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In this work, we study the canal surfaces foliated by pseudo spheres S12 along a Frenet curve in terms of their Gauss maps in Minkowski 3-space. Such kind of surfaces with pointwise 1-type Gauss maps are classified completely. For example, the canal surface with proper pointwise 1-type Gauss map of the first kind if and only if it is a part of a minimal surface of revolution.

International audience; In this paper we present a simple method to create general 3D digital surfaces of revolution based on a 2D implicit curve of revolution (therefore not limited to a circle) and a hand-drawn generatrix. Our method can handle any sequence of Euclidean 2D points, that represents a curve, as generatrix. One can choose the topology of the surface that may have 1-tunnels, 0-tunnels or no tunnels with applications in 3D printing for instance. An online tool that illustrates the method is proposed.

In this paper, we investigate the surfaces of revolution with light-like axis satisfying some equation in terms of a position vector field and the Laplacian with respect to the non-degenerate third fundamental form in Minkowski 3-space. As a result, we give some special example of the surfaces of revolution with light-like axis.

This paper proposes a practical evaluation approach to form deviation for surface of revolution based on discrete point coordinate data. An approximate curve composed of circular arc segments decided from an optimal interpolation process is accepted for computing the form deviation at measured points instead of the generating line of the surface to be inspected. This operation significantly simplifies the algorithm and improves the efficiency in the evaluation calculation. At the same time, it is also guaranteed that the difference of both form deviation values relative to the approximate surface or to the original surface is restricted within the specified accuracy in the circular arc interpolation operation. The effectiveness and efficiency of the approach are verified through a practical example of the surface of revolution with a generating line in the form of cubic Bézier curve.

In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic $3$-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution., Comment: 22 pages, 5 figures

We construct three kinds of complete embedded singly-peri-odic minimal surfaces in H 2 R. The rst one is a 1-parameter family ofminimal surfaces which is asymptotic to a horizontal plane and a verticalplane; the second one is a 2-parameter family of minimal surfaces whichhas a fundamental piece of nite total curvature and is asymptotic to a nite number of vertical planes; the last one is a 2-parameter family ofminimal surfaces which ll H 2 R by nite Scherk’s towers. 1. IntroductionIn this paper we construct complete embedded singly-periodic minimal sur-faces in H 2 R. The study of minimal surfaces in H 2 R was initiated by Nelliand Rosenberg [10, 12]. By several mathematicians, some interesting examplesare found as follows: the catenoid that is a surface of revolution about thevertical R-axis; the helicoid that is ruled by the horizontal geodesic; the Rie-mann type minimal surface that is foliated by horizontal circles and geodesics;the Jenkins-Serrin type minimal surface that is a minimal graph over a do-main bounded by an ideal polygon and is asymptotic to vertical planes; the

We study surfaces of revolution in the three dimensional Euclidean space with two distinct axes of revolution. As a result, we prove that if a connected surface in the three dimensional Euclidean space admits two distinct axes of revolution, then it is either a sphere or a plane.

The distance function to surfaces in three dimensions plays a key role in many geometric modeling applications such as medial axis approximations, surface reconstructions, offset computations and feature extractions among others. In many cases, the distance function induced by the surface can be approximated by the distance function induced by a discrete sample of the surface. The critical points of the distance functions are known to be closely related to the topology of the sets inducing them. However, no earlier theoretical result has found a link between topological properties of a geometric object and critical points of the distance to a discrete sample of it. We provide this link by showing that the critical points of the distance function induced by a discrete sample of a surface fall into two disjoint classes: those that lie very close to the surface and those that are near its medial axis. This closeness is precisely quantified and is shown to depend on the sampling density. It turns out that critical points near the medial axis can be used to extract topological information about the sampled surface. Based on this, we provide a new flow-complex-based surface reconstruction algorithm that, given a tight ε-sample of a surface, approximates the surface geometrically, both in distance and normals, and captures its topology. Furthermore, we show that the same algorithm can be used for curve reconstruction.

We describe some special surfaces in pseudo-Galilean spaces such as helical surfaces, ruled screw surfaces, surfaces of revolution and in particular tori of revolution. We define special surfaces and find their main properties.

In this paper a procedure for determining a plane section of the surface of revolution was described. The surface of revolution is given by an axis of the revolution and a meridian which is coplanar with the axis. The axis of revolution is parallel to the z axis of the coordinate system. The intersecting plane is given by its three points; these are the points where the plane intersects axes x, y and z. For the determination of a plane section of the surface of revolution we can use horizontal planes which intersect the surface of revolution on parallel circles and each auxiliary plane intersects the intersecting plane on the horizontal line. Two points of the intersecting space curve are given as intersecting points between the horizontal line and the circle and new points of the intersecting curve have been determined by using several auxiliary planes.

Isophote of a surface consists of a loci of surface points whose normal vectors form a constant angle with a given fixed vector. It also serves as a silhouette curve when the constant angle is given as π/2. We present efficient and robust algorithms to compute isophotes of a surface of revolution and a canal surface. For the two kinds of surfaces, each point on the isophote is derived by a closed-form solution. To find each connected component in the isophote, we utilize the feature of surface normals. Both surfaces are decomposed into a set of circles, where the surface normal vectors at points on each circle construct a cone. The vectors which form a constant angle with given fixed vector construct another cone. We compute the parametric range of the connected component of the isophote by computing the parametric values of the surface which derive the tangential intersection of these two cones.

In this paper the space intersecting curve between two surfaces of revolution with parallel axes of surfaces have been determined. Two mathematical models for determination of intersecting curve between two surfaces of revolution have been formed: auxiliary planes have been used in the first mathematical model and auxiliary spheres have been used in the second model (Obradovic 2000). In the first case each auxiliary plane intersected with each surface of revolution on circle and two points of intersecting curve are obtained as intersecting points between these two circles. In the second case centres of two locks of auxiliary spheres are put on axes of surfaces of revolution (centre of first lock is on axis of the first surface of revolution and centre of second lock is on axis of the second surface of revolution) on saine z coordinate (when axes of surfaces of revolution are parallel with z axis of coordinate system). First lock sphere intersects the first surface of revolution on w1 parallels and second lock corresponding sphere intersects the second surface of revolution on w2 circles. It is possible to find a relationship that for selected radius of the first lock sphere can determine the radius of second lock sphere and real points of intersecting curve have been determined by use of these two spheres. The points of intersecting curve between two surfaces of revolution are obtained by intersection between w1 circles from the first surface with w2 circles from the second surface (Obradovic 2000).

Results of number of geometric operations (often used in technical practise, as e.g. the operation of blending) are in many cases surfaces described implicitly. Then it is a challenging task to recognize the type of the obtained surface, find its characteristics and for the rational surfaces compute also their parameterizations. In this contribution we will focus on surfaces of revolution. These objects, widely used in geometric modelling, are generated by rotating a generatrix around a given axis. If the generatrix is an algebraic curve then so is also the resulting surface, described uniquely by a polynomial which can be found by some well-established implicitation technique. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is rotational or not. Motivated by this, our goal is to formulate a simple and efficient algorithm whose input is a polynomial with the coefficients from some subfield of $\mathbb{R}$ and the output is the answer whether the shape is a surface of revolution. In the affirmative case we also find the equations of its axis and generatrix. Furthermore, we investigate the problem of rationality and unirationality of surfaces of revolution and show that this question can be efficiently answered discussing the rationality of a certain associated planar curve.

In this paper we prove that two lines bounding an immersed minimal surface in a slab in R3 homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane.

An equation is derived describing small-amplitude vibrations of an arbitrary curved diaphragm, whose surface is considered as a two-dimensional Riemannian space. The derivation is based on the variational principle, from which the motion equation and conservation law follow in a form invariant with respect to arbitrary transformations of coordinates on the diaphragm surface. It has been shown that the wave equation, along with the two-dimensional Laplace-Beltrami operator, includes an additional term proportional to the scalar curvature of the diaphragm surface. As an example, the equations are considered for a spherical diaphragm and a catenoid-shaped diaphragm with a minimal surface of revolution.

The crista ampullaris is the epithelium at the end of the semicircular canals in the inner ear of vertebrates, which contains the sensory cells involved in the transduction of the rotational head movements into neuronal activity. The crista surface has the form of a saddle, or a pair of saddles separated by a crux, depending on the species and the canal considered. In birds, it was described as a catenoid by Landolt et al. (J Comp Neurol 159(2):257---287, doi:10.1002/cne.901590207, 1972). In the present work, we establish that this particular form results from principles of invariance maximization and energy minimization. The formulation of the invariance principle was inspired by Takumida (Biol Sci Space 15(4):356---358, 2001). More precisely, we suppose that in functional conditions, the equations of linear elasticity are valid, and we assume that in a certain domain of the cupula, in proximity of the crista surface, (1) the stress tensor of the deformed cupula is invariant under the gradient of the pressure, (2) the dissipation of energy is minimum. Then, we deduce that in this domain the crista surface is a minimal surface and that it must be either a planar, or helicoidal Scherk surface, or a piece of catenoid, which is the unique minimal surface of revolution. If we add the hypothesis that the direction of invariance of the stress tensor is unique and that a bilateral symmetry of the crista exists, only the catenoid subsists. This finding has important consequences for further functional modeling of the role of the vestibular system in head motion detection and spatial orientation.

Some non-coercive variational integrals are considered, including the classical time-of-transit functional arising in the problem of the brachistochrone, and the area functional in the problem of the minimal surface of revolution. A minimizer is constructed by means of the direct method. More precisely, each admissible curve is replaced by its convex envelope, and the functional is shown to decrease. Hence, there exists a minimizing sequence made up of convex curves, which in turn possesses a locally uniformly converging subsequence. The limiting curve is a minimizer because the functionals under consideration are continuous under such a convergence.

The vertical impact problem is considered for a body of revolution immersed in an ideal incompressible fluid bounded from below by a bottom in the shape of a surface of revolution. For a certain class of bodies it is proved that separation begins on the intersection between the wetted surface of the body and the meridional plane in which the shock impulse is located. As shown by the examples of spindle-shaped surfaces of revolution and an ellipsoid of revolution, separation can take place at one of two points on the boundary of the wetted surface of the body, one farther from and the other nearer to the point of application of the impulse.

The paper introduces two algorithms for the approximation of a digitized set of points from surfaces of revolution. In the first algorithm it is assumed that the rotation axis is known, in the second one the rotational axis is determined additionally.

In this paper, first, we provide a resultant-based implicitization method for revolution surfaces, generated by non necessarily rational curves. Secondly, we analyze the offsetting problem for revolution surfaces, proving that the offsetting and the revolution constructions are commutative. Finally, as a consequence of this, the (total and partial) degree formulas for the generic offset to an irreducible plane curve, given in our previous papers, are extended to the case of offsets to surfaces of revolution.

Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. However, many different extensions of the shortest path techniques to 3D have been previously proposed in which the 3D object is segmented via a collection of shortest paths rather than a minimal surface, leading to a solution which bears an uncertain relationship to the true minimal surface. Specifically, there is no guarantee that a minimal path between points on two closed contours will lie on the minimal surface joining these contours. We observe that an elegant solution to the computation of a minimal surface on a cellular complex (e.g., a 3D lattice) was given by Sullivan [47]. Sullivan showed that the discrete minimal surface connecting one or more closed contours may be found efficiently by solving a Minimum-cost Circulation Network Flow (MCNF) problem. In this work, we detail why a minimal surface properly extends a shortest path (in the context of a boundary) to three dimensions, present Sullivan's solution to this minimal surface problem via an MCNF calculation, and demonstrate the use of these minimal surfaces on the segmentation of image data.

The paper deals with volume-preserving approximations of surfaces of revolution. The approximating surfaces are generated only by line segments and circular arcs of a constant radius r . Further, for r > 0 , the approximating surfaces are visually C1 surfaces. For r = 0 , developable C0 surfaces are obtained (consisting of either congruent cylinders or frustums of cones of revolution). Two algorithms are discussed. The first algorithm preserves the volume enclosed by a surface of revolution and the planes of every two latitude circles; the approximating surface is, however, no longer a surface of revolution. The second algorithm applies an approximating surface of revolution; however, the volume preservation no longer holds globally.

We study a boundary value problem for Willmore surfaces of revolution, where the position and the mean curvature H = 0 are prescribed as boundary data. The latter is a natural datum when considering critical points of the Willmore functional in classes of functions where only the position at the boundary is fixed. For specific boundary positions, catenoids and a suitable part of the Clifford torus are explicit solutions. Numerical experiments, however, suggest a much richer bifurcation diagram. In the present paper we verify analytically some properties of the expected bifurcation diagram. Furthermore, we present a finite element method which allows the calculation of critical points of the Willmore functional irrespective of their stability properties.

A degree n rational plane curve rotating about an axis in the plane creates a degree 2n rational surface. Two formulas are given to generate 2n moving planes that follow the surface. These 2n moving planes lead to a 2n × 2n implicitization determinant that manifests the geometric revolution algebraically in two aspects. Firstly the moving planes are constructed by successively shifting terms of polynomials from one column to another of a spawning 3 × 3 determinant. Secondly the right half of the 2n × 2n implicitization determinant is almost an n-row rotation of the left half. As an aside, it is observed that rational parametrizations of a surface of revolution due to a symmetric rational generatrix must be improper.

A substantive part of the recent activity in the field of minimal surface theory has been the construction of new complete minimal surfaces immersed in E3. One approach in constructing new examples is to increase the genus of known minimal surfaces. In this paper, we do precisely this for certain minimal surfaces of finite total curvature whose ends are asymptotic to catenoids. We prove existence of surfaces of positive genus based on those in genus zero, with the feature that these higher genus examples maintain all the symmetry of their genus-zero counterparts. In these proofs we use the conjugate minimal surface construction and the maximum principle for minimal surfaces.

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles and that satisfy a Weingarten condition of type $a H+b K=c$, where $a,b$ and $c$ are constant and $H$ and $K$ denote the mean curvature and the Gauss curvature respectively. We prove that a such surface must be a surface of revolution, a Riemann minimal surface or a generalized cone., 17 pages

In this poster, we present a method for computing a piecewise linear approximation to the surface swept by a moving rotating elliptic cylinder. Our method is a generalization of the imprint point method we developed for computing points on a surface of revolution [1]. The method is based on on identifying grazing points on the surface of revolution at a sequence of positions, and for each position connecting the grazing points with a piecewise linear curve. A collection of grazing curves is joined to approximate the swept surface and stitched into a solid model. Previously this method has been tested on cylinders, toruses, and cones.We reduce the problem of finding grazing points on the elliptic cylinder to that of finding points on an ellipse by slicing the elliptic cylinder with planes perpendicular to the tool axis. The method for finding the grazing points an a circular slice of a surface of revolution works because the normals to the surface along each circle pass through the axis of revolution. This allowed us to express the direction of motion of a point on the surface as the sum of the motion of a point on the axis plus a rotation around that point on the axis.This simple method for finding grazing points fails for an elliptic cylinder because the normals of a slice of the elliptic cylinder do not pass through center of ellipse. Worse, in the case of a twisted cylinder, the normals to the surface do not lie in the plane of the elliptical slice. However, these problems are readily resolved by setting up a small system of equations that we can solve for the grazing points on an elliptic slice. We begin by solving the problem for the elliptic cylinder, after which we will show how to generalize the solution to the twisted elliptic cylinder.

We prove that any piece of a space-like minimal surface of the form z=z(x, y), in the pseudo-Euclidean space E3,1 with line element dx2+dy2−dz2 has the greatest area among close space-like surfaces with the same boundary. For time-like surfaces the following assertion is proved. Let V2 be a time-like minimal surface in E3,1. Then for any domain on V2, there exists a smooth variation in the class of time-like surfaces with fixed boundary that increases the area, and there exists a smooth variation in the same class of surfaces that decreases the area.

In many problems examined thus far we have required continuous differentiability of the function y (or Y) defining the classes for optimization. Already with the example of the minimal surface of revolution from §1.4(a) we have argued that for some configurations, the minimizing curve (if it exists) should exhibit “corners,” and it is natural to wonder whether cornered curves ŷ and Ŷ such as those shown in Figure 7.1 might not give improved results for other problems. Such curves are represented readily by functions which are piecewise continuously differentiable, or piecewise C1.

This chapter discusses examples of area-minimizing surfaces. The minimal surface equation gives the necessary condition that under smooth variations in the surface, the rate of change of the area is 0. This condition turns out to be equivalent to the vanishing of the mean curvature. A smoothly immersed surface which is locally the graph of a solution to the minimal surface equation (or, equivalently, which has mean curvature 0) is called a minimal surface. It presents figures of some famous minimal surfaces. The minimal surface theorem guarantees that small pieces of minimal surfaces are area mini- mizing, but larger pieces may not be. For example, the portion of Enneper's surface is not area minimizing. There are two area-minimizing surfaces with the same boundary and some systems of curves in R 3 bound infinitely many minimal surfaces. On a disc (or any other convex domain), there is a solution of the minimal surface equation with any given continuous boundary values. The proof is omitted because the minimal surface equation is not linear, this fact is not at all obvious, and it fails if the domain is not convex.

This paper exhibits some new features in one of the classical problems of the calculus of variations: finding minimal surfaces of revolution. We give a simple geometric construction to count the nu...

When dimension[codomain] = 1, the Jacobian of a single function is just the gradient of the function, and the zeros of the gradient determine the critical points of the function; in this case, the hessian classifies the critical points of a function. When dimension[domain] = 2, the Jacobian determines the first fundamental form of a parametric surface; in this case, the hessian, in vector form, determines the second fundamental form of a surface. The two fundamental forms together determine the Gaussian and mean curvatures of a surface. The analysis of the critical points involves looking at the values of the principal minors of this matrix at the critical points, namely, the determinants of the square submatrices running down the main diagonal in the upper left-hand side of the hessian. This chapter defines the critical points of a function as the points where the gradient is zero. If the hessian is positive definite at a critical point, then the critical point is a local minimum, and if it is negative definite, then the critical point is a local maximum. If a critical point is neither positive nor negative definite, then it is considered to be a saddle point. A minimal surface is one whose mean curvature is zero. The boundary of a two-dimensional piece of a surface is a closed curve. A surface is minimal if it has the least area of any surface with the same boundary.

In ray tracing a surface of revolution, the problem of finding the point of intersection of an arbitrary ray with the surface of revolution can be reduced to that of finding a point of intersection between two curves in a plane — the so called cut plane. In the cut plane, by using a local coordinate system (x′=x2,z′=z) instead of (x, z) where z is along the direction parallel to the axis of the surface of revolution, and then further translating the origin of the coordinate system along the x′ axis (depending upon the ray), the curve(s) of intersection of the surface of revolution with the cut plane are expressed exactly by the square radial curve defining the surface of revolution itself. We need to find the point(s) of intersection between this square radial curve and a transformed quadratic form of the ray in the cut plane. We represent the square radial curve of the surface of rotation by a variation of the usual strip_tree such that the direction of the sides of the rectangles corresponding to different nodes are parallel to the axes of the local coordinate system, instead of being in arbitrary directions. The terminal nodes in the Strip_tree contain the segments of the square radial curve divided approximately equally lengthwise. The method works efficiently for all cases with a smooth radial curve or a piecewise smooth radial curve.

This paper discusses the aberrational properties of nonspherical surfaces formed by the rotation of the involute of an evolute having the form of a circle. It is shown that the rotation of an arbitrary curve that is nonsymmetrical relative to the normal to it at some point around this normal results in the formation of two surfaces of revolution, which are called a binary surface. The exact equation is obtained for a nonspherical surface equidistant from a paraboloid of revolution. The resulting equation has the form of a power series in the form of an implicit function. © 2004 Optical Society of America

Let S be a smooth surface in Euclidean 3-space and let C be a smooth curve having its end points on S. Consider a surface x, bounded by the configuration , which minimizes Dirichlet's integral. We describe the local behaviour of x at the points where the arc C meets the boundary surface S.

Necessary and sufficient conditions are found such that the internally coalesced surface σ= S1 + S2 should have a parallel L ∃ S2 which divides the surface 2 into two parts so that the part σL, which does not contain a pole of the surface S2, should permit nontrivial bending slipping along L.

RECENTLY GREAT progress has been made in the classical theory of minimal surfaces in R3. For example, the proof of the embedding of the solution to Plateau’s problem for an extremal Jordan curve [lo] and the embedding of a solution to the free boundary value problem [ll], the proof of the bridge theorem [12], the regularity and finiteness of least area oriented surfaces bounding a smooth Jordan curve [5], the uniqueness and topological uniqueness of certain minimal surfaces 18, 9, 121, the proof that a complete stable minimal surface is a plane [l, 31, the existence of a collection of 4 Jordan curves which bound a continuous family of compact minimal surfaces [141, the existence of a complete minimal surface contained between two parallel planes [13] and a theorem [18] which states that if the Gauss map of a complete minimal surface misses 7 points on the ‘sphere, then the minimal surface is a plane. The aim of this work is to begin the exploration of the topological properties of complete minimal surfaces of finite total curvature. First we recall the fundamental classical result of Chern-Osserman [6] that states that such surfaces M are conformally equivalent to compact Riemann surfaces iii punctured in a finite number of points. Furthermore, they prove that the Gauss map on the surface M extends conformally to a. In particular, the theorem of Chern-Osserman implies that such surfaces have finite topological type with a finite number of topological ends and that the normal vectors at infinity on these ends are well defined. (Finite topological type means in this case that M is diffeomorphic to the interior of a compact surface with boundary.) Our first theorem gives a nice description of the topological placement in IX3 of a complete surface of finite topological type which has well defined normal vectors at infinity. We prove that the ends of such surfaces behave like the ends of the catenoid at infinity. More precisely, Theorem 1 shows that if f: M + R3 is an immersion of such a surface, then f is proper and the image f(M) viewed from infinity looks like a finite collection of flat planes (with multiplicity) that pass through the origin. For example, the catenoid viewed from infinity looks like two oppositely oriented copies of a plane passing through the origin. Theorem 1 has a natural generalization to submanifolds of arbitrary codimension in R”. This generalization which is Theorem 2 is discussed in

This paper presents a new and simple technique for a certain class of variational problems which includes many of the important problems of mathematical physics, e.g., the Brachistochrone, geodesies, and minimal surface of revolution problems. The technique uses Caratheodory's equivalent problems approach but combines two equivalent problems at the same time to get the sufficiency and uniqueness results. It does not use any of the classical sufficiency conditions such as the Weierstrass condition. The equations that we are led to by this new approach turn out to be the Hamilton-Jacobi and Euler-Lagrange equations for the problem, but here we have not had to use any of the classical Hamilton-Jacobi theory nor derivations to get its results (e.g., orthogonality of the extremals to the wave fronts) for this class of problems. The cases for one and n dependent variables are presented and illustrated. Implications and generalizations of the method are discussed.

It can be quite hard to prove that any particular surface is area minimizing. After all, it must compare favorably with all other surfaces with the given boundary. Fortunately, there are a number of beautiful examples of area minimizing surfaces. This chapter presents some examples of area minimizing surfaces. The minimal surface equation just gives the necessary condition that under smooth variations in the surface, the rate of change of the area is 0. This condition turns out to be equivalent to the vanishing of the mean curvature. A smoothly immersed surface that is locally the graph of a solution to the minimal surface equation is called a minimal surface. The chapter presents some pictures of some famous minimal surfaces. It discusses complex analytic varieties. Any compact portion of a complex analytic variety in C n = R 2n is area minimizing. The chapter further describes fundamental theorem of calibrations.

This chapter discusses the calculus of variations. One of the central problems in the classical analysis of functions of one or several variables was that of discovering the extrema of differentiable functions. Extremum problems play an important part in functional spaces. For example, the problem of determining the minimal surface of revolution bounded by two circles with a common axis is interpreted as the problem of finding the extremum of the function the argument y = y ( x ) of which is itself an element of a normed linear space. The calculus of variations has as its aim a generalization of the structures of classical analysis that will make possible the solution of similar extremum problems. More widely conceived, the calculus of variations is the analysis of infinitesimals in infinite-dimensional spaces. Hamilton's principle possesses a characteristic feature that its formulation omits any assumption of finiteness in respect of the number of degrees of freedom. It can, therefore, be applied even too mechanical systems with infinitely many degrees of freedom, and in particular, to problems with a continuous mass distribution, provided the potential and kinetic energies can be calculated for these systems.