implicit curve

In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation $x^2+y^2=1$. In general, every implicit curve is defined by an equation of the form
: $F(x,y)=0$
for some function ''F'' of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. ''Implicit'' means that the equation is not expressed as a solution for either ''x'' in terms of ''y'' or vice versa.

If $F(x,y)$ is a polynomial in two variables, the corresponding curve is called an ''algebraic curve'', and specific methods are available for studying it.

Plane curves can be represented in Cartesian coordinates (''x'', ''y'' coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation $y=f(x)$ in which the functional form is explicitly stated; this is called an ''explicit'' representation. The third essential description of a curve is the ''parametric'' one, where the ''x''- and ''y''-coordinates of curve points are represented by
two functions both of whose functional forms are explicitly stated, and which are dependent on a common parameter $t.$

Examples of implicit curves include:
# a line: $x+2y-3=0 ,$
# a circle: $x^2+y^2-4=0 ,$
# the semicubical parabola: $x^3-y^2=0 ,$
# Cassini ovals $(x^2+y^2)^2-2c^2(x^2-y^2)-(a^4-c^4)=0$ (see diagram),
# $\sin(x+y)-\cos(xy)+1=0$ (see diagram).
The first four examples are algebraic curves, but the last one is not algebraic. The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve.

The implicit function theorem describes conditions under which an equation $F(x,y)=0$ can be ''solved implicitly'' for ''x'' and/or ''y'' – that is, under which one can validly write $x=g(y)$ or $y=f(x)$. This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit curves make them essential tools in geometry and computer graphics.

An implicit curve with an equation $F(x,y)=0$ can be considered as the level curve of level 0 of the surface $z=F(x,y)$ (see third diagram).

Slope and curvature

In general, implicit curves fail the vertical line test (meaning that some values of ''x'' are associated with more than one value of ''y'') and so are not necessarily graphs of functions. However, the implicit function theorem gives conditions under which an implicit curve ''locally'' is given by the graph of a function (so in particular it has no self-intersections). If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.

There are several possible ways to compute these quantities for a given implicit curve. One method is to use implicit differentiation to compute the derivatives of ''y'' with respect to ''x''. Alternatively, for a curve defined by the implicit equation $F(x,y)=0$, one can express these formulas directly in terms of the partial derivatives of $F$. In what follows, the partial derivatives are denoted $F_x$ (for the derivative with respect to ''x''), $F_y$, $F_$ (for the second partial with respect to ''x''), $F_$ (for the mixed second partial), $F_.$

Tangent and normal vector

A curve point $(x_0, y_0)$ is ''regular'' if the first partial derivatives $F_x(x_0,y_0)$ and $F_y(x_0,y_0)$ are not both equal to 0.

The equation of the tangent line at a regular point $(x_0,y_0)$ is
: $F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)=0,$

so the slope of the tangent line, and hence the slope of the curve at that point, is
:$\text =-\frac.$

If $F_y(x,y)=0 \ne F_x(x,y)$ at $(x_0,y_0),$ the curve is vertical at that point, while if both $F_y(x,y)=0$ and $F_x(x,y)=0$ at that point then the curve is not differentiable there, but instead is a singular point – either a cusp or a point where the curve intersects itself.

A normal vector to the curve at the point is given by
:$\mathbf(x_0,y_0) = (F_x(x_0,y_0), F_y(x_0,y_0))$
(here written as a row vector).

Curvature

For readability of the formulas, the arguments $(x_0,y_0)$ are omitted. The curvature $\kappa$ at a regular point is given by the formula
:$\kappa = \frac$.

Derivation of the formulas

The implicit function theorem guarantees within a neighborhood of a point $(x_0,y_0)$ the existence of a function $f$ such that $F(x,f(x))=0$.
By the chain rule, the derivatives of function $f$ are
: $f'(x)=-\frac$ and $f''(x)=\frac$
(where the arguments $(x, f(x))$ on the right side of the second formula are omitted for ease of reading).

Inserting the derivatives of function $f$ into the formulas for a tangent and curvature of the graph of the explicit equation $y = f(x)$ yields
: $y=f(x_0)+f'(x_0)(x-x_0)$ (tangent)
: $\kappa(x_0)=\frac$ (curvature).

Advantage and disadvantage of implicit curves

Disadvantage

The essential disadvantage of an implicit curve is the lack of an easy possibility to calculate single points which is necessary for visualization of an implicit curve (see next section).

Advantages

#Implicit representations facilitate the computation of intersection points: If one curve is represented implicitly and the other parametrically the computation of intersection points needs only a simple (1-dimensional) Newton iteration, which is contrary to the cases ''implicit-implicit'' and ''parametric-parametric'' (see Intersection).
#An implicit representation $F(x,y)=0$ gives the possibility of separating points not on the curve by the sign of $F(x,y)$. This may be helpful for example applying the false position method instead of a Newton iteration.
#It is easy to generate curves which are almost geometrically similar to the given implicit curve $F(x,y)=0,$ by just adding a small number: $F(x,y)-c=0$ (see section #Smooth approximations).

Applications of implicit curves

Within mathematics implicit curves play a prominent role as algebraic curves.
In addition, implicit curves are used for designing curves of desired geometrical shapes. Here are two examples.

Smooth approximations

Convex polygons

A smooth approximation of a convex polygon can be achieved in the following way: Let $g_i(x,y)=a_ix+b_iy+c_i=0, \ i=1,\dotsc,n$ be the equations of the lines containing the edges of the polygon such that for an inner point of the polygon $g_i$ is positive. Then a subset of the implicit curve
:$F(x,y)=g_1(x,y)\cdots g_n(x,y)-c=0$
with suitable small parameter $c$ is a smooth (differentiable) approximation of the polygon.
For example, the curves
:$F(x,y)=(x+1)(-x+1)y(-x-y+2)(x-y+2)-c=0$ for $c= 0.03, \dotsc, 0.6$
contain smooth approximations of a polygon with 5 edges (see diagram).

Pairs of lines

In case of two lines
:$F(x,y)=g_1(x,y)g_2(x,y)-c=0$
one gets
:a pencil of ''parallel lines'', if the given lines are parallel or
:the pencil of hyperbolas, which have the given lines as asymptotes.
For example, the product of the coordinate axes variables yields the pencil of hyperbolas $xy-c=0, \ c\ne 0$, which have the coordinate axes as asymptotes.

Others

If one starts with simple implicit curves other than lines (circles, parabolas,...) one gets a wide range of interesting new curves. For example,
:$F(x,y)=y(-x^2-y^2+1)-c=0$
(product of a circle and the x-axis) yields smooth approximations of one half of a circle (see picture), and
:$F(x,y)=(-x^2-(y+1)^2+4)(-x^2-(y-1)^2+4)-c=0$
(product of two circles) yields smooth approximations of the intersection of two circles (see diagram).

Blending curves

In CAD one uses implicit curves for the generation of blending curves,E. Hartmann: ''Blending of implicit surfaces with functional splines'', CAD,Butterworth-Heinemann, Volume 22 (8), 1990, p. 500-507 which are special curves establishing a smooth transition between two given curves. For example,
: $F(x,y)=(1-\mu)f_1f_2-\mu (g_1g_2)^3 =0$
generates blending curves between the two circles
:$f_1(x,y)=(x-x_1)^2+y^2-r_1^2=0 ,$
:$f_2(x,y)=(x-x_2)^2+y^2-r_2^2=0 .$
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