implicitcurve

An implicit curve in the plane is the set of points (x,y) that satisfy an equation F(x,y)=0, where F: R^2 → R is a real-valued function. The curve is defined implicitly, meaning that the coordinates are not given as a function y=f(x); a given x can lie on more than one y-value, and the set may have multiple branches, cusps, or self-intersections.

When F is a polynomial, the curve is an algebraic curve. Classic examples include the circle x^2+y^2-1=0 and the line ax+by+c=0. More complex curves arise from higher-degree polynomials and from systems of implicit equations.

Properties: The gradient ∇F=(Fx,Fy) is normal to the curve at points where at least one partial derivative is nonzero. If Fy≠0, the implicit function theorem gives a local function y=f(x) and dy/dx=−Fx/Fy; points with Fx=Fy=0 are singular and may be cusps or crossings.

Non-polynomial F yield transcendental implicit curves, such as F(x,y)=cos(x)+cos(y)−1=0, which describes a curvilinear grid of level set points.

Computation and plotting: implicit curves are often drawn as level sets of F. Numerical methods include contouring and marching squares; solving intersections with other curves requires solving simultaneous equations F(x,y)=0 and G(x,y)=0.

Applications: implicit curves appear in algebraic geometry, CAD and computer graphics, shape analysis, and physics, where curves arise as level sets, projections, or as parts of larger implicit representations.

See also: implicit function, algebraic curve, level set, contour plot, curvature.