On strictly convex curves and linear monotonicity

1961 ◽  
Vol 65 (3) ◽  
pp. 213-219 ◽  
Author(s):  
Fr. Fabricius-Bjerre
Keyword(s):  
Strictly Convex ◽  
Convex Curves

On singular curves with Gaussian curvature bounds

Analysis ◽  
2018 ◽  
Vol 38 (3) ◽  
pp. 127-136
Author(s):  
Hao Fang ◽  
Weifeng Wo
Keyword(s):  
Gaussian Curvature ◽  
Short Note ◽  
Geodesic Curvature ◽  
Curvature Bounds ◽  
Strictly Convex ◽  
Convex Curves

Abstract In this short note, we consider piece-wise smooth strictly convex curves in {\mathbb{R}^{2}} with prescribed singular angles. Given some geodesic curvature bounds, we give the sharp length estimates.

scholarly journals Area Properties of Strictly Convex Curves

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 391 ◽  
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Yoon-Tae Jung
Keyword(s):  
Partial Answer ◽  
Level Curve ◽  
Conic Sections ◽  
Curve Segment ◽  
Level Curves ◽  
Strictly Convex ◽  
Convex Curves ◽  
Constant Area ◽  
Characterization Theorems

We study functions defined in the plane E 2 in which level curves are strictly convex, and investigate area properties of regions cut off by chords on the level curves. In this paper we give a partial answer to the question: Which function has level curves whose tangent lines cut off from a level curve segment of constant area? In the results, we give some characterization theorems regarding conic sections.

Lagrange’s theorem, convex functions and Gauss map

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 321-334
Author(s):  
Miodrag Mateljevic ◽  
Miloljub Albijanic
Keyword(s):  
Convex Functions ◽  
Gauss Map ◽  
Unique Property ◽  
The Other ◽  
Concave Functions ◽  
Strictly Convex ◽  
Convex Curves ◽  
Lagrange Theorem ◽  
Darboux Theorem

As one of the main results we prove that if f has Lagrange unique property then f is strictly convex or concave (we do not assume continuity of the derivative), Theorem 2.1. We give two different proofs of Theorem 2.1 (one mainly using Lagrange theorem and the other using Darboux theorem). In addition, we give a few characterizations of strictly convex curves, in Theorem 3.5. As an application of it, we give characterization of strictly convex planar curves, which have only tangents at every point, by injective of the Gauss map. Also without the differentiability hypothesis we get the characterization of strictly convex or concave functions by two points property, Theorem 4.2.

scholarly journals A robotic grasping approach with elliptical cone-based potential fields under disturbed scenes

2021 ◽  
Vol 18 (1) ◽  
pp. 172988142098573
Author(s):  
Wenjie Geng ◽  
Zhiqiang Cao ◽  
Zhonghui Li ◽  
Yingying Yu ◽  
Fengshui Jing ◽  
...  
Keyword(s):  
Principal Component ◽  
Major Axis ◽  
Target Object ◽  
Potential Fields ◽  
Single Shot ◽  
Robotic Grasping ◽  
Vertical Projection ◽  
Convex Curves ◽  
Functional Representation ◽  
Continuous Potential

Vision-based grasping plays an important role in the robot providing better services. It is still challenging under disturbed scenes, where the target object cannot be directly grasped constrained by the interferences from other objects. In this article, a robotic grasping approach with firstly moving the interference objects is proposed based on elliptical cone-based potential fields. Single-shot multibox detector (SSD) is adopted to detect objects, and considering the scene complexity, Euclidean cluster is also employed to obtain the objects without being trained by SSD. And then, we acquire the vertical projection of the point cloud for each object. Considering that different objects have different shapes with respective orientation, the vertical projection is executed along its major axis acquired by the principal component analysis. On this basis, the minimum projected envelope rectangle of each object is obtained. To construct continuous potential field functions, an elliptical-based functional representation is introduced due to the better matching degree of the ellipse with the envelope rectangle among continuous closed convex curves. Guided by the design principles, including continuity, same-eccentricity equivalence, and monotonicity, the potential fields based on elliptical cone are designed. The current interference object to be grasped generates an attractive field, whereas other objects correspond to repulsive ones, and their resultant field is used to solve the best placement of the current interference object. The effectiveness of the proposed approach is verified by experiments.

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