Tangent surface and surface normal. How to find the equations of the tangent plane and the surface normal at a given point

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Tangent planes play big role in geometry. The construction of tangent planes in practical terms is important, since their presence allows you to determine the direction of the normal to the surface at the point of contact. This task finds wide application in engineering practice. The help of tangent planes is also used to build essays 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 region of a tangent point.

Basic concepts and definitions

The plane tangent to the surface should be considered as the limit position of the secant plane (similar to the line tangent to the curve, which is also defined as the limit position of the secant).

The plane tangent to the surface at a given point on the surface is the set of all 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 a straight 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 limit position of the secant l j intersecting the surface at two points (MM 1, MM 2, ..., MM n), when the intersection points coincide (M ≡ M n , l n ≡ l M). Obviously (M 1 , M 2 , ..., M n ) ∈ g, since g ⊂ β. The following definition follows from the above: a tangent to a surface is a line tangent to any curve belonging to the surface.

Since the plane is defined by two intersecting straight lines, to set a plane tangent to the surface at a given point, it is enough to draw two arbitrary lines belonging to the surface (preferably simple in shape) through this point and construct tangents to each of them at the point of intersection of these lines . The constructed tangents uniquely determine the tangent plane. A visual representation of the holding 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 point of contact.

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 line of contact can be at the same time the line of intersection of the surface with the plane.

There are also cases when there are points on the surface where it is impossible to draw a tangent to the surface; such points are called singular. As an example of singular points, one can give points belonging to the cusp edge of the torso surface, 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 contact depend on the nature of the curvature of the surface.

surface curvature

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

To do this, in a plane tangent to the surface under consideration at point M (Fig. 205, 206), on tangents to normal sections on both sides of this point, segments are plotted equal to the square roots of the values ​​of the corresponding radii of curvature of these sections. The set of points - the ends of the segments define a curve called Dupin's 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 ∪ A 2 ∪ ... ∪ A n) is the Dupin indicatrix.

If the Dupin indicatrix of a surface is an ellipse, then the point M is called elliptical, 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 the same side of the tangent plane. An example 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 a 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. The Dupin indicatrix in this case is two parallel lines (Fig. 207*).

On fig. 208 shows a surface consisting of points in which

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

A loose 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 form of a saddle (oblique plane, one-sheeted hyperboloid, concave surfaces of revolution, etc.).

One surface can have points different types, for example, at the torso surface (Fig. 209) 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 considered section) have extreme values, located in two mutually perpendicular planes.

Such curvatures K 1 = 1/R max. K 2 \u003d 1 / R min are called the main ones, and the values ​​\u200b\u200bof H \u003d (K 1 + K 2) / 2 and K \u003d K 1 K 2 - respectively, the average curvature of the surface and the total (Gaussian) curvature of the surface at the point under consideration. For elliptic points K > 0, hyperbolic K

Setting the plane tangent to the surface on the Monge diagram

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

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

Option a (Fig. 210).

The tangent plane is defined by two tangents t 1 and t 2 drawn at the 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 || x axis. The horizontal projection of the tangent t "2 to the meridian d of the surface β, passing through the point M, will coincide with the horizontal projection of the meridian. To find the frontal projection of the tangent t" 2, the meridional plane γ (γ ∋ M) by rotating around the axis of the surface β is translated into position γ 1 parallel to the plane π 2 . In this case, the point M → M 1 (M "1, M" 1). The projection of the tangent t "2 rarr; t" 2 1 is determined by (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 M ∈ β define a 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: through a point outside the surface consisting of elliptical points, one can draw many planes tangent to the surface. The envelope of these surfaces will be some conical surface. Therefore, if there are no additional indications, then the problem has a set of solutions and in this case is reduced to carrying out conical surfaceγ tangent to the given surface β.

On 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 touch points 1 (1" and 1"), 2 (2" and 2"), 3 (3" and 3") and 4 (4" and 4"). The horizontal projection of the circle - the line of contact between the conical surface and the sphere will be projected into [ 1"2"] To find the points of the ellipse into which this circle is projected onto the frontal plane of projections, we will use the parallels of the sphere.

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

Some of the constructions that need to be done for this are shown in the following example.

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

As in Example 1, consider two solutions. a) point N ∈ β; b) point N ∉ β

Option a (rice 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. To construct it, you must:

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

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

3) draw and tangent t to d at point A.

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

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

* 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).

reaping given surface, necessary:

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

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

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

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

Intersecting lines t 1 , (AS) and t 2 , (BS) define the desired tangent planes α 1 and α 2

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

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

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

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

2) draw a tangent through the point K" 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 a plane γ through the point K and the axis of the surface, the horizontal projection t "2 will coincide 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 position γ 1 || π 2 . In this case, the meridional section by the plane γ will coincide 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. We return the plane γ 1 to its original position, point K" 1 → K "(point S" 1 ≡ S "). The frontal projection of the tangent t" 2 is determined by the points K" and S".

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

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

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

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

the thal projection of the circle d "2 - the neck of the surface of a one-sheeted hyperboloid of revolution; determine the points 1" and 2 at which t "1 and t" 2 intersect one of the surface guides d 1. From 1" and 2" we find 1" and 2", which together with K" determine the frontal projections of the desired lines.

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 correct surface.

In addition to 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:

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 given an analytical expression.

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 area is not simple surface. This necessitates a further generalization of the concept of a surface.

A subset of space in which each point has a neighborhood that is simple surface, is called correct 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 in an appropriate way, 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 and 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 a surface is its normal- unit vector perpendicular to the tangent plane at a given point:

The sign of the normal depends on the choice of coordinates.

The section of the surface by a plane containing the normal of the surface at a given point forms a certain curve, which is called normal section surfaces. The main normal for a 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 an angle with the surface normal θ (\displaystyle \theta ). Then the curvature k (\displaystyle k) curve is related to curvature k n (\displaystyle k_(n)) normal section (with the same tangent) Meunier's formula:

Normal vector coordinates 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 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 a minus sign if the directions of the normals are opposite.

Generally speaking, at every point on the surface there are two perpendicular directions e 1 (\displaystyle e_(1)) and e 2 (\displaystyle e_(2)), in which the normal curvature takes a minimum and maximum value; these directions are called 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) curvature.

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

called the Gaussian curvature, the total curvature, or simply the curvature of the surface. 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.

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

Let we have a surface given by an equation of the form

We introduce the following definition.

Definition 1. A straight line is called a tangent to the surface at some point if it is

tangent to some curve lying on the surface and passing through the point .

Since an infinite number of different curves lying on the surface pass through the point P, there will, in general, be an infinite number of tangents to the surface passing through this point.

Let us introduce the concept of singular and ordinary points of a surface

If at a point all three derivatives are equal to zero or at least one of these derivatives does not exist, then the point M is called a singular point of the surface. If at a point all three derivatives exist and are continuous, and at least one of them is different from zero, then the point M is called an ordinary point of the surface.

Now we can formulate the following theorem.

Theorem. All tangent lines to a given surface (1) at its ordinary point P lie in the same plane.

Proof. Let us consider a certain line L on the surface (Fig. 206) passing through a given point P of the surface. Let the curve under consideration be given by the parametric equations

The tangent to the curve will be tangent to the surface. The equations of this tangent have the form

If expressions (2) are substituted into equation (1), then this equation becomes an identity with respect to t, since curve (2) lies on surface (1). Differentiating it with respect to we get

The projections of this vector depend on - the coordinates of the point Р; note that since the point P is ordinary, these projections at the point P do not vanish at the same time, and therefore

tangent to the curve passing through the point P and lying on the surface. The projections of this vector are calculated on the basis of equations (2) with the value of the parameter t corresponding to the point Р.

Compute scalar product vectors N and which is equal to the sum of the products of the projections of the same name:

Based on equality (3), the expression on the right side is equal to zero, therefore,

It follows from the last equality that the LG vector and the tangent vector to the curve (2) at the point P are perpendicular. The above reasoning is valid for any curve (2) passing through the point P and lying on the surface. Consequently, each tangent to the surface at the point P is perpendicular to the same vector N, and therefore all these tangents lie in the same plane perpendicular to the vector LG. The theorem has been proven.

Definition 2. The plane in which all the tangent lines are located to the lines on the surface passing through its given point P is called the tangent plane to the surface at the point P (Fig. 207).

Note that in special points surface may not have a tangent plane. At such points, the tangent lines to the surface may not lie in the same plane. So, for example, the vertex of a conical surface is a singular point.

The tangents to the conical surface at this point do not lie in the same plane (they themselves form a conical surface).

Let us write the equation of the tangent plane to the surface (1) at an ordinary point. Since this plane is perpendicular to the vector (4), then, consequently, its equation has the form

If the surface equation is given in the form or the tangent plane equation in this case takes the form

Comment. If in formula (6) we set , then this formula will take the form

its right side is the total differential of the function . Consequently, . Thus, the total differential of a function of two variables at the point corresponding to the increments of the independent variables x and y is equal to the corresponding increment of the applicate of the tangent plane to the surface, which is the graph of this function.

Definition 3. A straight line drawn through a point of the surface (1) perpendicular to the tangent plane is called the normal to the surface (Fig. 207).