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Astroids and evolutes/involutes

In the article on Evolute, it is claimed that the involute of an evolute of a curve is the original curve again, and that the evolute of an ellipse is an astroid. These together imply that the involute of an astroid is an ellipse.

However, this article claims that the involute of a hypocycloid is a reduced version of the original hypocycloid.

These two statements are in obvious contradiction. Which of the two articles is correct? —Preceding unsigned comment added by 76.253.3.102 (talk) 19:29, 21 November 2010 (UTC)[reply]

Both are technically correct, a curve has only 1 evolute and infinitely many involutes depending on what point you start from. Zamadatix (talk) 17:26, 16 April 2011 (UTC)[reply]

Are the curves circular arcs - Flag of Portland

Are the curves, between the cusps, circular arcs (particularly for astroid, k=4)? If yes then can someone provide a reference ? if no then the claim that the flag of Portland, Oregon includes an astroid is false as the referenced city ordinance says it is formed from 4 quarter circles.

Since as k goes to infinity the curve tends to a cycloid it seems likely that the curves are not circular arcs for any k even 4. Astroid doen't claim the curves are circular so perhaps we should say here that the flag contains an astroid-like star. - Rod57 (talk) 10:24, 7 September 2012 (UTC)[reply]

"Smaller circle inside a larger circle"

Is it actually part of the definition of a hypocycloid that the rolling circle must be the smaller one? In the case where a larger circle rolls along a smaller circle in its interior, the parametric equations given still produce the correct curve. Would calling that curve a hypocycloid be incorrect? --Ian Maxwell (talk) 01:54, 16 September 2012 (UTC)[reply]

Hypocycloid rolling inside another

As shown in the picture, this still applies to the degenerate k=2 hypocycloid. Indeed, the deltoid is the smallest area in which it is possible to continuously rotate a line segment. --81.138.95.57 (talk) 12:32, 24 July 2014 (UTC)[reply]

@@ Line 20: / Line 20: @@
 Is it actually part of the definition of a hypocycloid that the rolling circle must be the smaller one? In the case where a larger circle rolls along a smaller circle in its interior, the parametric equations given still produce the correct curve. Would calling that curve a hypocycloid be incorrect? --[[User:Ian Maxwell|Ian Maxwell]] ([[User talk:Ian Maxwell|talk]]) 01:54, 16 September 2012 (UTC)
+== Hypocycloid rolling inside another ==
+As shown in the picture, this still applies to the degenerate k=2 hypocycloid. Indeed, the deltoid is the smallest area in which it is possible to continuously rotate a line segment. --[[Special:Contributions/81.138.95.57|81.138.95.57]] ([[User talk:81.138.95.57|talk]]) 12:32, 24 July 2014 (UTC)