Re: Area of an envelope of a curve

On Wed, 05 Dec 2007 21:43:36 GMT, rob@xxxxxxxxxxxxxx (Rob Johnson)
wrote:

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.

Could you explain this a little more?

The theorems of Pappus seem to relate to surfaces and volumes of
revolution, whereas the OP's problem concerns the area of a planar
region.

quasi
.

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