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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 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.

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.

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.

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.

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.

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

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.