I know that general second degree curve, i.e. $ax^2 + by^2 + 2hxy + 2gx + 2fy + c=0$ gives us the equation of different cross sections of a cone.

Similarly, what does a third degree* curve actually represent? Different cross sections of another solid structure, if so then which one? And what does a forth degree? ..fifth?

$^* =ax^3+ by^3 + 2dx^2y + 2ey^2x + 2jx^2 + 2ky^2 + 2hxy + 2gx + 2fy+ c=0$

  • You seem to be doing two things at once. The quadratic curve above does indeed give (possibly degenerate) conic sections. An easier progression may be to either: A) Consider quadric surfaces, algebraic surfaces of degree 2 (i.e., just adding a third variable), or B) Consider cubic curves, arbitrary curves (two variables) of degree $3$. Or do you really one to add an extra variable and degree at once? – pjs36 Jun 12 '15 at 17:32
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    Oh sorry, for the confusion. Actually I just wanted to increase power (without changing the number of variables) – Harshal Gajjar Jun 12 '15 at 17:38
  • Thanks for the clarification! That WolframMathWorld link has some interesting information on the curves you're talking about, and you can even copy/paste your equation into Desmos, fix the exponents, and vary parameters to see what some curves look like. You're venturing steadily into "algebraic geometry" territory, and I know practically nothing about that. – pjs36 Jun 12 '15 at 17:57
  • Thanks for sharing this @pjs36. The wolfram link was good but unfortunately didn't tell me anything about the structure.. Actually I'd tried varying all the parameters in Desmos, but couldn't get any clue from there. The graphs were strange, ~ three triangles approaching towards origin. – Harshal Gajjar Jun 12 '15 at 18:52

Here are two "visual" applications of implicit polynomial curves (also called plane algebraic curves).

Consider a surface mapping $(x(u,v), y(u,v), z(u,v))$ from the unit square to $\mathbb{R}^3$ where $x,$ $y,$ and $z$ are total degree $d$ polynomials for some $d>2.$ The intersection of a plane $ax+by+cz+d=0$ with this surface is a degree $d$ implicit polynomial curve in $uv$ space.

An easily visualized subset of this type of curves are the superellipses defined by $(x/a)^d + (y/b)^d = 1.$ As $d$ approaches infinity, the curve approaches the rectangle with sides $2a$ and $2b$ and centered at the origin.

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