Quartic plane curve

Quartic plane curve

A quartic plane curve is a plane curve of the fourth degree. It can be defined by a quartic equation:

:Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0.

This equation has fifteen constants. However, it can be multiplied by any non-zero factor without changing the shape of the curve. Therefore, quartic curves form a space of dimension fourteen. It also follows that there is exactly one quartic curve that passes through a set of fourteen distinct points in general position.

A quartic curve can have a maximum of:
* Four connected components
* Twenty-eight bi-tangents
* Three ordinary double points

Examples

*Toric section
*Klein quartic

* The bean curve is a special case of the crooked egg curve
* A spiric section is a special case of a toric section


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Plane curve — In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a …   Wikipedia

  • Algebraic curve — In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with… …   Wikipedia

  • Trott curve — In real algebraic geometry, the Trott curve is the set of points ( x , y ) satisfying the degree four polynomial equation:displaystyle 144(x^4+y^4) 225(x^2+y^2)+350x^2y^2+81=0.These points form a nonsingular quartic plane curve that has genus… …   Wikipedia

  • Bean curve — The bean curve is a quartic plane curve with the equation::x^4+x^2y^2+y^4=x(x^2+y^2) ,It is not the same as another curve also called the bean curve, or crooked egg curve, but both curves belong to a family of similar bean shaped curves.The bean… …   Wikipedia

  • Cruciform curve — The cruciform curve, or cross curve is a quartic plane curve given by the equation Cruciform with parameters (b,a) being (1,1) in red; (2,2) in green; (3,3) in blue …   Wikipedia

  • Ampersand curve — In mathematics, the ampersand curve is a quartic plane curve given by the equation:: (y^2 x^2)(x 1)(2x 3)=4(x^2+y^2 2x)^2.It is an algebraic curve of genus zero, with three ordinary double points, all in the real plane. By the Plücker formulas,… …   Wikipedia

  • Bow curve — A bow curve is a quartic plane curve with the equation::x^4=x^2y y^3. ,The bow curve has a single triple point at x=0, y=0, and consequently is a rational curve, with genus zero. Expanding around zero, the three branches have series y 1 =… …   Wikipedia

  • Bicuspid curve — The biscuspid is a quartic plane curve with the equation:(x^2 a^2)(x a)^2+(y^2 a^2)^2=0 ,where a determines the size of the curve.The bicuspid has only the two nodes as singularities, and hence is a curve of genus one, with j invariant −4096/11.… …   Wikipedia

  • Swastika curve — The swastika curve is the name given by Cundy and Rollett to the quartic plane curve with the Cartesian equation: y^4 x^4 = xy,, or, equivalently, the polar equation:r^2 = an(2 heta)/2. ,The curve looks similar to the right handed swastika, but… …   Wikipedia

  • Bézier curve — Cubic Bézier curve A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case. In… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”