Handbook of mathematics for engineers and scienteists part 59 ppt

374 DIFFERENTIAL GEOMETRY 9.1.1-4. Asymptotes. A straight line is called an asymptote of a curve Γ if the distance from a point M(x, y)of the curve to this straight line tends to zero as x 2 +y 2 →∞. The limit position of the tangent to a regular point of the curve is an asymptote; the converse assertion is generally not true. For a curve given explicitly as y = f(x), vertical asymptotes are determined as points of discontinuity of the function y = f(x), while horizontal and skew asymptotes have the form y = kx + b,where k = lim t→∞ f(x) x , b = lim t→∞ [f(x)–kx]; both limits must be finite. Example 12. Let us find the asymptotes of the curve y = x 3 /(x 2 + 1). Since the limits k = lim t→∞ f(x) x = lim t→∞ x 3 x(x 2 + 1) = 1, b = lim t→∞ [f(x)–kx] = lim t→∞  x 3 x 2 + 1 – x  = lim t→∞  –x x 2 + 1  = 0 exist, the asymptote is given by the equation y = x. To find an asymptote of a parametrically defined curve x = x(t), y = y(t), one should find the values t = t i for which x(t) →∞or y(t) →∞. If x(t i )=∞ but y(t i )=c ≠ ∞, then the straight line y = c is a horizontal asymptote. If y(t i )=∞ but x(t i )=a ≠ ∞, then the straight line x = a is a vertical asymptote. If x(t i )=∞ and y(t i )=∞, then one should calculate the following two limits: k = lim t→t i y(t) x(t) and b = lim t→t i [y(t)–kx(t)]. (9.1.1.4) If both limits exist, then the curve has the asymptote y = kx + b. Example 13. Let us find the asymptote of the Folium of Descartes x = 3at t 3 + 1 , y = 3at 2 t 3 + 1 (–∞ ≤ t ≤ ∞). Since x(–1)=∞ and y(–1)=∞, one should use formulas (9.1.1.4), k = lim t→–1 y(t) x(t) = lim t→–1 3at 2 t 3 + 1 3at t 3 + 1 = lim t→–1 t =–1, b = lim t→–1 [y(t)–kx(t)] = lim t→–1  3at 2 t 3 + 1 + 3at t 3 + 1  = lim t→–1 3at(t + 1) (t + 1)(t 2 – t + 1) =–a, which imply that the asymptote is given by the equation y =–x – a. 9.1. THEORY OF CURVES 375 Suppose that the function F(x, y) in the equation F (x, y)=0 is a polynomial in the variables x and y. We choose the terms of the highest order in F (x, y). By Φ(x, y)we denote the set of highest-order terms and solve the equation for the variables x and y: x = ϕ(y), y = ψ(x). The values y i = c for which x = ∞ give the horizontal asymptotes y = c;thevaluesx i = a for which y = ∞ give the horizontal asymptotes x = a. To find skew asymptotes, one should substitute the expression y = kx + b into F (x, y). We write the resulting polynomial F (x, kx + b)as F (x, kx + b)=f 1 (k, b)x n + f 2 (k, b)x n–1 + If the system of equations f 1 (k, b)=0, f 2 (k, b)=0 is consistent, then its solutions k, b are the parameters of the asymptotes y = kx + b. 9.1.1-5. Osculating circle. The osculating circle (circle of curvature) of a curve Γ at a point M 0 is defined to be the limit position of the circle passing through M 0 and two neighboring points M 1 and M 2 of the curve as M 1 → M 0 and M 2 → M 0 (Fig. 9.10). C M M M 2 1 0 Figure 9.10. The osculating circle. The center of this circle (the center of curvature of the curve Γ at the point M 1 ) is called the center of the osculating circle and lies on the normal to this curve (Fig. 9.10). The coordinates of the center of curvature can be found by the following formulas: for a curve defined explicitly, x c = x 0 – y  x  1 +(y  x ) 2  y  xx , y c = y 0 + 1 +(y  x ) 2 y  xx ; for a curve defined implicitly, x c = x 0 – F x  F x 2 + F y 2  2F xy F x F y – F 2 x F yy – F 2 y F xx , y c = y 0 + F y  F x 2 + F y 2  2F xy F x F y – F 2 x F yy – F 2 y F xx ; 376 DIFFERENTIAL GEOMETRY for a curve defined parametrically, x c = x 0 – y  t  (x  t ) 2 +(y  t ) 2  x  t y  tt – y  t x  tt , y c = y 0 + x  t  (x  t ) 2 +(y  t ) 2  x  t y  tt – y  t x  tt ; for a curve in polar coordinates, x c = r 0 cos ϕ 0 –  r 2 0 +  r  ϕ  2  r 0 cos ϕ 0 + r  ϕ sin ϕ 0  r 2 0 + 2  r  ϕ  2 – r 0 r  ϕϕ , y c = r 0 sin ϕ 0 –  r 2 0 +  r  ϕ  2  r 0 sin ϕ 0 – r  ϕ cos ϕ 0  r 2 0 + 2  r  ϕ  2 – r 0 r  ϕϕ , x 0 = r 0 cos ϕ 0 , y 0 = r 0 sin ϕ 0 , where all derivatives are evaluated at x = x 0 , y = y 0 , t = t 0 ,andϕ = ϕ 0 . The radius of the osculating circle is called the radius of curvature of the curve at the point M 0 (x 0 , y 0 ); its length in a Cartesian coordinate system is ρ =  1 +(y  x ) 2  3/2   y  xx   =  F x 2 + F y 2  3/2   2F xy F x F y – F 2 x F yy – F 2 y F xx   =  x  t  2 +  y  t  2  3/2   x  t y  tt – y  t x  tt   , and in the polar coordinate system it is ρ =  r 2 0 +  r  ϕ  2  3/2   r 2 0 + 2  r  ϕ  2 – r 0 r  ϕϕ   , where all derivatives are evaluated at x = x 0 , y = y 0 , t = t 0 ,andϕ = ϕ 0 . 9.1.1-6. Curvature of plane curves. The limit ratio of the tangent rotation angle Δϕ to the corresponding arc length Δs of the curve Γ as Δs → 0 (Fig. 9.11), k = lim Δs→0    Δϕ Δs    , is called the curvature of Γ at the point M 1 . M 1 α α+dα d=αφΔ Δs X Y Figure 9.11. The curvature of the curve. 9.1. THEORY OF CURVES 377 The curvature and the radius of curvature are reciprocal quantities, k = 1 ρ . The more bent a curve is near a point, the larger k is and the smaller ρ is at this point. For a circle of radius a, the radius of curvature is ρ = a and the curvature is k = 1/a (they are constant at all points of the circle); for a straight line, ρ = ∞ and k = 0; for all other curves, the curvature varies from point to point. Remark. All points of inflection are points of zero curvature. 9.1.1-7. Fr ´ enet formulas. To each point M of a plane curve, one can naturally assign a local coordinate system. The role of the origin O is played by point M itself, and the role of the axes OX and OY are played by the tangent and normal at this point. The unit tangent and normal vectors to the curve are usually denoted by t and n, respectively (Fig. 9.12). M t n Figure 9.12. The unit tangent t and normal n vectors to the curve. Suppose that the arc length is taken as a (natural) parameter on the curve: r = r(s); then the Fr ´ enet formulas t  s = kn, n  s =–kt hold, where k is the curvature of the curve. With first-order accuracy, the Fr ´ enet formulas determine the rotation of the vectors t and n when translated along the curve to a close point, s → s + Δs. 9.1.1-8. Envelope of a family of curves. A one-parameter family of curves is the set of curves defined by the equation F (x, y,C)=0,(9.1.1.5) which is called the equation of the family.HereC is a parameter varying in a certain range, C 1 ≤ C ≤ C 2 ; in particular, the range can be –∞ ≤ C + ∞. A curve that is tangent at each point to some curve of a one-parameter family of curves (9.1.1.5) is called the envelope of the family. The point of tangency of the envelope to a curve of the family is called a characteristic point of the curve of the family (Fig. 9.13). Figure 9.13. The envelope of the family of curves. 378 DIFFERENTIAL GEOMETRY The equation of the envelope of a one-parameter family is obtained by elimination of the parameter C from the system of equations F (x, y,C)=0, ∂F(x, y, C) ∂C = 0, (9.1.1.6) which determines the discriminant curve of this family. The discriminant curve (9.1.1.6) of a one-parameter family is an envelope if it does not consist of singular points of the curves. X Y Figure 9.14. The envelope of the family of semicubical parabolas. Example 14. Consider the family of semicubical parabolas (Fig. 9.14) 3(y – C) 2 – 2(x – C) 3 = 0. Differentiating with respect to the parameter C, we obtain y – C –(x – C) 2 = 0. Solving the system 3(y – C) 2 – 2(x – C) 3 = 0, y – C –(x – C) 2 = 0, we obtain x – C = 0, y – C = 0; x – C = 2 3 , y – C = 4 9 . Eliminating the parameter C, we see that the discriminant curve splits into the pair of straight lines x = y and x – y = 2/9. Only the second of these two straight lines is an envelope, because the first straight line is the locus of singular points. 9.1.1-9. Evolute and evolvent. The locus of centers of curvature of a curve is called its evolute.Ifacurveisdefined via a natural parameter s, then the position vector of a point of the evolute of a plane curve p can be expressed in terms of the radius vector r of a point of this curve, the normal vector n, and its radius of curvature ρ as follows: p = r + ρn. To obtain the equation of the evolute, it also suffices to treat x c and y c in the equations for the coordinates of the center of curvature as the current coordinates of the evolute. 9.1. THEORY OF CURVES 379 Geometric properties of the evolute: 1. The normals to the original curve coincide with the tangents to the evolute at the corresponding points. 2. If the radius of curvature ρ varies monotonically on a given part of the curve, then its increment is equal to the distance passed by the center of curvature along the evolute. 3. At the points of extremum of the radius of curvature ρ, the evolute has a cusp of the first kind. 4. Since the radius of curvature ρ is always positive, any point of the evolute lies on a normal to the curve on the concave side. 5. The evolute is the envelope of the family of normals to the original curve. A curve that intersects all curves of a family at the right angle is called an orthogonal trajectory of a one-parameter family of curves. A trajectory orthogonal to tangents to a given curve is called an evolvent of this curve. If a curve is defined via its natural parameter s, then the vector equation of its evolvent has the form p = r +(s 0 – s)t, where s 0 is an arbitrary constant. Basic properties of the evolvent: 1. The tangent to the original curve at each point is the normal to the evolvent at the corresponding point. 2. The distance between the corresponding points of two evolvents of a given curve is constant. 3. For s = s 0 , the evolvent has cusps of the fi rst kind. The evolute and the evolvent are related to each other. The original curve is the evolvent of its evolute. The converse assertion is also true; i.e., the original curve is the evolute of its evolvent. The normal to the evolvent is tangent to the evolute. 9.1.2. Space Curves 9.1.2-1. Regular points of space curve. A space curve Γ is in general determined parametrically or in vector form by the equations x = x(t), y = y(t), z = z(t)orr = r(t)=x(t)i + y(t)j + z(t)k, where i, j,andk are the unit vectors (see (4.5.2.2)), t is an arbitrary parameter (t [t 1 , t 2 ]), and t 1 and t 2 can be –∞ and +∞, respectively. A point M(x(t), y(t), z(t)) is said to be regular if the functions x(t), y(t), and z(t)have continuous first derivatives in a sufficiently small neighborhood of this point and these derivatives are not simultaneously zero, i.e., if dr/dt ≠ 0. If the functions x(t), y(t), and z(t) have continuous derivatives with respect to t and dr/dt ≠ 0 for all t [t 1 , t 2 ], then Γ is a regular arc. For the parameter t it is convenient to take the arc length s, that is, the length of the arc from a point M 0 (x(t 0 ), y(t 0 ), z(t 0 )) to M 1 (x(t 1 ), y(t 1 ), z(t 1 )), s =  Γ ds =  Γ √ dr ⋅ dr =  t 1 t 0  (x  t ) 2 +(y  t ) 2 +(z  t ) 2 dt.(9.1.2.1) 380 DIFFERENTIAL GEOMETRY The sign of ds is chosen arbitrarily, and it determines the positive sense of the curve and the tangent. A space curve can also be defined as the intersection of two surfaces F 1 (x, y, z)=0, F 2 (x, y, z)=0.(9.1.2.2) For a curve determined as the intersection of two planes, a point M 0 is regular if the vectors ∇F 1 and ∇F 2 are not linearly dependent at this point. 9.1.2-2. Tangents and normals. A straight line is called the tangent to a curve Γ at a regular point M 0 if it is the limit position of the secant passing through M 0 and a point M 1 infinitely approaching the point M 0 . At a regular point M 0 , the equation of the tangent has the form r = r 0 + λr  t (t 0 ), (9.1.2.3) where λ is a variable parameter. Eliminating the parameter λ from (9.1.2.3), we obtain the canonical equation x – x 0 x  t (t 0 ) = y – y 0 y  t (t 0 ) = z – z 0 z  t (t 0 ) of the tangent. The equation of the tangent at a point M 0 of the curve Γ obtained as the intersec- tion (9.1.2.2) of two planes is x – x 0 (F 1 ) y (F 2 ) z –(F 1 ) z (F 2 ) y = y – y 0 (F 1 ) z (F 2 ) x –(F 1 ) x (F 2 ) z = z – z 0 (F 1 ) x (F 2 ) y –(F 1 ) y (F 2 ) x , where all derivatives are evaluated at x = x 0 and y = y 0 . A perpendicular to the tangent at the point of tangency is called a normal to a space curve. Obviously, at each point of the curve, there are infinitely many normals that form the plane perpendicular to the tangent. The plane passing through the point of tangency and perpendicular to the tangent is called the normal plane (Fig. 9.15). r t Figure 9.15. The normal plane. . larger k is and the smaller ρ is at this point. For a circle of radius a, the radius of curvature is ρ = a and the curvature is k = 1/a (they are constant at all points of the circle); for a straight. suffices to treat x c and y c in the equations for the coordinates of the center of curvature as the current coordinates of the evolute. 9.1. THEORY OF CURVES 379 Geometric properties of the evolute: 1 center of this circle (the center of curvature of the curve Γ at the point M 1 ) is called the center of the osculating circle and lies on the normal to this curve (Fig. 9.10). The coordinates of