# Array variable of envelope function (parameter representation)

• A
• mk3
In summary, you should be able to consider the envelope of a family of curves by considering the curve defined by the parameter within that range.

#### mk3

Hi, I have a question regarding the envelope function in parameter representation.

Let an array of curves in cartesian coordinates be given in parameter representation, with curve parameter π‘ and array variable π
π₯=π₯(π‘,π)
π¦=π¦(π‘,π)

Condition for envelope is:
π/ππ‘ π₯(π‘,π) π/ππ π¦(π‘,π)=π/ππ π₯(π‘,π) π/ππ‘ π¦(π‘,π)

Solving this equation will give a relationship how the array variable π depends on the curve parameter π‘ along the envelope:
π_πππ£ (π‘)=Ξ¨(π‘)

But now here comes the question.

What if i want not the envelope of all array variables. Instead I want for example c from 0.5...1.0. How can I consider this in the function?
The target would be to give a limit for c.

Would be great to get a hint for that.

BG
MK

mk3 said:
Hi, I have a question regarding the envelope function in parameter representation.

mk3 said:
Let an array of curves in cartesian coordinates be given in parameter representation, with curve parameter π‘ and array variable π
What do you mean by 'array' and 'array variable'?

mk3 said:
Condition for envelope is:
π/ππ‘ π₯(π‘,π) π/ππ π¦(π‘,π)=π/ππ π₯(π‘,π) π/ππ‘ π¦(π‘,π)
What do you mean by ## \dfrac \partial {\partial c} ##? You have said that ## c ## is an 'array variable'; whatever that is it doesn't sound like something you could differentiate with respect to.

mk3 said:
What if i want not the envelope of all array variables. Instead I want for example c from 0.5...1.0. How can I consider this in the function?
Ok, now you've completely lost me: what is an 'array variable' that can take non-integer values?

Oh I think I see where the confusion lies, let me check.

You are talking about the envelope of a family of curves. We don't call this an 'envelope function' because the envolope is often not a function.

I think you are confused by the reference to a 'parameter' in relation to the family of curves. This has nothing to do with any parametric representation of individual curves within the family. I am not sure where you learned this stuff from but I think you need to go back over it.

When you have done that you should see that your final question translates to "what if i want not the envelope of all curves in the family. Instead I want for example the curves defined by the parameter ## c \in [0.5, 1 ]##?"
You should be able to answer this question by considering the definition of an envelope as a curve satisfying $$F(c, x, y) = \dfrac \partial {\partial c} F(c, x, y) = 0$$
If we restrict the values of ## c ## to a range then how does this affect the curve?

Note that I have kept to your use of the parameter ## c ## instead of the more usual ## t ##. Also rather than writing ## x = x(c, t) ## etc it would be better to specify an individual member of the family of curves as
$$f_c = \begin{cases} x = x_c(t) \\ y = y_c(t) \end{cases}$$

BvU