Perfect Hexagons, Elementary Triangles, and the Center of a Cubic Curve

2012 ◽  
pp. 115-130
Author(s):  
Raymond R. Fletcher
Keyword(s):  
Cubic Curve

Cubic curve update

1989 ◽  
Vol 9 (6) ◽  
pp. 70-72
Author(s):  
J.F. Blinn
Keyword(s):  
Cubic Curve

On the remarkable quadrics of a space cubic curve

1930 ◽  
Vol 26 (2) ◽  
pp. 206-219 ◽  
Author(s):  
R. Vaidyanathaswamy
Keyword(s):  
Cubic Curve ◽  
Special Relations

In the geometry of the cubic curve Γ in space, we have to study the quadrics Q which stand in certain special relations to the curve. We shall find it convenient to refer to these relations as A, B, B′, C, C′; these are defined below.

scholarly journals The characterization of elliptic curves over finite fields

1988 ◽  
Vol 45 (2) ◽  
pp. 275-286 ◽  
Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Maximum Size ◽  
Desarguesian Plane ◽  
Cubic Curve ◽  
Odd Order ◽  
Even Order ◽  
Curves Over Finite Fields

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

Experimental Study of Mechanism of Rock Breakage with High-Pressure Cavitating Water Jets

2011 ◽  
Vol 243-249 ◽  
pp. 2130-2137
Author(s):  
Zhao Long Ge ◽  
Yi Yu Lu ◽  
Ji Ren Tang ◽  
Ke Hu ◽  
Wen Feng Zhang
Keyword(s):  
High Pressure ◽  
Confining Pressure ◽  
Rock Breaking ◽  
Pressure Increase ◽  
Rock Breakage ◽  
Bubble Cloud ◽  
Cubic Curve ◽  
Water Jets ◽  
Erosion Efficiency ◽  
Pump Pressure

To explore the relationship among the erosion ability of high-pressure cavitating water jets, hydraulic parameters and rock nature with a series of experiments relating to the efficiency of rock-breaking with cavitating water jets for different porosity of rock under different confining pressures and pump pressures. The results show that the erosion efficiency (erosion mass and erosion depth) of cavitating water jets is fitted a conic curve with pump pressure and confining pressure. It increases with the pump pressure increases while decreases with the confining pressure increases; the length of the bubble cloud decreases with the confining pressure increase and the length increases with the pump pressure increase, which is accorded with cubic curve. The bubble cloud length influences the rock-breaking efficiency by deciding the valid stand-off distance directly. Under the experimental condition, the cavitation happens once the pump pressure reaches 7MPa, and the cavitating water jets can crushing the sandstones which the uniaxial compressive strength is 96MPa. On the other hand, the porosity of rock is another main factor of rock breakage with high pressure cavitating water jets. The higher the porosity of rock is, the easier the rock can be broken.

scholarly journals Matrices of integers associated with self-transformations of surfaces

1948 ◽  
Vol 193 (1032) ◽  
pp. 25-43
Keyword(s):  
Rational Curve ◽  
Pell Equation ◽  
Quartic Surface ◽  
Twisted Cubic ◽  
Cubic Curve ◽  
The Matrix ◽  
Quartic Surfaces ◽  
Infinite Sequences ◽  
Square Matrix

Let c = ( c 1 , . . . , c r ) be a set of curves forming a minimum base on a surface, which, under a self-transformation, T , of the surface, transforms into a set T c expressible by the equivalences T c = Tc, where T is a square matrix of integers. Further, let the numbers of common points of pairs of the curves, c i , c j , be written as a symmetrical square matrix Г. Then the matrix T satisfies the equation TГT' = Г. The significance of solutions of this equation for a given matrix Г is discussed, and the following special surfaces are investigated: §§4-7. Surfaces, in particular quartic surfaces, wìth only two base curves. Self-transformations of these depend on the solutions of the Pell equation u 2 - kv 2 = 1 (or 4). §8. The quartic surface specialized only by being made to contain a twisted cubic curve. This surface has an involutory transformation determined by chords of the cubic, and has only one other rational curve on it, namely, the transform of the cubic. The appropriate Pell equation is u 2 - 17 v 2 = 4. §9. The quartic surface specialized only by being made to contain a line and a rational curve of order m to which the line is ( m - 1)⋅secant (for m = 1 the surface is made to contain two skew lines). The surface has two infinite sequences of self-transformations, expressible in terms of two transformations R and S , namely, a sequence of involutory transformations R S n , and a sequence of non-involutory transformations S n .

435. [M. 5.] Surfaces generated by the Motion of an Invariable Cubic Curve

1914 ◽  
Vol 7 (112) ◽  
pp. 370-371
Author(s):  
Harold Hilton
Keyword(s):  
Cubic Curve

On the Theory of Rational Derivation on a Cubic Curve

1880 ◽  
Vol 3 (4) ◽  
pp. 356
Author(s):  
William E. Story
Keyword(s):  
Cubic Curve
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