Obtaining the curve through envelope?

[ebola_virus]

I was looking through how to obtain the curve if you have a family of straight lines.. and I found that you 'obtain the envelope' promptly looking this up i got:The form of envelope treated here is a manifold that manages to be tangent to some point of each member of a family of manifolds. Curves are the usual manifolds involved.

The simplest formal expression for an envelope of curves in the (x,y)-plane is the pair of equations

F(x,y,t)=0\qquad\qquad(1)\,

{\partial F(x,y,t)\over\partial t}=0\qquad\qquad(2)\,

where the family is implicitly defined by (1). Obviously the family has to be "nicely" -- differentiably -- indexed by t.

The logic of this form may not be obvious, but in the vulgar: solutions of (2) are places where F(x,y,t), and thus (x,y), are "constant" in t -- ie, where "adjacent" family members intersect, which is another feature of the envelope. [wikipedia]

any idea on how this could be done? a tip on how enveloping would be helpfull.. thank you

 

[mighty2000]

Hello, thank you for your interest in this topic. Enveloping is a mathematical concept that is used to find a curve that is tangent to each member of a family of curves. This curve is called the envelope. To obtain the envelope, you can follow these steps:

1. Define the family of curves: The first step is to define the family of curves that you want to find the envelope for. This can be done by using a parameter, such as t, to represent the different curves in the family.

2. Write the equation for the envelope: As mentioned in the forum post, the envelope can be represented by the pair of equations (1) and (2). Equation (1) represents the curve itself, while equation (2) represents the condition for tangency.

3. Solve the equations: Once you have the equations, you can solve them to find the envelope. This can be done by using methods of calculus, such as finding the derivative and setting it equal to zero.

4. Verify the solution: After solving the equations, it is important to check if the solution is indeed the envelope of the family of curves. This can be done by substituting the solution back into the original equations and checking if they hold true.

I hope this helps in understanding the concept of enveloping and how it can be done. If you have any further questions, please let me know. Thank you.