scholarly journals A conjecture concerning rational points on cubic curves

1954 ◽  
Vol 2 ◽  
pp. 49 ◽  
Author(s):  
Ernst S. Selmer
Keyword(s):  
Rational Points ◽  
Cubic Curves

scholarly journals Counting rational points on cubic curves

2010 ◽  
Vol 53 (9) ◽  
pp. 2259-2268 ◽  
Author(s):  
Roger Heath-Brown ◽  
Damiano Testa
Keyword(s):  
Rational Points ◽  
Cubic Curves ◽  
Counting Rational Points

Rational Points on Cubic Curves and Surfaces

1944 ◽  
Vol 51 (6) ◽  
pp. 332-339
Author(s):  
L. J. Mordell
Keyword(s):  
Rational Points ◽  
Cubic Curves ◽  
Curves And Surfaces

Rational Points on Cubic Curves and Surfaces

1944 ◽  
Vol 51 (6) ◽  
pp. 332
Author(s):  
L. J. Mordell
Keyword(s):  
Rational Points ◽  
Cubic Curves ◽  
Curves And Surfaces

scholarly journals Counting rational points on smooth cubic curves

2018 ◽  
Vol 189 ◽  
pp. 138-146
Author(s):  
Manh Hung Tran
Keyword(s):  
Rational Points ◽  
Cubic Curves ◽  
Counting Rational Points

scholarly journals Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

2021 ◽  
Vol 2 (5) ◽  
Author(s):  
Soroosh Tayebi Arasteh ◽  
Adam Kalisz
Keyword(s):  
Computer Graphics ◽  
Geometric Design ◽  
The Other ◽  
Control Points ◽  
Complex Surfaces ◽  
Linear Transformations ◽  
Computer Aided Geometric Design ◽  
Cubic Curves ◽  
Original Curve ◽  
Computer Aided

AbstractSplines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

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