normal curvature in all directions of the same (for example, in the sphere or on the end of the ellipsoid of rotation), then all directions at the point are the main ones.Surfaces with negative (left), zero (center) and positive (right) curvature.

Normal curvatures in the main directions are called

The main curvators ; Denote themκ 1 (\\ displaystyle \\ kappa _ (1)) κ 2 (\\ displaystyle \\ kappa _ (2)) and . Value:K \u003d κ 1 κ 2 (\\ displaystyle k \u003d \\ kappa _ (1) \\ kappa _ (2))

The tangent planes play a large role in geometry. The construction of tangential planes is important, since the presence of them allows you to determine the direction of normal to the surface at the touch point. This task is widely used in engineering practice. The help of tangential planes also apply to construct essays of geometric figures limited by closed surfaces. In the theoretical plan of the plane tangent to the surface, used in differential geometry when studying the surface properties in the touch point area.

Basic concepts and definitions

The plane tangent to the surface should be considered as the limit position of the securing plane (by analogy with a straight-tangent to the curve, which is also defined as the limit position of the section).

Plane tangent to the surface in a point specified on the surface, there are many of all direct-tangent, spent to the surface through a specified point.

In differential geometry, it is proved that the PSS tangents to the surface carried out in an ordinary point are companed (belong to the same plane).

Find out how direct is carried out tangent to the surface. Tangent T to the surface β in the point specified on the surface m (Fig. 203) represents the limit position of the section LJ, crossing 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. The following definition implies from the above: tangent to the surface is called straight, tangent to any curve belonging to the surface.

Since the plane is determined by two intersecting straight, then to set a plane tangent to the surface at a given point, it is enough to spend two arbitrary lines, belonging to the surface (preferably simple in shape), and to build tangents at the point of intersection of these lines . Constructed tangents uniquely determine the tangent plane. A visual representation of the plane α, tangent to the surface β at a given point M, gives rice. 204. In this figure, N is also shown to the surface β.

Normal to the surface at a given point is a direct, perpendicular to the tangent plane and passing through the touch point.

The line intersection of the surface with the plane passing through the normal is called the normal surface cross section. Depending on the surface of the surface, the tangent plane may have, both alone and a plurality of points (line). The touch line can be at the same time and the line intersection of the surface with the plane.

There are also cases when there are points on the surface on which it is impossible to conduct a tangent to the surface; Such points are called special. As an example of singular points, it is possible to bring points belonging to the edge of returning the torch surface, or the point of intersection of the meridian of the rotation surface with its axis, if the meridian and the axis intersect not at right angles.

Types of touch depend on the nature of the curvature of the surface.

The curvature of the surface

The problems of the curvature of the surface were investigated by the French mathematician F. Dupin (1784-1873), which suggested a visual way to image changes in curvature of normal surface sections.

To do this, in the plane tangent to the surface under consideration at the point M (Fig. 205, 206), for tangents to normal sections on both sides of this point, segments are deployed equal to roots square from the values \u200b\u200bof the corresponding radii of curvature of these sections. Many points - the ends of the segments set the curve called indicator Dupina. Algorithm for constructing Indicatrix Dupos (Fig. 205) You can write:

1. m ∈ α, m ∈ β ∧ α β;

2. \u003d √ (R l 1), \u003d √ (R l 2), ..., \u003d √ (r l n)

where R is the radius of curvature.

(A 1 ∪ A 2 ∪ ... ∪ A N) - Indicator Dupos.

If the Indicator is duped surface - ellipse, then the point m is called an elliptical, and the surface is a surface with elliptic dots(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 in the point under consideration are located one way from the tangent plane. An example of surfaces with elliptic dots can be: a rotational paraboloid, an ellipsoid of rotation, the sphere (in this case, the indictrication of dupene is a circle, etc.).

When conducting a tangent plane to a torch surface, the plane will touch this surface in a straight line forming. Points of this direct are called parabolic, and surface - surface with parabolic dots. Indicatrix dupene in this case is two parallel straight (Fig. 207 *).

In fig. 208 shows the surface consisting of points in which

* The second order curve is Parabola - under certain conditions it can decay into two valid parallel straight, two imaginary parallel straight, two coinciding straight. In fig. 207 We are dealing with two valid parallel straight.

ryy tangent plane crosses the surface. This surface is called hyperbolic, and the points belonging to it - hyperbolic dots. Indicator dupene in this case - hyperbole.

The surface, all points of which are hyperbolic, has the shape of the saddle (oblique plane, single-graded hyperboloid, concave surfaces of rotation, etc.).

One surface may have points of different species, for example, in a torch surface (Fig. 209), the point M is elliptical; point n - parabolic; Point to - hyperbolic.

The course of differential geometry is proved that normal sections in which the curvature K j \u003d 1 / r j are proved (where R j, the radius of the curvature of the section under consideration) have extreme values, are located in two mutually perpendicular planes.

Such curvatures to 1 \u003d 1 / R max. K 2 \u003d 1 / R min is called the main, and the values \u200b\u200bof H \u003d (K 1 + K 2) / 2 and K \u003d K 1 to 2 - respectively, the average curvature surface and the total (Gaussian) curvature surface in the point under consideration. For elliptic points to\u003e 0, hyperbolic to

Setting the plane tangent to the surface on the Monta Eplere

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

Example 1. To construct a plane α, tangent to the surface of rotation β, with elliptic points. Consider two options for solving this problem, a) point m ∈ β and b) point M ∉ β

Option A (Fig. 210).

The tangent plane is determined by two tangents T 1 and T 2, carried out at the point M to the parallel and meridian of the surface β.

Projections of tangent T 1 to parallel H surface β will be t "1 ⊥ (s" m ") and t" 1 || axis x. 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 tangent T" 2, the meridional plane γ (γ ∋ M) by rotating around the axis of the surface β is translated into the γ position 1, parallel plane π 2. In this case, the point M → M 1 (M "1, M" 1). Projection of tangent T "2 RARR; T" 2 1 is determined (M "1 S"). If we now return the plane Γ 1 to the original position, then the point S "will remain in place (as belonging to the axis of rotation), and M" 1 → M "and the front projection of tangent T" 2 will be determined (M "S")

Two intersecting at point m ∈ β tangents t 1 and t 2 determine the plane α, tangent to the surface β.

Option B (Fig. 211)

To construct a plane tangent to the surface passing through a point that does not belong to the surface, it is necessary to proceed from the following considerations: through the point outside the surface consisting of elliptic points, we can carry out a plurality of planes tanging to the surface. The envelope of these surfaces will be some conic surface. Therefore, if there is no additional instructions, the task has many solutions and, in this case, it is reduced to the conical surface Γ, tangent of this surface β.

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

To construct the projections of the surface Γ from points M "and M" we carry out tangents to the circumferences H "and F" - the projections of the sphere. We celebrate the touch points 1 (1 "and 1"), 2 (2 "and 2"), 3 (3 "and 3") and 4 (4 "and 4"). Horizontal circle of the circle - the touch line of the conical surface and the sphere is designed in [1 "2"] to find the point of the ellipse, into which this circle is sprocketing to the frontal plane of projections, we use the parallels of the sphere.

In fig. 211 In this way, the front projections of points E and F (E "and F" are determined. Having a conical surface Γ, we build a tangent plane α. Character and sequence of graphical

kih constructions that need to be performed for this are given in the following example.

Example 2 To construct a plane α, tangent to the surface β with parabolic dots

As in Example 1, we consider two variants of the solution. And a point n ∈ β; b) point n ∉ β

Option A (Fig. 212).

The conical surface belongs to surfaces with parabolic dots (see Fig. 207.) The plane tangent to the conical surface relates to its straightforward forming. For its construction, it is necessary:

1) through this point n to carry out the sn (s "n" and s "n");

2) note the intersection point of forming (Sn) with a guide D: (Sn) ∩ D \u003d A;

3) Conducts to the ascent T to D at point A.

Forming (SA) and intersecting it tangent T determine α, tangent to the conical surface β at a given point N *.

To carry out the plane α, tangent to the conical surface β and passing through the point N, not

* Since the surface β consists of parabolic points (except the vertex S), the tangent of it the plane α will have a common one with it n, and the direct (SN).

through a given surface, you need:

1) through this point N and the vertex of the conical surface β spend direct a (a "and a");

2) to determine the horizontal trail of this direct N A;

3) through H a to carry out tangents t "1 and t" 2 curve H 0β - horizontal trail of the conical surface;

4) Touch points A (A "and A") and in (in "and B") to connect from the vertex of the conical surface S (S "and S").

Intersecting straight lines T 1, (AS) and T 2, (BS) determine the desired tangent plane α 1 and α 2

Example 3. To construct a plane α, tangent to the surface β with hyperbolic dots.

The point K (Fig. 214) is on the surface of the global (the inner surface of the ring).

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

1) through the point to the parallel of the surface β h (h ", h");

2) through the point to "Conductance t" 1 (t "1 ≡ h");

3) To determine the directions of projections of the tangential to the meridional section, it is necessary to carry out through the point K and the surface axis, the plane Γ, the horizontal projection T "2 coincides with H 0γ; to build a frontal projection of tangent T" 2, the plane γ is pre-rotated by rotating it around the axis of the rotation surface in position Γ 1 || π 2. In this case, the meridional cross section of the plane γ is combined with the left essay arc front projection - a semi-rareness G. "

The point K (K ", K"), belonging to the curve of the meridional section, moves to the position K 1 (K "1, K" 1). Through to "1 We carry out the front projection of tangent T" 2 1, in a combined with a plane γ 1 || π 2 position and note the point of its intersection with the front projection of the axis of rotation S "1. We return the plane Γ 1 to its original position, the point to" 1 → K "(point s" 1 ≡ s "). Frontal projection of tangent T" 2 Determined by points To "and s".

Tangents T 1 and T 2 determine the desired tangent plane α, which crosses the surface β by curve L.

Example 4. To construct the plane α, tangent to the surface β at the point K. The point K is on the surface of the single-graded rotation hyperboloid (Fig. 215).

This task can be solved by adhering to the algorithm used in the previous example, but considering that the surface of the single-graded rotation hyperboloid is a line surface, which has two families of straightforward forming, and each of the forming one family crosses all forming another family (see § 32, rice . 138). Through each point of this surface, two intersecting straight lines can be carried out - forming, which will be simultaneously tangent to the surface of the single-graded rotation hyperboloid.

These tangents determine the tangent plane, t e. The plane tangent to the surface of the single-graded rotation hyperboloid, crosses this surface along two straight g 1 and g 2. To build projections of these directly, a rather IT horizontal projection point to carry tangents t "1 and t" 2 to horizontal

the total projection of the circumference D "2 is the throat of the surface of the single-graded rotation hyperboloid; determine points 1" and 2, in which T "1 and T" 2 crosses one IT guide surface D 1. 1 "and 2" find 1 "and 2", which together with K "determine the frontal projections of the desired direct.

Definition. The point lying on the second order surface specified relative to the ODC in the overall equation (1) is called non-singular, if among the three numbers: there is at least one, not equal to zero.

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

Definition. Tangent direct to the second order surface in this not a special point is called direct, passing through this point, crossing the second order surface in a two-fold point or is a straightforward surface forming.

Theorem 3. Tangential direct to the second order surface in this not a special point is lying in the same plane, called the tangent plane to the surface in the point under consideration. The equation of the tangent plane has

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

Since the point lies on the surface (1), then from equation (3) we find (this value corresponds to the point). In order for the intersection point to the straight with the surface (1) was double, or to lie entirely on the surface, it is necessary and enough to make the equality:

If at the same time:

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

That directly lies on the surface (1).

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

Back if the coordinates of some point different from satisfying this equation, the coordinates, vector, satisfy the relation (4), which means that direct is tangent to the surface under consideration.

Since the point is a non-singing point of the surface (1), then among numbers, there is at least one, not equal to zero; Therefore, equation (5) is the equation of the first degree relative. This is the equation of a plane tangent to the surface (1) in this not a special point on it.

Based on the canonical equations of the second-order surfaces, it is easy to make the equations of the tangent planes to the ellipsoid, hyperboloid, etc. In this point on them.

one). Tangent plane to ellipsoid:

2). Tangent plane to one and bispsing hyperboloids:

3). Tangency plane to elliptical and hyperbolic paraboloids:

§ 161. Transfer the tangent plane with the second order surface.

We will take a non-singular point of the second order surface for the beginning of the coordinates of the ODC, the axes and place in the plane tangent to the surface at the point. Then in the overall surface equation (1), the free member is zero:, and the equation of flat-bone concerning the surface at the beginning of the coordinates should be of the form :.

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

And, since this equation should be equivalent to an equation, then,.

So, in the selected coordinate system, the surface equation (1) must be viewed:

Back, if, equation (6) is an equation of the surface passing through the origin of the coordinates, and the plane is a tangent plane to this surface at the point. The equation of the line along which the tangent plane to the surface at the point crosses the surface (6) is:

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

This is the second order line. According to this line, the invariant, so:

With here two imaginary intersecting straight lines.

When - two valid intersecting straight lines.

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

Finally, if, then the plane

it is part of this surface, and the surface itself breaks down, therefore, a pair of planes

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

1. Suppose the tangent plane to the second order surface at the point crosses it in two MNIs intersecting direct. In this case, the point is called an elliptic surface point.

2. Suppose the tangent plane to the second order surface at the point crosses it along two valid direct, intersecting at the touch point. In this case, the point is called a hyperbolic surface point.

3. Suppose the tangent plane to the second order surface at the point crosses it in two coinciding straight. In this case, the point is called a parabolic point of the surface.

Theorem 4. Let the second order surface relative to the ODASC set by equation (1) and this equation (1) is the equation of the actual non-exposed second order surface. Then, if; All the points of the surface are elliptical.

Evidence. We introduce a new coordinate system, choosing for the beginning of the coordinates any non-singular point of this surface and placing the axis and in the plane tangent to the surface at the point. Equation (1) in the new coordinate system is converted to mind:

Where. Calculate the invariant for this equation.

Since when moving from one ODASC to another union, the sign does not change, then signs are opposite, therefore, if, then; And, as follows from the classification (see § 161) the tangent plane to the surface at the point crosses the surface along two imaginary intersecting straight, i.e. - Elliptic point.

2) single-graded hyperboloid and hyperbolic paraboloid consist of hyperbolic dots.

3) A valid second order cone (the peak is excluded), elliptical (valid), hyperbolic and parabolic cylinders consist of parabolic dots.

Parabolic cylinder.

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

1) the plane of symmetry parallel to the cylinder forming;

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

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

In the event that the general equation determines the parabolic cylinder, it can be rewritten in the form:

Putister m. so that the plane

would be mutually perpendicular:

In this sense m. plane

it is a plane of symmetry parallel to the cylinder forming.

Plane

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

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

The graph of the function of 2 variables z \u003d f (x, y) is the surface designed to the Xoy plane in the definition area of \u200b\u200bthe function D.
Consider the surface σ given by the equation z \u003d 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, i.e. z 0 \u003d f (x 0, y 0). Purpose. Online calculator is designed to find equations of the tangent plane and normal to the surface. The solution is issued in Word format. If it is necessary to find the equation tangent to the curve (y \u003d f (x)), then it is necessary to use this service.

Rules for entering functions:

Rules for entering functions:

1. All variables are expressed in X, Y, Z

Tangential surface plane σ At its point M. 0 is called the plane in which there are tangents to all curves spent on the surface σ Through the point M. 0 .
The equation of the tangent plane to the surface specified by the equation z \u003d f (x, y) at the point M 0 (x 0, y 0, z 0) is:

z - z 0 \u003d f 'x (x 0, y 0) (x - x 0) + f' y (x 0, y 0) (y - y 0)

The vector is called the vector of normal to the surface σ At point M 0. Normal vector perpendicular to tangent plane.
Normal to the surface σ At point M. 0 is called straight, passing through this point and having the direction of the vector N.
The canonical equations of normal to the surface given by the equation z \u003d f (x, y) at the point M 0 (x 0, y 0, z 0), where Z 0 \u003d f (x 0, y 0), have the form:

Example number 1. The surface is set by the equation x 3 + 5y. Find the equation of the tangent plane to the surface at point M 0 (0; 1).
Decision. We write the equation of tangent in general form: z - z 0 \u003d f "x (x 0, y 0, z 0) (x - x 0) + f" y (x 0, y 0, z 0) (y - y 0 )
Under the condition of problem x 0 \u003d 0, y 0 \u003d 1, then z 0 \u003d 5
We find private derivatives of the functions z \u003d x ^ 3 + 5 * y:
f "x (x, y) \u003d (x 3 +5 y)" x \u003d 3 x 2
f "x (x, y) \u003d (x 3 +5 y)" y \u003d 5
At point M 0 (0.1), values \u200b\u200bof private derivatives:
f "x (0; 1) \u003d 0
F "Y (0; 1) \u003d 5
Taking advantage of the formula, we obtain the equation of the tangent plane to the surface at the point M 0: Z - 5 \u003d 0 (x - 0) + 5 (y - 1) or -5 y + z \u003d 0

Example number 2. The surface is defined implicitly y 2 -1 / 2 * x 3 -8z. Find the equation of the tangent plane to the surface at point M 0 (1; 0; 1).
Decision. We find private derivatives. Since the function is defined in implicit form, then derivatives are looking for by the formula:

For our function:

Then:

At point m 0 (1,0,1), values \u200b\u200bof private derivatives:
f "x (1; 0; 1) \u003d -3 / 16
f "y (1; 0; 1) \u003d 0
Using the formula, we obtain the equation of the tangent plane to the surface at the point M 0: Z - 1 \u003d -3 / 16 (x - 1) + 0 (y - 0) or 3/16 x + z- 19/16 \u003d 0

Example. Surface σ Posted by equation z.\u003d y / x + xY. – 5x. 3. Find the equation of the tangent plane and normal to the surface σ At point M. 0 (x. 0 , y. 0 , Z. 0) belonging to her if x. 0 = –1, y. 0 = 2.
Find private derived functions z.= f.(x., y.) \u003d y / x + xY. – 5x. 3:
f X '( x., y.) \u003d (y / x + xY. – 5x. 3) 'x \u003d - y / x 2 + y. – 15x. 2 ;
f y '( x., y.) \u003d (y / x + xY. – 5x. 3) 'y \u003d 1 / x + x..
Point M. 0 (x. 0 , y. 0 , Z. 0) belongs to the surface σ , so you can calculate z. 0, substituting the specified x. 0 \u003d -1 and y. 0 \u003d 2 in the surface equation:

z.\u003d y / x + xY. – 5x. 3

z. 0 = 2/(-1) + (–1) 2 – 5 (–1) 3 = 1.
At point M. 0 (-1, 2, 1) values \u200b\u200bof private derivatives:
f X '( M. 0) \u003d -1 / (- 1) 2 + 2 - 15 (-1) 2 \u003d -15; F y '( M. 0) = 1/(-1) – 1 = –2.
Using the formula (5) we obtain the equation of the tangent plane to the surface σ At point M. 0:
z. – 1= –15(x. + 1) – 2(y. – 2) z. – 1= –15x. – 15 – 2y +.4 15x. + 2y. + z. + 10 = 0.
Using the formula (6) we obtain canonical equations of normal to the surface σ At point M. 0: .
Answers: equation of the tangent plane: 15 x. + 2y. + z. + 10 \u003d 0; Equations Normal: .

Example number 1. The function z \u003d f (x, y) and two points A (x 0, y 0) and in (x 1, y 1) are given. Required: 1) Calculate the value Z 1 function at point in; 2) Calculate the approximate value of z 1 of the function at the point in accordance with the value Z 0 of the function at a point A, replacing the increment of the function when moving from point A to the point in the differential; 3) Make the equation of the tangent plane to the surface z \u003d f (x, y) at the point C (x 0, y 0, z 0).
Decision.
We write the equation tangent in general form:
z - z 0 \u003d f "x (x 0, y 0, z 0) (x - x 0) + f" y (x 0, y 0, z 0) (y - y 0)
Under the condition of problem x 0 \u003d 1, y 0 \u003d 2, then z 0 \u003d 25
We find private derivatives z \u003d f (x, y) x ^ 2 + 3 * x * y * + y ^ 2:
f "x (x, y) \u003d (x 2 +3 x y + y 2)" x \u003d 2 x + 3 y 3
f "x (x, y) \u003d (x 2 +3 x y + y 2)" y \u003d 9 x y 2
At point m 0 (1.2), values \u200b\u200bof private derivatives:
f "x (1; 2) \u003d 26
f "Y (1; 2) \u003d 36
Taking advantage of the formula, we obtain the equation of the tangent plane to the surface at the point M 0:
z - 25 \u003d 26 (x - 1) + 36 (Y - 2)
or
-26 x-36 y + z + 73 \u003d 0

Example number 2. Write the equations of the tangent plane and the normal to the elliptical paraboloid Z \u003d 2x 2 + y 2 at the point (1; -1; 3).