Wikipedia:Reference desk/Archives/Mathematics/2016 June 18

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June 18

Degenerate annuli

Are the degenerate annuli the disks, where the inner circle has radius 0; and empty annuli, where the inner circle is the same as the outer one? GeoffreyT2000 (talk) 03:30, 18 June 2016 (UTC)

Article Annulus (mathematics) doesn't discuss those cases. Have you a reference to where these terms were used? -- SGBailey (talk) 22:53, 18 June 2016 (UTC)

How to find this plane

PLANES

I have a panel defined by three points(Green Triangle).The panel is having fixed size.I want to find the new coordinates(Red Triangle) of the triangle when its normal passing through the centroid of triangle intersects the origin. My GOAL is to move the green triangle as much so that its normal passes through the origin.red triangle shows what i want. Please help me solve the problem. How i approached is like this: 1.i need six equations to solve six variables-> 1,2,3.Three equations are obtained by keeping sides of triangle contstant. 4.The normal would be perpendicular to plane after rotation.

An6y other simpler method of approach would be much appreaciated.

i would share matlab code also... if someone helps. please help me greats

SD — Preceding unsigned comment added by Sameerdubey.sbp (talk • contribs) 05:18, 18 June 2016 (UTC)

This does not appear to be adequately constrained to produce a specific solution. Any solution can be rotated around the axis through the origin and the fixed point of the triangle to produce another solution. Is there another constraint that you would like to apply? —Quondum 14:26, 18 June 2016 (UTC)

Indeed, for every plane that does not go through the origin, you can draw a normal to the origin from some point; let that point be the centroid of the new triangle, and draw the new triangle around that centroid, rotated any way you like within that plane. Loraof (talk) 14:48, 18 June 2016 (UTC)

In fact, we could place the red triangle in the same plane as the green one. If we specify the problem that way, and if we require movement of the green to the red triangle via translation only (i.e., with no rotation), the problem is well-defined. I'd like to see how that is done to find (a, b, c) etc. Loraof (talk) 14:52, 18 June 2016 (UTC)

Not quite: the diagram implies the constraint that one of the vertices of the triangle is fixed, which eliminates all translations (other than zero displacement). We are confined to rotations and reflections around the fixed vertex of the triangle. One could find all solutions in terms of a parameter to represent the residual angle of rotation (plus a discrete indication of reflection), but I'd first like to see whether the intention was a fully constrained problem. —Quondum 15:51, 18 June 2016 (UTC)

Sorry I missed that fixed vertex. In your earlier post, Quondum, you said Any solution can be rotated around the axis through the origin and the fixed point of the triangle to produce another solution. But the original post said its normal passing through the centroid of triangle intersects the origin. Doesn't your rotation violate that? If so, maybe the problem is well-specified? Loraof (talk) 16:42, 18 June 2016 (UTC)

No, I don't see that it would violate that. Since the whole figure (triangle, centroid and normal) all rotate around the origin, any line that passes through the origin (the normal through the centriod in this instance) will remain passing through the origin. —Quondum 17:15, 18 June 2016 (UTC)

I don't see that. The normal to the plane from the origin intersects the plane at exactly one point. If that point is the centroid of one triangle, then it's not the centroid of a triangle formed by rotating that triangle about a point other than the centroid (in this case one of the vertices).Loraof (talk) 18:18, 18 June 2016 (UTC)

Or in other words, you're not rotating about the origin (or about the centroid)—you're rotating about the fixed vertex, which is not on a normal line to the origin. Loraof (talk) 18:29, 18 June 2016 (UTC)

You seem to be thinking only of rotations about lines that are normal to the plane of the triangle. I am referring to a rotation about a line (defined by the vertex and the origin) that is not normal to the plane of the triangle. It is a rotation around a line that contains the origin, and it changes the plane of the triangle. —Quondum 19:42, 18 June 2016 (UTC)

Okay, I see. It occurred to me that you could finish the problem specification by requiring the shortest, or longest, distance from the origin O to the centroid G. But that distance is always the same: the distance AO from the fixed vertex A to the origin is fixed; the distance AG from the fixed vertex to the centroid is fixed; and the triangle OAG has a right angle at G because of the constraint that the line GO is normal to the plane containing the original triangle and particularly containing GA. So GO² = AO² – GA² and hence is fixed. Is that right? Loraof (talk) 00:48, 19 June 2016 (UTC)

Yes. Every metric (distance, angle) on the entire configuration is unchanged by the rotation. The only possible constraint would be with reference to something external to the rotated configuration, and the only sensible reference would be the original position of the triangle. So one could, for example, ask for the solution that minimizes the distance of the centroid from its original position. Thus would result in the rotation being selected in which the plane containing the origin, the fixed point and the centriod contains the original centroid (from before the normal intersected the origin). Actually there'd be two solutions: this and also its reflection in this plane. —Quondum 01:20, 19 June 2016 (UTC)