Array variable of envelope function (parameter representation)

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  • #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
 

Answers and Replies

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

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'?

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.

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
pbuk
Science Advisor
Gold Member
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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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