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Юриспунденкция

C U R V E S

Definition and equations of a curve. In ordinary three-dimensional space let us establish a left-handed orthogonal cartesian coordinate system with the same unit of distance for all three axes. In this system any point P has coordinates x, y, z.

A curve may be described qualitatively as the locus of a point moving with one degree of freedom. A curve is also sometimes said to be the locus of a one-parameter family of points or the locus of a single infinity of points.

Definition 1. Let the coordinates x, y, z of a point P be given as single-valued real-valued analytic functions of a real independent variable t on an interval T of t-axis, by equations of the form:

x = x (t), y = y (t), z = z (t). (2.1)

Further suppose that the functions x (t), y (t), z (t) are not all constant on T. Then the locus of the point P, as t varies on the interval T, is a real proper analytic curve C.

Some comments on the foregoing definition will perhaps clarify its meaning. Equations (2.1) are called the parametric equations of the curve C, the parameter being the variable t. We reserve the right to permit the parame-ter t to take on complex values. Moreover, one or more of the coordinates x, y, z may, under suitable conditions, be allowed to be complex. The curve C would in this case be called complex, or perhaps, or suitable conditions, imaginary. To say that a curve is proper means that it does not reduce to a single fixed point , as it would do if the coordinates x, y, z were all constant. It is clear that at an ordinary point of a real proper analytic curve, i. e. , a point where nothing exceptional occurs, the inequality.

x'² + y'² + z'² > 0 (x' ; … ) (2.2)

holds. Any point of such a curve where this inequality fails to hold is called singular, although the singularity may belong to the parametric representation being used for the curve defined as a point-locus, or may belong to the curve itself. A curve, or portion of a curve, which is free of singular points may be called nonsingular. Furthermore, we assume that the interval T is so small that values of the parameter t on the interval T and points (x, y, z) on the curve C are in one-to-one correspondence, so that the parameter t is a coordinate of the corresponding point (x, y, z) on the curve C.

To say that the functions are analytic means, roughly, that they can be expanded into power series. More precisely, this statement means that, at each point t_o within the interval T, each of these functions can be expanded into a Taylor’s series of power of the difference t – t_owhich converges when the absolute value t – t_o is sufficiently small. It would be possible to study differential geometry under the hypothesis that the functions considered possess only a definite, and rather small number of derivatives; but we assume analyticity in the interests of simplicity. So the word “ function” will mean for us “analytic function”, and the word “curve” will mean a real proper nonsingular analytic curve unless the contrary is indicated.

Some examples of parametric equations of curves will now be adduced. First of all, the equations (2.1) may be linear, of the form

x = a + Lt, y = b + mt, z = c + nt (2.3)

in which a, b, c and l, m, n are constants. Then the curve C is a straight line through the fixed points (a, b, c) and with direction cosines proportional to l, m, n. If t is the algebraic distance from the fixed point (a, b, c) to the variable point (x, y, z) on the line then L, m, n are the direction cosines of the line and satisfy the equation

l² + m² + n² = 1 (2.4)

As a second example, equations (2.1) may take the form

x = t, y = t², x = t³ (2.5)

The curve C is then a cubical parabola. This is one form of a twisted cubic which can be defined as the residual intersection of two quadric surfaces that intersect elsewhere in a straight line. Finally, if equations (2.1) have the form

x = a cos t, y = a sin t, z = bt (a> 0, b <C) (2.6)

the curve C is a left-handed circular helix, or machine sorew. This may be described as the locus of a point which revolves around the z – axis at a constant distance a from it and at the same time moves parallel to the z – axis at a rate proportional to the angle t of revolution. If we had supposed b < 0, then the helix would have been right-handed.

A curve can be represented analytically in other ways than by its parametric equations. For example, it is known that one equation in x, y, z represents a surface, and that two independent simultaneous equations in x, y, z, say.

F (x, y, z) = 0, C (x, y, z) = 0 (2.7)

represent the intersection of two surfaces, which is a curve. Equations (2.7) are called implicit equations of this curve. Sometimes it is convenient to represent a curve by implicit equations, when really the curve under consideration is only part of the intersection of the two surfaces represented by the individual equations.

If the implicit equations (2.7) be solved for two of the variables in terms of the third, say for y and z in terms of x, the result can be written in the form

y = y (x), z = z (x). (2.8)

These equations represent the same curve as equations (2.7); and they, or the equations which similarly express any two of the coordinates of a variable point on the curve as functions of the third coordinate, are called explicit equations of the curve. Each of equations (2.8) separately represents a cylinder projecting the curve onto one of the coordinate planes. So equations (2.8) are a special form of equations (2.7) for which the two surfaces are projecting cylinders.

If the first of the parametric equations (2.1) of a curve C be solved for t as a function of x, and if the result is substituted in the remaining two of these equations, the explicit equations (2.8) of the curve C are obtained. From one point of view the explicit equations (2.8) of a curve, when supplemented by identity x = x, are parametric equations

x = x, y = y (x), z = z (x). (2.9)

of the curve, the parameter now being the coordinate x.

I. Learn the following word combination:

Ordinary three-dimensional space, a left-handed orthogonal coordinate system, the same unit of distance, one degree of freedom, the locus of a point, a one-parameter family of curves, a single infinity of points, a real proper analytic space curve, a single-valued real-valued analytic function, a real independent variable, under a suitable condition, to reduce to a single fixed point, the function considered (under consideration), at the same time, in other ways, in terms of, from one point of view, in one-to-one correspondence, fail to hold, a point-locus, under the hypothesis, unless the contrary is indicated, direction cosines, a twisted cubic, a left-handed circular helix, independent simultaneous equations.

II. Form nouns from the verbs using the suffixes:

-ation, -tion, -ion -ment -ence

to direct to establish to differ

to represent to state to occur

to substitute to develop to converge

to intersect to improve to correspond

to consider to move to depend

to vary

to define

III. Form adverbs from the adjectives, using the suffix–ly:

Clear, real, final, absolute, individual, indefinite, independent, convenient, implicit, explicit, proportional, proper, sufficient, simultaneous, separate, special.

IV. Learn the following nouns:

identity, difficulty, reality, simplicity, analyticity, singularity, inequality, infinity.

V. Use “under consideration” or “in question” instead of “considered” in order to express the same idea:

The theorem considered, the figure considered, the problem considered, the function considered, the equation considered, the point considered, the curve considered.

VI. Use “to hold” instead of “to be valid”, “to be true”:

1. This inequality is valid for all cases. 2. This theorem is valid in the case of the uniform convergence. 3. This formula is valid for a single-valued analytic function too. 4. These relations are true under suitable conditions. 5. For a = b = 1 the given property is true.

VII. Use “to fail to” instead of “do not”:

1. I did not solve the problem given by the professor. 2. These properties do not hold for real numbers. 3. We did not expand these functions into power series. 4. He did not prove the theorem correctly. 5. We do not represent this curve by an implicit equation. 6. I did not understand your question. 7. The boy did not add these two numbers correctly.

VIII. Read and translate these sentences, paying attention to the new worlds and word combinations from the text.

1. In a left-handed orthogonal cartesian coordinate system any point has three coordinates. 2. A curve may be defined as the locus of a one-parameter family of points. 3. The coordinates x, y, z are given here as single-valued real-valued analytic functions by the equations of the form x = x (t), y = y (t), z = z (t). 4. These equations are called parametric equations. 5. The parameter may take on complex values. 6. The curve is proper when it does not reduce to a single fixed point. 7. A curve which is free of singular points is called nonsingular. 8. The functions are analytic if they can be expanded into power series. 9. Sometimes a curve may be represented be implicit equations. 10. Two variables in this implicit equation may be solved in terms of the third. 11. The curve under consideration is a real proper analytic space curve.

IX. State the type of these conditional sentences and translate them:

1. A curve is called nonsingular if it is free of singular points. 2. The curve would reduce to a single fixed point if the coordinates x, y, z were all constant. 3. If the equations x = x (t), y = y (t), z = z (t) had taken the form x = t, y = t², z = t², then the curve C would have been a cubic parabola. 4. The curve is called complex, if one or more of the coordinates x, y, z are complex. 5. If “t” is the algebraic distance from the fixed point (x, y, z) on the line, then l, m, n in the given equations are direction cosines. 6. If the endpointes are included, the interval is called closed. 7. If these implicit equations were solved for two of the variables in terms of the third, they could be writtebn in another form. 8. The result would have been written in the form y = y (x), z = z (x) if the implicite quations had been solved for the two of the variables in terms of te third.

X. Change the conditional sentences of type I to those of Type II.

1. If the coordinates of the point satisfy the equation, the point lies on the curve. 2. The curve is called imaginary if one or more of the coordinates are complex. 3. If a is less than 0, then this formula holds. 4. If the equation F (x, y, z) = 0 is homogeneous in x, y, z, it represents a cone. 5. We cannot solve the system of equations unless we eliminate the unknowns. 6. If we suppose that b<0, the helix is right-handed. 7. We obtain the explicit equations of the curve C if we solve one of the parametric equations of the curve for t as a function of x.

XI. Analyze the following sentences and translate them:

1. One factor of the product being equal to 0, the product must be equal to 0. 2. A function being continuous at every point of the set, it is continuous throught the set. 3. The resulting equations are parametric equations of the curve C, the parameter being t. 4. We call such an equation linear, any other type being called non-linear. 5. The equation x – y = 0 being given, we can rewrite it in the for y = x. 6. We study polygons now, this type of geometric figures being very important in studying geometry.

XII. Answer the following questions:

1. What is this text about? 2. In what way may a curve be described? 3. Can you give the definition of a real proper analytic curve? 4. What do we call the equations of the form x = x (t), y = y (t), z = z (t)? 5. What does the letter t denote in these equations? 6. May the parameter t take on complex values? 7. What do the letters x, y, z denote in these equations? 8. In what case is the curve C called complex or imaginary? 9. When do we call the curve C pro- per? 10. When does the curve C reduce to a single point? 11. What point is called singlar? 12. What curve is called nonsinglar? 13. When are coordinates x, y, z analytic? 14. What forms can the equations x = x (t), y = y (t), z = z (t) take? 15. In which case is the curve C a atraight line, a cubic parabola and a left-handed helix? 16. Can a curve be represented analytically? 17. By what equations can it be represented analytically? 18. What equations are called implicit or explicit equations?

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