Implicit curve

Cassini curves:
(1) a = 1.1, c = 1 (top),
(2) a = c = 1 (middle),
(3) a = 1, c = 1.05 (bottom)

Implicit curve: ${\displaystyle \sin(x+y)-\cos(xy)+1=0}$

the implicit curve as the level line of the surface ${\displaystyle \sin(x+y)-\cos(xy)+1=0}$ ${\displaystyle z=\sin(x+y)-\cos(xy)+1}$

An implicit curve in mathematics is a curve in the Euclidean plane that is represented by an equation of form

${\displaystyle F(x,y)=0}$

is described. An implicit curve is the totality of the zeros of a function of two variables. Implicit means that the equation of the curve is not for or solved. ${\displaystyle x}$ ${\displaystyle y}$

Function graphs are usually described by an equation and are therefore explicitly represented curves. The third important description of curves is the parameter representation : . The and coordinates of curve points are described by two functions that are dependent on a common parameter . The transition from one representation to another is usually only easy if there is an explicit representation : (implicit), (parameterized). ${\displaystyle y=f(x)}$ ${\displaystyle (x(t),y(t))}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x(t)\,,y(t)}$ ${\displaystyle y=f(x)}$ ${\displaystyle y-f(x)=0}$ ${\displaystyle (t,f(t))}$

Examples of implicit curves:

1. a straight line : ${\displaystyle x+2y-3=0}$
2. a circle : ${\displaystyle x^{2}+y^{2}-4=0}$
3. the semicubical parabola : ${\displaystyle x^{3}-y^{2}=0}$
4. Cassini curves (see picture), ${\displaystyle (x^{2}+y^{2})^{2}-2c^{2}(x^{2}-y^{2})-(a^{4}-c^{4})=0}$
5. ${\displaystyle \sin(x+y)-\cos(xy)+1=0}$ (see image).

While the first three examples also have simple parametric representations, this is not the case with the 4th and 5th examples. Example 5) shows that an implicit curve can consist of a confusing number of sub-curves.

With the theorem about implicit functions one can show that under certain conditions an equation can (theoretically) be solved gradually and / or gradually . However, the resolution is mostly practically impossible. However, this theoretical result is the key to calculate essential geometric properties such as tangents , normals and curvatures in known curve points using the given function (see below). The fact that implicit curves are not very popular in practice is due to a major disadvantage: While you can easily calculate any number of points for a parameterized curve or function graph, this is usually not the case for implicit curves. However, implicit representations of curves also have their advantages (see below). ${\displaystyle F(x,y)=0}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle F}$

If a polynomial is in and , the associated curve is called algebraic . ${\displaystyle F(x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$

Example 5) is not algebraic.

Remark: For a better understanding, an implicit curve with the equation can also be understood as a level line of height 0 of the surface (see 3rd illustration). ${\displaystyle F(x,y)=0}$ ${\displaystyle z=F(x,y)}$

Formulas

For the following formulas, the implicit curve is always described by an equation , whereby the function fulfills the necessary differentiability requirements. The partial derivatives of be with , , called so. ${\displaystyle F(x,y)=0}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle F_{x}}$ ${\displaystyle F_{y}}$ ${\displaystyle F_{xx}}$

Tangent and normal vector

A curve point is called regular, if ${\displaystyle (x_{0},y_{0})}$

• ${\displaystyle (F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))\neq (0,0)}$

holds, otherwise the point is called singular.

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

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

${\displaystyle \mathbf {n} (x_{0},y_{0})=(F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))^{T}}$is a normal vector .

curvature

In order to keep the formula clear, the arguments have been left out: ${\displaystyle (x_{0},y_{0})}$

• ${\displaystyle \kappa ={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_{y}^{2})^{3/2}}}}$is the curvature of the curve at a regular point.

Derivation of the formulas

The theorem about implicit functions (in the simplest case) says:

• If both and are valid for a sufficiently often differentiable function of two variables at one point , then a function with exists in a neighborhood of . ${\displaystyle F}$ ${\displaystyle (x_{0},y_{0})}$ ${\displaystyle F(x_{0},y_{0})=0}$ ${\displaystyle F_{y}(x_{0},y_{0})\neq 0}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle F(x,f(x))=0}$

The derivatives of the function result from implicit differentiation with the chain rule : ${\displaystyle f}$

• ${\displaystyle f'(x)=-{\frac {F_{x}(x,f(x))}{F_{y}(x,f(x))}}}$

${\displaystyle f''={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{F_{y}^{3}}}}$ (The arguments have been omitted here.)

→ Main article : Implicit differentiation

If the derivatives of calculated in this way are put into the formulas for the tangent and curvature of a function graph : ${\displaystyle f}$ ${\displaystyle (x,f(x))}$

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

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

one, then the above formulas result for the tangent and curvature of an implicit curve.

Note: If a resolution for x is possible, the same formulas result for the tangent and curvature of the implicit curve.

Advantages and disadvantages of implicit curves

disadvantage

The major disadvantage of implicit curves already mentioned above is the fundamental difficulty in calculating individual curve points . B. is absolutely necessary for the visualization of a curve. See the next section.

advantages

1. Implicit representations of curves have great advantages , especially when calculating the intersection points of two curves: If one curve is implicit and the other is parameterized, then only the usual one-dimensional Newton method has to be used to determine the intersection point. In the cases implicit-implicit or parameterized-parameterized, the use of the two-dimensional Newton method is necessary. See also: Intersection .
2. An implicit representation offers the possibility of dividing the points of the plane into two subsets based on the sign of . So you can z. B. using the sign of whether a point is inside or outside the unit circle. This can be important if you are looking for starting points for determining the intersection of curves or if you want to use the falsi rule instead of the Newton method . ${\displaystyle F(x,y)=0}$ ${\displaystyle F(x,y)}$ ${\displaystyle F(x,y)=x^{2}+y^{2}-1}$
3. Curves similar to an implicitly given curve can easily be given by considering curves with small numbers . (See section Smooth Approximations of Convex Polygons . ) ${\displaystyle F(x,y)=0}$ ${\displaystyle F(x,y)-c=0}$ ${\displaystyle c}$

Applications of implicit curves

Smooth approximation of a convex polygon

Smooth approximation of
(1) a semicircle,
(2) a circular triangle

In mathematics , implicit curves play an important role in the field of algebraic curves . In addition to this classic area of ​​application, implicit curves offer simple options for creating new curves. There are two methods:

Smooth approximations of convex polygons

To smooth approximation of convex polygons are implicit curves particularly suitable: if the sides of a convex polygon on the straight lines with equations so that the inside of the polygon in the positive areas of the functions is so describes the implicit curve ${\displaystyle n}$ ${\displaystyle g_{i}(x,y)=a_{i}x+b_{i}y+c_{i}=0,\ i=1,\dotsc ,n}$ ${\displaystyle g_{i}}$

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

for suitable positive numbers, smooth (differentiable) approximations of the polygon. For example surrender ${\displaystyle c}$

${\displaystyle F(x,y)=(x+1)(-x+1)y(-x-y+2)(x-y+2)-c=0}$ For ${\displaystyle c=0{,}03,\dotsc ,0{,}6}$

the smooth approximations of a pentagon shown in the picture.

Note 1:

If you include the borderline case “two straight lines”, you also get according to the construction described

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

either a family of parallel lines, if the given lines are parallel,
or the family of hyperbolas with the given lines as asymptotes, if the lines intersect. For example, the product of the x-axis with the y-axis yields:, the family of all hyperbolas with the coordinate axes as asymptotes. ${\displaystyle xy-c=0,\ c\neq 0}$

Note 2:

If you use other simple implicit curves (circles, parabolas, ...) instead of straight lines, you can also design interesting curves in a targeted manner. E.g.

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

(Product of a circle with the x-axis) provides smooth approximations of a semicircle (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) are smooth approximations of a circle-two-point (see picture).

Transition curves

Transition curves (red) for two circles

In the field of computer design, implicit curves are used to produce transition curves of particularly high quality ( geometric continuity ). For example, the following provides a simple construction

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

Continuous curvature transition curves between the two implicitly given circles

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

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

(see image). The two straight lines

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

determine the contact points of the transition curves on the circles. The parameter is a design parameter. In the picture is . Only the middle parts of the curve serve as transition curves. A vehicle could therefore drive along the transition curve from one circular arc to the other without jerking. ${\displaystyle \mu }$ ${\displaystyle \mu =0{,}05,\dotsc ,0{,}2}$

Equipotential lines of two equal point charges

Equipotential lines of two equal point charges in the blue points

The equipotential lines of two equal point charges in the points can be implicitly passed through ${\displaystyle P_{1}=(1,0);\;P_{2}=(-1,0)}$

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

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

describe. For the curve results from the origin and has a colon. The curves look like Cassini curves , but they are not. ${\displaystyle c=2}$

Visualization of an implicit curve

To visualize a curve, a polygon is usually calculated from curve points and this polygon is drawn. This is not a problem with a parameterized curve: the associated sequence of curve points can be calculated directly for a given sequence of parameters. With an implicit curve one has to solve two sub-problems:

1. calculate a curve point for a starting point near the curve,
2. Determine a starting point for another curve point from a known curve point.

For the solution of both problems it is favorable to assume that is not the zero vector. This seems like a severe limitation. As a rule, this requirement is only violated in isolated points and in practice it is rather unlikely that you will encounter exactly such a point. ${\displaystyle \operatorname {grad} F}$

Point algorithm

An implicit curve requires a computer program that calculates a curve point for a starting point near the curve : ${\displaystyle {\mathsf {KPunkt}}}$ ${\displaystyle Q_{0}=(x_{0},y_{0})}$ ${\displaystyle P}$

(P1) For the starting 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 the 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 is small enough. ${\displaystyle (x_{j+1},y_{j+1}),\,(x_{j},y_{j})}$

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

Tracking algorithm

Regarding the tracking algorithm: starting points are green

A polygon on the implicit curve having a step size to produce ${\displaystyle \approx s}$

(V1) one chooses a suitable starting point near the curve.

(V2) calculated with the first point on the curve . ${\displaystyle {\mathsf {KPunkt}}}$ ${\displaystyle P_{1}}$

(V3) calculates the tangent (see above), selects a further starting point on the tangent with the step size and calculates the second point on the curve . ${\displaystyle s}$ ${\displaystyle {\mathsf {KPunkt}}}$ ${\displaystyle P_{2}}$

${\displaystyle \cdots }$

Since the algorithm follows the course of the curve, it is called the tracking algorithm . The algorithm only ever delivers individual components of the implicit curve. You may have to go through it several times with suitable starting points.

Raster algorithm for implicit curves

Raster algorithm

If the implicit curve consists of many sub-curves, the following raster algorithm provides a good visualization of the curve:

(R1) Create a network (grid) in the area of ​​the xy plane in question.

(R2) Use each point of the grid as the starting point for the point algorithm and mark the curve point obtained in this way. ${\displaystyle {\mathsf {KPunkt}}}$

If you make the mesh very dense, you get a good impression of the implicit curve and you can then process interesting parts with the tracking algorithm.

Example: The picture shows a (rough for demonstration) grid with the associated calculated curve points for the implicit curve with the equation:

${\displaystyle F(x,y)=(3x^{2}-y^{2})^{2}y^{2}-(x^{2}+y^{2})^{4}=0}$

(An optimized algorithm was used to produce the image, in which not every grid point is used as the starting point.)

software

Implicit curves can be displayed graphically with the help of suitable visualization programs , for example with the free software

• GeoGebra (only for algebraic curves) or
• Gnuplot .

References to other free software can be found in the Web Links section .

Implicit space curves

A curve in space given by 2 equations

${\displaystyle {\begin{matrix}F(x,y,z)=0,\\G(x,y,z)=0\end{matrix}}}$

is called an implicit space curve.

A curve point is called regular if the cross product of the gradients from and at this point does not result in the zero vector: ${\displaystyle (x_{0},y_{0},z_{0})}$ ${\displaystyle F}$ ${\displaystyle G}$

${\displaystyle \mathbf {t} (x_{0},y_{0},z_{0})=\operatorname {grad} F(x_{0},y_{0},z_{0})\times \operatorname {grad} G(x_{0},y_{0},z_{0})\neq (0,0,0)}$

is otherwise singular. The vector is a tangent vector at the point on the space curve. ${\displaystyle \mathbf {t} (x_{0},y_{0},z_{0})}$ ${\displaystyle (x_{0},y_{0},z_{0})}$

Sphere-cylinder intersection curve

Examples:

${\displaystyle (1)\quad x+y+z-1=0\ ,\ x-y+z-2=0}$

describes a straight line.

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

describes a plane section of a sphere, i.e. a circle.

${\displaystyle (3)\quad x^{2}+y^{2}-1=0\ ,\ x+y+z-1=0}$

describes an ellipse (plane cylinder section).

${\displaystyle (4)\quad x^{2}+y^{2}+z^{2}-16=0\ ,\ (y-y_{0})^{2}+z^{2}-9=0}$

describes the intersection of a sphere with a cylinder.

For the calculation of curve points and the visualization of an implicit space curve, see intersection curve .

See also

• Implicit area

Web links

• M. Erne: Mathematics III for civil engineers. Chapter 10.6 .: Implicit plane curves and tangents. (PDF), University of Hanover.
• COMPUTER-aided descriptive and constructive geometry. P. 33–37 (implicit curves) and P. 200 (transition curves), Uni Darmstadt (PDF; 3.4 MB).
• Desmos graphics calculator shows implicit curves.
• Famous Curves.

literature

• Hoffmann, Marx, Vogt: Mathematics for Engineers 1st Pearson Studies, 2005, ISBN 3-8273-7113-9 , p. 519.
• G. Taubin: Distance Approximations for Rastering Implicit Curves. ACM Transactions on Graphics, Vol. 13, No. 1, 1994.
• A. Gomes, I. Voiculescu, J. Jorge, B. Wyvill, C. Galbraith: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms. 2009, Springer-Verlag London, ISBN 978-1-84882-405-8 .
• R. Goldman: Curvature formulas for implicit curves and surfaces.
• CL Bajaj, CM Hoffmann, RE Lynch: Tracing surface intersections. Comp. Aided Geom. Design 5 (1988), 285-307.