Re: Area of an envelope of a curve

In article <a08ee452-d34b-41fd-9f97-b5945686d079@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Martin <sleziak@xxxxxxxxxxxxx> wrote:
Let C be a smooth curve in a plane.
If I move a line segment of length 2d along this curve in a such way
that
* the center of the line segment is always on the curve
* the line segment is perpendicular to the curve
What can be said about area of the curve?

My guess (what the intuition suggests);
If the radius of curvature at each point is at least d, then the area
is
A = l*2d
where l denotes the length of the curve.
If this condition is not fulfilled, then the inequality
A <= l*2d
holds.

What you have stated is the First Theorem of Pappus

<http://mathworld.wolfram.com/PappussCentroidTheorem.html>

It is quite well known.

If I want to formulate similar problem also for piecewise continuous
curves (i.e. I do not have curvature in each point) one possibility
would be to take the points such that distance from the curve is at
most d. (I am in the similar situation as above, but I have added two
half-circles.)
My guess is that again a similar inequality should be valid.

As long as the curves sweeping out area don't overlap you are okay.
If they do overlap, the overlap is counted more than once.

Rob Johnson <rob@xxxxxxxxxxxxxx>
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