Implicit curve

A curve point ${\displaystyle (x_{0},y_{0})}$ is regular if the first partial derivatives ${\displaystyle F_{x}(x_{0},y_{0})}$ and ${\displaystyle F_{y}(x_{0},y_{0})}$ are not both equal to 0.

The equation of the tangent line at a regular point ${\displaystyle (x_{0},y_{0})}$ is

${\displaystyle F_{x}(x_{0},y_{0})(x-x_{0})+F_{y}(x_{0},y_{0})(y-y_{0})=0,}$

so the slope of the tangent line, and hence the slope of the curve at that point, is

${\displaystyle {\text{slope}}=-{\frac {F_{x}(x_{0},y_{0})}{F_{y}(x_{0},y_{0})}}.}$

If ${\displaystyle F_{y}(x,y)=0\neq F_{x}(x,y)}$ at ${\displaystyle (x_{0},y_{0}),}$ the curve is vertical at that point, while if both ${\displaystyle F_{y}(x,y)=0}$ and ${\displaystyle F_{x}(x,y)=0}$ 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

${\displaystyle \mathbf {n} (x_{0},y_{0})=(F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))}$

(here written as a row vector).

For readability of the formulas, the arguments ${\displaystyle (x_{0},y_{0})}$ are omitted. The curvature ${\displaystyle \kappa }$ at a regular point is given by the formula

${\displaystyle \kappa ={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_{y}^{2})^{3/2}}}}$.[1]

The implicit function theorem guarantees within a neighborhood of a point ${\displaystyle (x_{0},y_{0})}$ the existence of a function ${\displaystyle f}$ such that ${\displaystyle F(x,f(x))=0}$. By the chain rule, the derivatives of function ${\displaystyle f}$ are

${\displaystyle f'(x)=-{\frac {F_{x}(x,f(x))}{F_{y}(x,f(x))}}}$ and ${\displaystyle f''(x)={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{F_{y}^{3}}}}$

(where the arguments ${\displaystyle (x,f(x))}$ on the right side of the second formula are omitted for ease of reading).

Inserting the derivatives of function ${\displaystyle f}$ into the formulas for a tangent and curvature of the graph of the explicit equation ${\displaystyle y=f(x)}$ yields

${\displaystyle y=f(x_{0})+f'(x_{0})(x-x_{0})}$ (tangent)

${\displaystyle \kappa (x_{0})={\frac {f''(x_{0})}{(1+f'(x_{0})^{2})^{3/2}}}}$ (curvature).

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).

1. 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).
2. An implicit representation ${\displaystyle F(x,y)=0}$ gives the possibility of separating points not on the curve by the sign of ${\displaystyle F(x,y)}$. This may be helpful for example applying the false position method instead of a Newton iteration.
3. It is easy to generate curves which are almost geometrically similar to the given implicit curve ${\displaystyle F(x,y)=0,}$ by just adding a small number: ${\displaystyle F(x,y)-c=0}$ (see section #Smooth approximations).

A smooth approximation of a convex polygon can be achieved in the following way: Let ${\displaystyle g_{i}(x,y)=a_{i}x+b_{i}y+c_{i}=0,\ i=1,\dotsc ,n}$ be the equations of the lines containing the edges of the polygon such that for an inner point of the polygon ${\displaystyle g_{i}}$ is positive. Then a subset of the implicit curve

${\displaystyle F(x,y)=g_{1}(x,y)\cdots g_{n}(x,y)-c=0}$

with suitable small parameter ${\displaystyle c}$ is a smooth (differentiable) approximation of the polygon. For example, the curves

${\displaystyle F(x,y)=(x+1)(-x+1)y(-x-y+2)(x-y+2)-c=0}$ for ${\displaystyle c=0.03,\dotsc ,0.6}$

contain smooth approximations of a polygon with 5 edges (see diagram).

In case of two lines

${\displaystyle F(x,y)=g_{1}(x,y)g_{2}(x,y)-c=0}$

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 ${\displaystyle xy-c=0,\ c\neq 0}$, which have the coordinate axes as asymptotes.

If one starts with simple implicit curves other than lines (circles, parabolas,...) one gets a wide range of interesting new curves. For example,

${\displaystyle F(x,y)=y(-x^{2}-y^{2}+1)-c=0}$

(product of a circle and the x-axis) yields smooth approximations of one half of a circle (see picture), and

${\displaystyle F(x,y)=(-x^{2}-(y+1)^{2}+4)(-x^{2}-(y-1)^{2}+4)-c=0}$

(product of two circles) yields smooth approximations of the intersection of two circles (see diagram).

Blending curve (red) of two circles

In CAD one uses implicit curves for the generation of blending curves,[2][3] which are special curves establishing a smooth transition between two given curves. For example,

${\displaystyle F(x,y)=(1-\mu )f_{1}f_{2}-\mu (g_{1}g_{2})^{3}=0}$

generates blending curves between the two circles

${\displaystyle f_{1}(x,y)=(x-x_{1})^{2}+y^{2}-r_{1}^{2}=0,}$

${\displaystyle f_{2}(x,y)=(x-x_{2})^{2}+y^{2}-r_{2}^{2}=0.}$

The method guarantees the continuity of the tangents and curvatures at the points of contact (see diagram). The two lines

${\displaystyle g_{1}(x,y)=x-x_{1}=0,\ g_{2}(x,y)=x-x_{2}=0}$

determine the points of contact at the circles. Parameter ${\displaystyle \mu }$ is a design parameter. In the diagram, ${\displaystyle \mu =0.05,\dotsc ,0.2}$.

Equipotential curves of two point charges at the blue points

Equipotential curves of two equal point charges at the points ${\displaystyle P_{1}=(1,0),\;P_{2}=(-1,0)}$ can be represented by the equation

${\displaystyle f(x,y)={\frac {1}{|PP_{1}|}}+{\frac {1}{|PP_{2}|}}-c}$

${\displaystyle ={\frac {1}{\sqrt {(x-1)^{2}+y^{2}}}}+{\frac {1}{\sqrt {(x+1)^{2}+y^{2}}}}-c=0.}$

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 ${\displaystyle {\mathsf {CPoint}}}$), which, when given a point ${\displaystyle Q_{0}=(x_{0},y_{0})}$ near an implicit curve, finds a point ${\displaystyle P}$ that is exactly on the curve:

(P1) for the start point is ${\displaystyle j=0}$

(P2) repeat

${\displaystyle (x_{j+1},y_{j+1})=(x_{j},y_{j})-{\frac {F(x_{j},y_{j})}{F_{x}(x_{j},y_{j})^{2}+F_{y}(x_{j},y_{j})^{2}}}\,\left(F_{x}(x_{j},y_{j}),F_{y}(x_{j},y_{j})\right)}$

( Newton step for function ${\displaystyle g(t)=F\left(x_{j}+tF_{x}(x_{j},y_{j}),y_{j}+tF_{y}(x_{j},y_{j})\right)\ .}$)

(P3) until the distance between the points ${\displaystyle (x_{j+1},y_{j+1}),\,(x_{j},y_{j})}$ is small enough.

(P4) ${\displaystyle P=(x_{j+1},y_{j+1})}$ is the curve point near the start point ${\displaystyle Q_{0}}$.

to the tracing algorithm: starting points are green

In order to generate a nearly equally spaced polygon on the implicit curve one chooses a step length ${\displaystyle s}$ and

(T1) chooses a suitable starting point in the vicinity of the curve

(T2) determines a first curve point ${\displaystyle P_{1}}$ using program ${\displaystyle {\mathsf {CPoint}}}$

(T3) determines the tangent (see above), chooses a starting point on the tangent using step length ${\displaystyle s}$ (see diagram) and determines a second curve point ${\displaystyle P_{2}}$ using program ${\displaystyle {\mathsf {CPoint}}}$ .

${\displaystyle \cdots }$

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.

Example: An illustration of the raster algorithm applied to the implicit curve ${\displaystyle F(x,y)=(3x^{2}-y^{2})^{2}y^{2}-(x^{2}+y^{2})^{4}=0}$. The curve (red) is what the algorithm is trying to draw. The raster points (black) are used as starting points to find the closest points on the curve (red circles). The spacing between each raster point is exaggerated to show the individual curve points; to more accurately trace the curve, more raster points would be used.[4]

If the implicit curve consists of several or even unknown parts, it may be better to use a rasterisation 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 ${\displaystyle P}$ in the raster, run the point algorithm ${\displaystyle {\mathsf {CPoint}}}$ 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.