What is a Family of Curves?

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

Classification of a Curve Family

One-Parameter Family of Curves

If a location on a curve is defined, an equation for a specific member of that set of polynomial functions can be calculated. One-parameter curve family would be learning about using differential equations to locate the curves which cross the specified curves at right angles, a challenge that comes up often in applications. If an equation F (x, y, c) = 0 denotes a curve within the xy-plane with each given real value of c, and if vector c describes an infinite number of curves, then the totality of such curves is considered a one-variable family of curves, where c is considered the parameter of the family F.

A set of plane curves with only one variable is described mostly by equation f(x,y,C) = 0 in which C is a variable. The context of this set of curves is a curve that reaches one of its family's curves most tangentially for each point. This curve is called the envelope of the family of curves. The envelope's parameterized equations are described by a set of equations f(x,y,C) = 0 and f’C(x,y,C) = 0, which consists of the initial equation and also the equation derived by differentiating the initial equation w.r.t. the variable C. We could get the envelope equation in explicit (direct) or implicit (indirect) form by removing the variable C from such equations. It’s commonly described in differential equation formulations and solutions. The existence of such an envelope is predicated from the above system in the form of differential equations. Aside from the envelope curve, the approach of this scheme could have singular points of its family's curves that do not belong to the envelope. The discriminant curve is the set of all the solutions for the above system of equations. As a result, the envelope is a member of such a discriminant curve.

In contrast to the above system of differential equations, we assume that perhaps the following inequalities were fulfilled:

f x f y f ' C x f ' C y 0, 2 f C 2 0

It's worth noting that not every one-parameter family has an envelope. The set of concentric circles, that is defined by equation x 2 + y 2 = C 2 , is indeed a classic counter-instance.

Polynomial Family of Curves

A set of polynomial functions with the same zeroes is referred to as a family of curves. The x-intercepts of the graphs of the polynomial curves belonging to the same group are the same, but there is a possibility of y-intercepts being distinct (unless zero is one such x-intercept). The stability conditions are being used to determine the envelope equation specifically.

Orthogonal Family of Curves

At a given argument, two curves (graphs) are called orthogonal when their tangent lines are vertical or perpendicular to each other. This means that the two tangent lines' slopes are negative reciprocals of one another. If all of the curves within one family G(x,y,C1)=0 converge orthogonally all the curves in another family H(x,y,C2)=0, the families are called to be orthogonal trajectories. Mostly in the development of meteorological charts and the study or observation of electricity and magnetism, orthogonal trajectories arise obviously. The orthogonal direction of the given set of curves must be determined. The differential equation which defines the set of curves should be found. The second, and orthogonal, group differential equations are therefore further solved by following the different techniques to solve that set of curve equations.

Uses of Family Curves 

Orthogonal trajectories can be used in a variety of physics fields. The lines of force of the electrostatic field are orthogonal to a line of the same or constant potential. In aerodynamic performance, streamlines are orthogonal trajectories with the velocity-equipotential curve. It can be used in a wide range of topics in mathematics, such as the outline of a group as well as the caustic of a specific curve. Graphs are commonly used for science and technology. Graphs are extremely useful to historians, researchers, and economists.

Concept and Steps to be Followed While Solving

Consider the one-parameter curve family of orthogonal trajectories that always begins with the differential equation and let us make our way towards its solution. The argument is that developing the right differential equation is always a part of the problem. Such as, in real-life circumstances, people are always presented with a situation that they hope could be turned into a differential equation, and then they hope the equation could be solved. Define all curves with the property that the slope of the curve for each location on the curve would be equal to the square of the y-coordinate of the point from the geometric concept. The key fact of implementing the envelope principle is to demonstrate one source of a singular solution to a differential equation.

If one-parameter family y= f(x,C) has an envelope E, say y = g(x), then the envelope is a solution of that first differential equation (prove by differentiating these equations), so each location here on the envelope is a location on at least one of members of the one-parameter family. E can be found at any location (x0,y0) here on the envelope. It’s also found that the particular member of the family is uniquely determined by differentiating these equations.

Assuming that a set of curves is parametrized by t, i.e., a curve F x,y,t =0, for each value of t, let’s demonstrate how to locate the envelopes of such a family curve. That envelope seems to be the projection of a tangent plane's vertical locations. Whenever the normal or perpendicular to a tangent may have no vertical part, that tangent plane becomes vertical. The term dF dt is the vertical component of the normal or perpendicular plane. As a result, the envelope would be the position x,y,t where F x,y,t =0 and dF dt =0.

For Example: If a set of curves is generated by the term of y= 1 C cos Cx+α , whereas C is a variable and α is an arbitrary angle. A family of planes must be defined in a differential equation form.

y= 1 C cos Cx+α y'= dy dx = 1 C sin Cx+α C=y=sin Cx+α

By squaring both sides and add then we get;

y' 2 +C y 2 = cos 2 Cx+α + sin 2 Cx+α y' 2 +C y 2 =1 C= 1 y' 2 y

Thus, the group of curves is described as an implicit differential equation in the form:

y'=sin Cx+α =sin x 1 y' 2 y +α .

Examples of Family of Curves

  • Y=mx, where m is real, represents a family of lines.
  • Y2=4ax, where a is real represents a family of parabolas.
  • x=rcosθ,y=rsinθ, where θ is a variable of the curve, and r is a real number, represents a family of circles in a polar form.
  • The curve’s equation which involves x, y, and no other variables is a family of curves. For example, a circle having radius 3, and center at (0,3), x 2 + y3 2 =9; parabolas of the form x 2 =y and y 2 =x.
  • x 2 =y shows that y is a function of x, so dy dx can be found anywhere on that curve that shows y is said explicitly as a function of x. Similarly, y 2 =x shows x is a function of y , and here dx dy can be found anywhere on that curve that shows x is stated explicitly as a y function.
  • x 2 + y3 2 =9 shows neither y is expressed explicitly in x, nor x is expressed explicitly in y, which shows the link between x and y is a signified implicit function.

Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

  • Bachelor of Science in Mathematics
  • Master of Science in Mathematics

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