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.
Let us calculate the scalar product of the 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 the tangent plane may not exist at singular points of the surface. 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 . Hence, . 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).
Let's write the normal equations. Since its direction coincides with the direction of the vector N, then its equations will have the form
Definition. A point lying on a second-order surface given by the general equation (1) with respect to the ODSC 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 second-order surface 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 straight line tangent 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 a second-order surface at a given non-singular point on it lie in the same plane, called the tangent plane to the surface at the point under consideration. The tangent plane equation has
Proof. Let , , be parametric equations of a 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 line with the surface (1) to be double, or for the line to lie entirely on the surface, it is necessary and sufficient that the equality be satisfied:
If at the same time:
Then the point of intersection of the straight line with the surface (1) is double. And if:
Then the line lies entirely 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 , , of the vector satisfy relation (4), which means that the line is tangent to the surface under consideration.
Since the 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; so 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 nonsingular point given on it.
Based on the canonical equations of second-order surfaces, it is easy to compose the equations of tangent planes to an ellipsoid, hyperboloid, etc. at a given point on them.
1). Tangent plane to ellipsoid:
2). Tangent plane to one and two-sheeted hyperboloids:
3). Tangent plane to elliptic and hyperbolic paraboloids:
§ 161. Intersection of a tangent plane with a surface of the second order.
We take a non-singular point of the surface of the second order as the origin of coordinates of the ODSC, 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 touching the surface at the origin should look like: .
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 chosen coordinate system, the surface equation (1) should look like:
Conversely, if , then equation (6) is the equation of the surface passing through the origin of coordinates , and the plane is the tangent plane to this surface at the 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 invariant theory for second-order lines. Equation (7)
This is the second line. By the form of this line, the invariant is , therefore:
For , here are two imaginary intersecting lines.
When - two real intersecting lines.
If , but at least one of the coefficients , , is not equal to zero, then the line of intersection (7) is two coinciding lines.
Finally, if , then the plane
is part of the given surface, and the surface itself breaks up, 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 an 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 intersecting at the point of contact. 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 second-order surface with respect to the ODSC be given by equation (1) and this equation (1) be the equation of a real non-decomposing surface of the second order. Then if ; then all points of the surface are elliptic.
Proof. Let us introduce a new coordinate system , choosing any non-singular point of the given surface as the origin of coordinates and placing the axes and in the plane tangent to the surface at the point . Equation (1) in the new coordinate system is transformed to the form:
Where . Let us calculate the invariant for this equation .
Since the sign does not change during the transition from one ODSC to another, the signs and 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-sheeted hyperboloid and a hyperbolic paraboloid consist of hyperbolic points.
3) The 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 a parabolic cylinder, it is enough to know:
1) a plane of symmetry parallel to the generators of the cylinder;
2) a 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 general equation defines a parabolic cylinder, it can be rewritten as:
Let's pick m so that the plane
would be mutually perpendicular:
With this value m plane
will be a plane of symmetry parallel to the generators of the cylinder.
Plane
will be the tangent plane to the cylinder, perpendicular to the indicated plane of symmetry, and the vector
will be perpendicular to the found tangent plane and directed towards the concavity of the cylinder.
Download from Depositfiles
4. THEORY OF SURFACES.
4.1 EQUATIONS OF SURFACES.
A surface in 3D space can be defined:
1) implicitly: F ( x , y , z ) =0 (4.1)
2) explicitly: z = f ( x , y ) (4.2)
3) parametrically: (4.3)
or: 
(4.3’)
where are scalar arguments 
sometimes called curvilinear coordinates. For example, a sphere 
it is convenient to set in spherical coordinates: 
.
4.2 TANGENT PLANE AND NORMAL TO THE SURFACE.
If the line lies on the surface (4.1), then the coordinates of its points satisfy the surface equation:
Differentiating this identity, we get:
(4.4)
or 
(4.4
’
)
at every point on the curve on the surface. Thus, the gradient vector at non-singular points of the surface (at which the function (4.5) is differentiable and 
) is perpendicular to the tangent vectors to any lines on the surface, i.e. can be used as a normal vector to formulate the equation of the tangent plane at point M 0
(x
0
,
y
0
,
z
0
) surfaces
(4.6)
and as a direction vector in the normal equation:
(4.7)
In the case of an explicit (4.2) assignment of the surface, the equations of the tangent plane and the normal, respectively, take the form:
(4.8)
And 
(4.9)
In the parametric representation of the surface (4.3), the vectors 
lie in the tangent plane and the equation of the tangent plane can be written as:
(4.10)
and their vector product can be taken as the directing normal vector:
and the normal equation can be written as:
(4.11)
Where 
- parameter values corresponding to point M 0
.
In what follows, we confine ourselves to considering only those points of the surface where the vectors
are not equal to zero and are not parallel.
Example 4.1 Compose the equations of the tangent plane and the normal at the point M 0
(1,1,2) to the surface of the paraboloid of revolution 
.
Solution: Since the paraboloid equation is given explicitly, according to (4.8) and (4.9) we need to find 
at point M 0
:
, and at the point M 0 
. Then the equation of the tangent plane at the point M 0 will take the form:
2(x
-1)+2(y
-1)-(z-2)=0 or 2 x
+2
y
-z - 2=0, and the normal equation 
.
Example 4.2 Compose the equations of the tangent plane and the normal at an arbitrary point on the helicoid 
, .
Solution. Here ,
Tangent plane equation:
or
Normal equations:
.
4.3 THE FIRST QUADRATIC FORM OF THE SURFACE.
If the surface is given by the equation
then the curve 
on it can be given by the equation 
(4.12)
Radius vector differential 
along the curve corresponding to the displacement from the point M 0
to a nearby point M, is equal to
(4.13)
Because 
is the differential of the curve arc corresponding to the same displacement), then
(4.14)
Where .
The expression on the right side of (4.14) is called the first quadratic form of the surface and plays a huge role in the theory of surfaces.
Integrating differentialds ranging from t 0 (corresponds to point M 0 ) to t (corresponds to point M), we obtain the length of the corresponding segment of the curve
(4.15)
Knowing the first quadratic form of the surface, one can find not only the lengths, but also the angles between the curves.
If du
,
dv
are the differentials of curvilinear coordinates corresponding to an infinitesimal displacement along one curve, and 
— on the other hand, then, taking into account (4.13):
(4.16)
Using the formula
(4.17)
the first quadratic form makes it possible to calculate the area of a region 
surfaces.
Example 4.3 On a helicoid, find the length of the helix 
between two points.
Solution. Because on a helix 
, That . Find at a point 
the first quadratic form. Denoting andv
=
t
,
we obtain the equation of this helix in the form . Quadratic shape:
= - the first quadratic form.
Here . In formula (4.15) in this case 
and arc length:
=
4.4 SECOND QUADRATIC FORM OF THE SURFACE.
Denote 
- unit normal vector to the surface 
:
(4.18)
.
(4.23)
A line on a surface is called a line of curvature if its direction at each point is the principal direction.
4.6 THE CONCEPT OF GEODETIC LINES ON THE SURFACE.
Definition 4.1 . A curve on a surface is called a geodesic if its principal normal is 
at every point where the curvature is nonzero, coincides with the normal 
to the surface.
Through each point of the surface in any direction passes, and only one geodesic. On a sphere, for example, great circles are geodesics.
A parametrization of a surface is called semi-geodesic if one family of coordinate lines consists of geodesics and the other is orthogonal to it. For example, on the sphere meridians (geodesics) and parallels.
A geodesic on a sufficiently small segment is the shortest among all curves close to it connecting the same points.
A surface is defined as a set of points whose coordinates satisfy a certain type of equation:
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 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:
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 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 is the unit vector perpendicular to the tangent plane at 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 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:
k n = ± k cos θ (\displaystyle k_(n)=\pm k\,\cos \,\theta )The coordinates of the normal vector for different ways of specifying the surface 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 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) = | 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 the minimum and maximum values; 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:
K = κ 1 κ 2 (\displaystyle K=\kappa _(1)\kappa _(2)) 1°1°. Equations of the tangent plane and the normal for the case of an explicit specification of the surface.
Consider one of the geometric applications of partial derivatives of a function of two variables. Let the function z = f(x;y ) differentiable at a point (x0; at 0) some area DÎ R2. Let's cut the surface S , depicting function z, planes x = x 0 And y = y 0(Fig. 11).
Plane X = x0 crosses the surface S along some line z 0 (y ), whose equation is obtained by substitution into the expression of the original function z==f(x;y ) instead of X numbers x 0 . Dot M 0 (x 0 ;y0,f(x 0 ;y 0)) belongs to the curve z 0 (y ). Due to the differentiable function z at the point M 0 function z 0 (y ) is also differentiable at the point y = y 0 . Therefore, at this point in the plane x = x 0 to the curve z 0 (y ) tangent can be drawn l 1 .
Carrying out similar reasoning for the section at = y 0 , construct a tangent l 2 to the curve z 0 (x ) at the point X = x 0 - Direct 1 1 And 1 2 define a plane called tangent plane to the surface S at the point M 0 .
Let's make an equation for it. Since the plane passes through the point Mo(x 0 ;y0;z0), then its equation can be written as
A (x - ho) + B (y - yo) + C (z - zo) \u003d 0,
which can be rewritten like this:
z -z 0 \u003d A 1 (x - x 0) + B 1 (y - y 0) (1)
(dividing the equation by -C and denoting 
).
Let's find A 1 and B1.
Tangent Equations 1 1 And 1 2 look like
respectively.
Tangent l 1 lies in the plane a , hence the coordinates of all points l 1 satisfy equation (1). This fact can be written as a system
Resolving this system with respect to B 1 , we obtain that . Carrying out similar reasoning for the tangent l 3, it is easy to establish that .
Substituting the values A 1 and B 1 into equation (1), we obtain the desired equation of the tangent plane:
A line passing through a point M 0 and perpendicular to the tangent plane constructed at this point on the surface is called its normal.
Using the condition of perpendicularity of a line and a plane, it is easy to obtain the canonical equations of the normal:
Comment. The formulas for the tangent plane and the normal to the surface are obtained for ordinary, i.e., not singular, points on the surface. Dot M 0 surface is called special, if at this point all partial derivatives are equal to zero or at least one of them does not exist. We do not consider such points.
Example. Write the equations of the tangent plane and the normal to the surface at its point M(2; -1; 1).
Solution. Find the partial derivatives of this function and their values at the point M
Hence, applying formulas (2) and (3), we will have: z-1=2(x-2)+2(y+1) or 2x+2y-z-1=0- tangent plane equation and 
are the normal equations.
2°. Tangent plane and normal equations for the case of implicit surface specification.
If the surface S given by the equation F(x; y;z)= 0, then equations (2) and (3), taking into account the fact that partial derivatives can be found as derivatives of an implicit function.