Envelope of a family of curves.

[ELESSAR TELKONT]

I have to justify that the envelope of a uniparametric family represented by $$f(x,y,c)=0$$ is the solution to the next system
$$f(x,y,c)=0, \frac{\partial f(x,y,c)}{\partial c}=0$$.

How I justify it, I don't know how to justify at all!

 

[EnumaElish]

What is the definition of an envelope?

 

[ELESSAR TELKONT]

the definition of an envelope is a curve that at every point of it there's a curve of the family tangential to the envelope. In other words, if the family is a result of a differential equation, the envelope is a singular solution (one that violates uniqueness in all of its points). But why this curve is obtained taking the derivative about the parameter of the family and the family itself, please help me

 

[EnumaElish]

But why this curve is obtained taking the derivative about the parameter of the family and the family itself, please help me

Based on your description, "the family itself" should be obvious: any point on the envelope belongs to some member of the family and must satisfy the family equation.

The derivative w/r/t/ the family parameter is less obvious. An envelope is called that because it "wraps" the family "from the outside" as it were. For any (x,y) combination, the envelope is tangent to either the "lowest" member (for a "cupping envelope") or the "highest" member (for a "capping envelope") of the family at that point. Letting E be the envelope, E(x,y) = F(x,y,c*) such that either F(x,y,c*) < F(x,y,c) or F(x,y,c*) > F(x,y,c) for all c. That condition is satisfied when dF/dc = 0.