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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an implicit curve is a plane curve defined by an
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
relating two coordinate variables, commonly ''x'' and ''y''. For example, the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
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 In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
'', and specific methods are available for studying it. Plane curves can be represented in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(''x'', ''y'' coordinates) by any of three methods, one of which is the implicit equation given above. The
graph of a function In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subs ...
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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
: x+2y-3=0 , # a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
: x^2+y^2-4=0 , # the
semicubical parabola In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form : y^2 - a^2 x^3 = 0 (with ) in some Cartesian coordinate system. Solving for leads to the ''explicit form'' : y = ...
: 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:
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
s, normals, and
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
. 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 In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, ''y'', for each unique input, ''x''. If a vertical line intersects a curve on an ''xy''-plane mor ...
(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 derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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 Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
– either a
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...
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 In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
\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 In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, 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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
). #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 In mathematics, the ''regula falsi'', method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and ...
instead of a Newton iteration. #It is easy to generate curves which are almost
geometrically similar In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wi ...
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 mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic pl ...
. 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 In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
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 Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
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 . The method guarantees the continuity of the tangents and curvatures at the points of contact (see diagram). The two lines :g_1(x,y)=x-x_1=0 , \ g_2(x,y)=x-x_2=0 determine the points of contact at the circles. Parameter \mu is a design parameter. In the diagram, \mu= 0.05, \dotsc, 0.2 .


Equipotential curves of two point charges

Equipotential curves of two equal
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
s at the points P_1=(1,0), \; P_2=(-1,0) can be represented by the equation :f(x,y)=\frac+\frac-c ::: =\frac+\frac-c=0 . The curves are similar to Cassini ovals, but they are not such curves.


Visualization of an implicit curve

To visualize an implicit curve one usually determines a polygon on the curve and displays the polygon. For a parametric curve this is an easy task: One just computes the points of a sequence of parametric values. For an implicit curve one has to solve two subproblems: # determination of a first curve point to a given starting point in the vicinity of the curve, # determination of a curve point starting from a known curve point. In both cases it is reasonable to assume \operatorname F \ne (0,0) . In practice this assumption is violated at single isolated points only.


Point algorithm

For the solution of both tasks mentioned above it is essential to have a computer program (which we will call \mathsf), which, when given a point Q_0=(x_0,y_0) near an implicit curve, finds a point P that is exactly on the curve: :(P1) for the start point is j=0 :(P2) repeat ::(x_,y_)= (x_j,y_j)- \frac\, \left( F_x(x_j,y_j),F_y(x_j,y_j)\right) :::( Newton step for function g(t)=F\left(x_j+tF_x(x_j,y_j),y_j+tF_y(x_j,y_j)\right) \ . ) :(P3) until the distance between the points (x_,y_),\, (x_j,y_j) is small enough. :(P4) P=(x_,y_) is the curve point near the start point Q_0.


Tracing algorithm

In order to generate a nearly equally spaced polygon on the implicit curve one chooses a step length s and :(T1) chooses a suitable starting point in the vicinity of the curve :(T2) determines a first curve point P_1 using program \mathsf :(T3) determines the tangent (see above), chooses a starting point on the tangent using step length s (see diagram) and determines a second curve point P_2 using program \mathsf . : \cdots Because the algorithm traces the implicit curve it is called a ''tracing algorithm''. The algorithm traces only connected parts of the curve. If the implicit curve consists of several parts it has to be started several times with suitable starting points.


Raster algorithm

If the implicit curve consists of several or even unknown parts, it may be better to use a
rasterisation In computer graphics, rasterisation (British English) or rasterization (American English) is the task of taking an image described in a vector graphics format (shapes) and converting it into a raster image (a series of pixels, dots or lines, whi ...
algorithm. Instead of exactly following the curve, a raster algorithm covers the entire curve in so many points that they blend together and look like the curve. :(R1) Generate a net of points (raster) on the area of interest of the x-y-plane. :(R2) For every point P in the raster, run the point algorithm \mathsf starting from P, then mark its output. If the net is dense enough, the result approximates the connected parts of the implicit curve. If for further applications polygons on the curves are needed one can trace parts of interest by the tracing algorithm.


Implicit space curves

Any
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
which is defined by two equations :\begin F(x,y,z)=0, \\ G(x,y,z)=0 \end is called an ''implicit space curve''. A curve point (x_0,y_0,z_0) is called ''regular ''if the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of the gradients F and G is not (0,0,0) at this point: :\mathbf t(x_0,y_0,z_0)=\operatornameF(x_0,y_0,z_0)\times \operatornameG(x_0,y_0,z_0)\ne (0,0,0); otherwise it is called ''singular''. Vector \mathbf t(x_0,y_0,z_0) is a ''tangent vector'' of the curve at point (x_0,y_0,z_0). ''Examples:'' (1)\quad x+y+z-1=0 \ ,\ x-y+z-2=0 ::is a line. (2)\quad x^2+y^2+z^2-4=0 \ , \ x+y+z-1=0 ::is a plane section of a sphere, hence a circle. (3)\quad x^2+y^2-1=0 \ , \ x+y+z-1=0 ::is an ellipse (plane section of a cylinder). (4)\quad x^2+y^2+z^2-16=0 \ , \ (y-y_0)^2+z^2-9=0 ::is the intersection curve between a sphere and a cylinder. For the computation of curve points and the visualization of an implicit space curve see
Intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
.


See also

*
Implicit surface In mathematics, an implicit surface is a surface in Euclidean space defined by an equation : F(x,y,z)=0. An ''implicit surface'' is the set of zeros of a function of three variables. ''Implicit'' means that the equation is not solved for ...


References

* Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: ''Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms'', 2009, Springer-Verlag London, *C:L: Bajaj, C.M. Hoffmann, R.E. Lynch: ''Tracing surface intersections'', Comp. Aided Geom. Design 5 (1988), 285-307.
''Geometry and Algorithms for COMPUTER AIDED DESIGN''


External links

{{commons category, Implicit curves

Curves Computer-aided design