As-Rigid-As-Possible Stereo under Second Order Smoothness Priors

2014 ◽  
pp. 112-126 ◽  
Author(s):  
Chi Zhang ◽  
Zhiwei Li ◽  
Rui Cai ◽  
Hongyang Chao ◽  
Yong Rui
Keyword(s):  
Second Order ◽  
Order Smoothness

A Simple Practical Criterion to Guarantee Second Order Smoothness of Blend Surfaces

1989 ◽  
Author(s):  
J. Pegna ◽  
F.-E. Wolter
Keyword(s):  
Second Order ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Common Boundary ◽  
First Order ◽  
Computer Aided ◽  
The Common ◽  
Blend Surfaces ◽  
Continuous Curves ◽  
Order Smoothness

Abstract Computer Aided Geometric Design of surfaces sometimes presents problems that were not envisioned by mathematicians in differential geometry. This paper presents mathematical results that pertain to the design of second order smooth blending surfaces. Second order smoothness normally requires that normal curvatures agree along all tangent directions at all points of the common boundary of two patches, called the linkage curve. The Linkage Curve Theorem proved here shows that, for the blend to be second order smooth when it is already first order smooth, it is sufficient that normal curvatures agree in one direction other than the tangent to a first order continuous linkage curve. This result is significant for it substantiates earlier works in computer aided geometric design. It also offers simple practical means of generating second order blends for it reduces the dimensionality of the problem to that of curve fairing, and is well adapted to a formulation of the blend surface using sweeps. From a theoretical viewpoint, it is remarkable that one can generate second order smooth blends with the assumption that the linkage curve is only first order smooth. This property may be helpful to the designer since linkage curves can be constructed from low order piecewise continuous curves.

Dupin Cyclide and Second-Order Curves. Part 1

2016 ◽  
Vol 4 (2) ◽  
pp. 19-28 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Second Order ◽  
Descriptive Geometry ◽  
Analytical Equation ◽  
Not Given ◽  
Academic Literature ◽  
Order Curve ◽  
Plane Passing ◽  
Dupin Cyclide ◽  
The One ◽  
Order Smoothness

Making smooth shapes of various products is caused by the following requirements: aerodynamic, structural, aesthetic, etc. That’s why the review of the topic of second-order curves is included in many textbooks on descriptive geometry and engineering graphics. These curves can be used as a transition from the one line to another as the first and second order smoothness. Unfortunately, in modern textbooks on engineering graphics the building of Konik is not given. Despite the fact that all the second-order curves are banded by a single analytical equation, geometrically they unites by the affiliation of the quadric, projective unites by the commonality of their construction, in the academic literature for each of these curves is offered its own individual plot. Considering the patterns associated with Dupin cyclide, you can pay attention to the following peculiarity: the center of the sphere that is in contact circumferentially with Dupin cyclide, by changing the radius of the sphere moves along the second-order curve. The circle of contact of the sphere with Dupin cyclide is always located in a plane passing through one of the two axes, and each of these planes intersects cyclide by two circles. This property formed the basis of the graphical constructions that are common to all second-order curves. In addition, considered building has a connection with such transformation as the dilation or the central similarity. This article considers the methods of constructing of second-order curves, which are the lines of centers tangent of the spheres, applies a systematic approach.

scholarly journals Depth Map Inpainting under a Second-Order Smoothness Prior

2013 ◽  
pp. 555-566 ◽  
Author(s):  
Daniel Herrera C. ◽  
Juho Kannala ◽  
L’ubor Ladický ◽  
Janne Heikkilä
Keyword(s):  
Depth Map ◽  
Second Order ◽  
Order Smoothness

scholarly journals Global Stereo Reconstruction under Second-Order Smoothness Priors

2009 ◽  
Vol 31 (12) ◽  
pp. 2115-2128 ◽  
Author(s):  
O. Woodford ◽  
P. Torr ◽  
I. Reid ◽  
A. Fitzgibbon
Keyword(s):  
Second Order ◽  
Stereo Reconstruction ◽  
Order Smoothness

A Systematic Approach to Well Surveying Calculations

1972 ◽  
Vol 12 (06) ◽  
pp. 474-488 ◽  
Author(s):  
Howard L. Taylor ◽  
Mack C. Mason
Keyword(s):  
Second Order ◽  
Mathematical Framework ◽  
Mathematical Form ◽  
Directional Drilling ◽  
Smooth Approximation ◽  
Systematic Analysis ◽  
Special Cases ◽  
Bearing Angle ◽  
Tools And Techniques ◽  
Order Smoothness

Abstract The information used in well surveying calculations is studied and formalized. A general, natural, mathematical approach to the problem is presented and four special cases are developed: presented and four special cases are developed:triangular or trapezoidal,secant,quadratic, andminimum curvature. These methods are compared and their properties analyzed for reasonableness. Two of these methods are new and have appeared promising when applied to real and test data. Introduction Several new approaches have been proposed to the problem of performing well surveying calculations; and it has been pointed out that improvements in directional drilling tools and techniques justify a better treatment than the old tangential methods. However, a systematic analysis of the problem in a standard mathematical form has not been published. This paper will try to cover several of the more useful methods in a natural mathematical framework and to suggest several alternative approaches and directions for improvement. FORMULATION OF THE PROBLEM In performing a directional survey of a well, a tool measures an inclination angle with respect to the vertical and a bearing angle with respect to North at a number of points, Pi, in the well. Thus, we know three things:(1)the location of the first point, Po, at the surface,(2)the distance between point, Po, at the surface,(2)the distance between any two points, Pi - and Pi+1, along the arc length, and(3)the direction of the wellbore at each point, Pi (as given by the inclination and bearing). Pi (as given by the inclination and bearing).If one considers using a smooth approximation to the wellbore rather than straight-line segments joined at angles, the question of what degree of smoothness is to be used arises. For example, cubic spline approximations could be used to insure second-order smoothness. There are two difficulties that arise. In the first place, the authors know of no reason why a wellbore should have second-order smoothness. Second, the calculations become very difficult, partly because more than two points must be dealt with at a time. In this paper we shall follow the common practice of only analyzing the wellbore from one point, say P0, to the next point, P1, and assume that the process is repeated until P1, and assume that the process is repeated until the final point, Pn, is located. Referring to Fig. 1, we assume that the Z-axis is vertical and the X-axis is in the direction North. Thus the Y-axis will point East. The angle of inclination will be 0 and the angle from North, which is essentially the bearing, will be, where and . Hence, we can state our problem mathematically as follows. GivenThe point, Po = (Xo, Yo, Zo)The distance, S1 between Po and P1The direction of the wellbore determined byo and o at Po and similarly 1 and 1 at P1 SPEJ P. 474

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