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Chapter 3: Problem 10

(Contact of Curves and Surfaces.) A curve \(C\) and a surface \(S\), which have a common point \(p\), have contact of order \(\geq n(n\) integer \(\geq 1)\) at \(p\) if there exists a curve \(\bar{C} \subset S\) passing through \(p\) such that \(C\) and \(\bar{C}\) have contact of order \(\geq n\) at \(p\). Prove that a. If \(f(x, y, z)=0\) is a representation of a neighborhood of \(p\) in \(S\) and \(\alpha(t)=(x(t), y(t), z(t))\) is a parametrization of \(C\) in \(p\), with \(\alpha(0)=p\), then \(C\) and \(S\) have contact of order \(\geq n\) if and only if $$ f(x(0), y(0), z(0))=0, \quad \frac{d f}{d t}=0, \ldots, \frac{d^{n} f}{d t^{n}}=0 $$ where the derivatives are computed for \(t=0\). b. If a plane has contact of order \(\geq 2\) with a curve \(C\) at \(p\), then this is the osculating plane of \(C\) at \(p\). c. If a sphere has contact of order \(\geq 3\) with a curve \(C\) at \(p\), and \(\alpha(s)\) is a parametrization by arc length of this curve, with \(\alpha(0)=p\), then the center of the sphere is given by $$ \alpha(0)+\frac{1}{k} n+\frac{k^{\prime}}{k^{2} \tau} b \text {. } $$ Such a sphere is called the osculating sphere of \(C\) at \(p\).

Short Answer

Expert verified
Validation via initial parametrization near point p and derivative zero constraints upto Nth order for condition satisfaction of both osculating planes and spheres.

Step by step solution

01

Contact Definition and Function Representation

Contact of order ( 鈮� n) is defined by 鈭冡菇 passing through p such that C 黏厳 鈮� p. To show this equality for surfaces, start by representing surface S as 饾憮(饾懃, 饾懄, 饾懅)=0 in a neighborhood near p.
02

Parametrization and Simplifications

Parametrize the curve C as 伪(t) = (x(t), y(t), z(t)) near point p, with 伪(0) = p. Substitute 伪(t) into 饾憮( 饾懃(t), 饾懄(t), 饾懅(t) ).
03

Initial Conditions

Consider conditions at 饾憽 = 0. Set 饾憮( 饾懃(0), 饾懄(0), 饾懅(0) ) = 0 and all derivatives upto order 饾憶i.e., 饾惙饾憮 / 饾惙饾憽 = 饾惙^饾憶饾憮 / 饾惙饾憽^饾憶 = 0 at 饾憽 = 0.
04

Plane Contact Order Validation

Investigate if the osculating plane having contact of order 2 鈮� with a curve 饾憹demonstrates the equations 饾拠( 饾懃(饾憽), 饾懄(饾憽), 饾懅(饾憽) = 0 (饾憹 = 0).Thus, validate plane conditions via tangent and normal constraints at 饾憽 = 0.
05

Validating Order 3 Sphere Contact

For a sphere having contact order 鈮� 3 with a parametrized arc length curve near 饾憹, set center 饾憼defined by 饾浖(0) + 1/饾憳(饾憶) + 饾憳'/饾憳^2 饾湉(饾憦) value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametrization of Curves
Parametrization is a method of defining a curve using a parameter, typically denoted as \(t\). This parameter helps to describe the position on the curve as a function of \(t\). If we have a curve \(C\), we can represent it using a parametrization \(\alpha(t) = (x(t), y(t), z(t))\). Here, \(x(t)\), \(y(t)\), and \(z(t)\) are functions that give the coordinates of the points on the curve for different values of \(t\).

For example, let's consider a curve passing through a point \(p\). If the parameter \(t = 0\) corresponds to this point, we can write \(\alpha(0) = p\). By substituting different values of \(t\), we can parametrize the entire curve. This method is essential because it allows us to use calculus and other mathematical tools to analyze the curve.
Osculating Plane
The osculating plane at a point \(p\) of a curve \(C\) is the plane that best approximates the curve near that point. It is called osculating because it 'kisses' or touches the curve more closely than any other plane.

To determine the osculating plane, we need two things: the tangent vector and the normal vector. The tangent vector is the direction of the curve at \(p\), while the normal vector is perpendicular to the tangent.

If a plane has contact of order \(\geq 2\) with a curve at point \(p\), this plane is considered the osculating plane. This means that not only is the curve tangent to the plane, but their centers of curvature also coincide.
Osculating Sphere
An osculating sphere at a point \(p\) of a curve \(C\) is a sphere that best fits the curve near that point. This sphere touches the curve at \(p\) and has a contact called the contact of order \(\geq 3\).

For a curve parametrized by arc length \(\alpha(s)\), the center of the osculating sphere can be found using the formula:
\[ \alpha(0) + \frac{1}{k} n + \frac{k'}{k^2 \tau} b \]
Here, \( k \) is the curvature, \(\tau\) is the torsion, \( n \) is the principal normal, and \( b \) is the binormal vector. These quantities are derived from the Frenet-Serret formulas and describe how the curve bends and twists in space.
Derivatives of Curves
To analyze the behavior of a curve near a point, we use derivatives of the parametrization functions. If \(\alpha(t) = (x(t), y(t), z(t))\) represents a curve, the first derivative \(\alpha'(t)\) gives the tangent vector to the curve. The higher-order derivatives provide more information about the curve's shape.

For instance, the second derivative generally relates to the curvature of the curve, and the third derivative gives information about torsion. Evaluating these derivatives at \(t = 0\) (or any specific point) helps to understand how the curve progresses through that point.

When the derivatives of \(f(x(t), y(t), z(t))\), where \(f\) represents a surface, are zero up to order \(n\) at \(t = 0\), it means the curve and the surface have contact of order \(\geq n\).
Contact Order
The order of contact between a curve and a surface at a point \(p\) is a measure of how closely they follow each other near that point. If a curve and surface have contact of order \(\geq n\), they share not only the point but also the n-th derivatives at that point.

To quantify this, suppose \(S\) is a surface represented by \(f(x,y,z) = 0\) near \(p\), and \(C\) is a curve parametrized by \(\alpha(t) = (x(t), y(t), z(t))\) with \(\alpha(0) = p\). The curve and surface have contact of order \(\geq n\) if
\[ f(x(0), y(0), z(0)) = 0, \quad \frac{df}{dt} = 0, \quad \ldots, \quad \frac{d^n f}{dt^n} = 0 \] evaluated at \( t = 0 \). This implies that not just the positions but also their corresponding directional behaviors align up to the n-th derivative.

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Most popular questions from this chapter

Let \(C \subset S\) be a regular curve in \(S\). Let \(p \in C\) and \(\alpha(s)\) be a parametrization of \(C\) in \(p\) by are length so that \(\alpha(0)=p\). Choose in \(T_{p}(S)\) an orthonormal positive basis \(\\{t, h\\}\), where \(t=\alpha^{\prime}(0)\). The geodesic torsion \(\tau_{g}\) of \(C \subset S\) at \(p\) is defined by $$ \tau_{g}=\left\langle\frac{d N}{d s}(0), h\right\rangle . $$ Prove that a. \(\tau_{g}=\left(k_{1}-k_{2}\right) \cos \varphi \sin \varphi\), where \(\varphi\) is the angle from \(e_{1}\) to \(t\) and \(t\) is the unit tangent vector corresponding to the principal curvature \(k_{1}\). b. If \(\tau\) is the torsion of \(C, n\) is the (principal) normal vector of \(C\) and \(\cos \theta=\langle N, n\rangle\), then $$ \frac{d \theta}{d s}=\tau-\tau_{g} . $$ c. The lines of curvature of \(S\) are characterized by having geodesic torsion identically zero.

Let \(p\) be an elliptic point of a surface \(S\), and let \(r\) and \(r^{\prime}\) be conjugate directions at \(p\). Let \(r\) vary in \(T_{p}(S)\) and show that the minimum of the angle of \(r\) with \(r^{\prime}\) is reached at a unique pair of directions in \(T_{p}(S)\) that are symmetric with respect to the principal directions.

Consider the parametrized surface \(\mathbf{x}(u, v)=\left(\sin u \cos v, \sin u \sin v, \cos u+\log \tan \frac{u}{2}+\varphi(v)\right)\), where \(\varphi\) is a differentiable function. Prove that a. The curves \(v=\) const. are contained in planes which pass through the \(z\) axis and intersect the surface under a constant angle \(\theta\) given by $$ \cos \theta=\frac{\varphi^{\prime}}{\sqrt{1+\left(\varphi^{\prime}\right)^{2}}} $$ Conclude that the curves \(v=\) const. are lines of curvature of the surface. b. The length of the segment of a tangent line to a curve \(v=\) const., determined by its point of tangency and the \(z\) axis, is constantly equal to 1 . Conclude that the curves \(v=\) const. are tractrices (cf. Exercise 6).

(Contact of Order \(\geq 2\) of Surfaces.) Two surfaces \(S\) and \(\bar{S}\), with a common point \(p\), have contact of order \(\geq 2\) at \(p\) if there exist parametrizations \(\mathbf{x}(u, v)\) and \(\overline{\mathbf{x}}(u, v)\) in \(p\) of \(S\) and \(\bar{S}\), respectively, such that $$ \mathbf{x}_{u}=\overline{\mathbf{x}}_{u}, \quad \mathbf{x}_{v}=\overline{\mathbf{x}}_{v}, \quad \mathbf{x}_{u u}=\overline{\mathbf{x}}_{u u}, \quad \mathbf{x}_{u v}=\overline{\mathbf{x}}_{w v}, \quad \mathbf{x}_{v v}=\overline{\mathbf{x}}_{v v} $$ at \(p\). Prove the following: *a. Let \(S\) and \(\bar{S}\) have contact of order \(\geq 2\) at \(p ; \mathbf{x}: U \rightarrow S\) and \(\overline{\mathbf{x}}: U \rightarrow\) \(\bar{S}\) be arbitrary parametrizations in \(p\) of \(S\) and \(\bar{S}\), respectively; and \(f: V \subset R^{3} \rightarrow R\) be a differentiable function in a neighborhood \(V\) of \(p\) in \(R^{3}\). Then the partial derivatives of order \(\leq 2\) of \(f \circ \overline{\mathbf{x}}: U \rightarrow R\) are zero in \(\overline{\mathbf{x}}^{-1}(p)\) if and only if the partial derivatives of order \(\leq 2\) of \(f \circ \mathbf{x}: U \rightarrow R\) are zero in \(\mathbf{x}^{-1}(p)\). *b. Let \(S\) and \(\bar{S}\) have contact of order \(\geq 2\) at \(p\). Let \(z=f(x, y)\), \(z=\bar{f}(x, y)\) be the equations, in a neighborhood of \(p\), of \(S\) and \(\bar{S}\), respectively, where the \(x y\) plane is the common tangent plane at \(p=(0,0)\). Then the function \(f(x, y)-\bar{f}(x, y)\) has all partial derivatives of order \(\leq 2\), at \((0,0)\), equal to zero. c. Let \(p\) be a point in a surface \(S \subset R^{3}\). Let \(O x y z\) be a Cartesian coordinate system for \(R^{3}\) such that \(O=p\) and the \(x y\) plane is the tangent plane of \(S\) at \(p\). Show that the paraboloid $$ z=\frac{1}{2}\left(x^{2} f_{x x}+2 x y f_{x y}+y^{2} f_{y y}\right) $$ obtained by neglecting third- and higher-order terms in the Taylor development around \(p=(0,0)\), has contact of order \(\geq 2\) at \(p\) with \(S\) (the surface (*) is called the osculating paraboloid of \(S\) at \(p\) ). *d. If a paraboloid (the degenerate cases of plane and parabolic cylinder are included) has contact of order \(\geq 2\) with a surface \(S\) at \(p\), then it is the osculating paraboloid of \(S\) at \(p\). e. If two surfaces have contact of order \(\geq 2\) at \(p\), then the osculating paraboloids of \(S\) and \(\bar{S}\) at \(p\) coincide. Conclude that the Gaussian and mean curvatures of \(S\) and \(\bar{S}\) at \(p\) are equal. f. The notion of contact of order \(\geq 2\) is invariant by diffeomorphisms of \(R^{3}\); that is, if \(S\) and \(\bar{S}\) have contact of order \(\geq 2\) at \(p\) and \(\varphi: R^{3} \rightarrow R^{3}\) is a diffeomorphism, then \(\varphi(S)\) and \(\varphi(\bar{S})\) have contact of order \(\geq 2\) at \(\varphi(p)\). g. If \(S\) and \(\bar{S}\) have contact of order \(\geq 2\) at \(p\), then $$ \lim _{r \rightarrow 0} \frac{d}{r^{2}}=0 $$ where \(d\) is the length of the segment cut by the surfaces in a straight line normal to \(T_{p}(S)=T_{p}(\bar{S})\), which is at a distance \(r\) from \(p\).

Let \(\lambda_{1}, \ldots, \lambda_{m}\) be the normal curvatures at \(p \in S\) along directions making angles \(0,2 \pi / m, \ldots,(m-1) 2 \pi / m\) with a principal direction, \(m>2\). Prove that $$ \lambda_{1}+\cdots+\lambda_{m}=m H, $$ where \(H\) is the mean curvature at \(p\).
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