Array variable of envelope function (parameter representation)

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.
  • #1

mk3

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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 𝑑 and array variable 𝑐
π‘₯=π‘₯(𝑑,𝑐)
𝑦=𝑦(𝑑,𝑐)

Condition for envelope is:
πœ•/πœ•π‘‘ π‘₯(𝑑,𝑐) πœ•/πœ•π‘ 𝑦(𝑑,𝑐)=πœ•/πœ•π‘ π‘₯(𝑑,𝑐) πœ•/πœ•π‘‘ 𝑦(𝑑,𝑐)

Solving this equation will give a relationship how the array variable 𝑐 depends on the curve parameter 𝑑 along the envelope:
𝑐_𝑒𝑛𝑣 (𝑑)=Ξ¨(𝑑)

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
 
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  • #2
mk3 said:
Hi, I have a question regarding the envelope function in parameter representation.
When you say 'envelope function' are you talking about this or this?

mk3 said:
Let an array of curves in cartesian coordinates be given in parameter representation, with curve parameter 𝑑 and array variable 𝑐
What do you mean by 'array' and 'array variable'?

mk3 said:
Condition for envelope is:
πœ•/πœ•π‘‘ π‘₯(𝑑,𝑐) πœ•/πœ•π‘ 𝑦(𝑑,𝑐)=πœ•/πœ•π‘ π‘₯(𝑑,𝑐) πœ•/πœ•π‘‘ 𝑦(𝑑,𝑐)
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?
 
  • #3
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
$$
f_c = \begin{cases}
x = x_c(t) \\
y = y_c(t)
\end{cases}
$$
 
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