Implicit curve

of the surface z=\sin(x+y)-\cos(xy)+1 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.…

of the surface z=\sin(x+y)-\cos(xy)+1 ]] 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). 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_{xx} (for the second partial with respect to x), F_{xy} (for the mixed second partial), F_{yy}. 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{slope} =-\frac{F_x(x_0,y_0)}{F_y(x_0,y_0)}. 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{n}(x_0,y_0) = (F_x(x_0,y_0), F_y(x_0,y_0)) (here written as a row vector). \kappa = \frac{-F_y^2F_{xx}+2F_xF_yF_{xy}-F_x^2F_{yy}}{(F_x^2+F_y^2)^{3/2}} . 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{F_x(x,f(x))}{F_y(x,f(x))} and f(x)=\frac{-F_y^2F_{xx}+2F_xF_yF_{xy}-F_x^2F_{yy}}{F_y^3} (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{f(x_0)}{(1+f'(x_0)^2)^{3/2}} (curvature). 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).

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). 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 equal point charges at the points P_1=(1,0), \; P_2=(-1,0) can be represented by the equation f(x,y)=\frac{1} +\frac{1} -c =\frac{1}{\sqrt{(x-1)^2+y^2}}+\frac{1}{\sqrt{(x+1)^2+y^2}}-c=0 . The curves are similar to Cassini ovals, but they are not such curves. For the solution of both tasks mentioned above it is essential to have a computer program (which we will call \mathsf{CPoint} ), 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, up to the accuracy of computation: (P1) for the start point is j=0 (P2) repeat (x_{j+1},y_{j+1})= (x_j,y_j)- \frac{F(x_j,y_j)}{F_x(x_j,y_j)^2+F_y(x_j,y_j)^2}\, \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_{j+1},y_{j+1}),\, (x_j,y_j) is small enough. (P4) P=(x_{j+1},y_{j+1}) is the curve point near the start point Q_0 . F(x,y,z)=0, \\ G(x,y,z)=0 \end{matrix} is called an implicit space curve. A curve point (x_0,y_0,z_0) is called regular if the cross product of the gradients F and G is not (0,0,0) at this point: \mathbf t(x_0,y_0,z_0)=\operatorname{grad}F(x_0,y_0,z_0)\times \operatorname{grad}G(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.

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

Text derived from https://en.wikipedia.org/w/index.php?oldid=1238136465 under CC BY-SA 4.0; modifications have been made.