Talk:List of centroids

Triangle

It should be noted that there is a more general formula for the x coordinate of the triangle's centroid, as illustrated here, but the present illustration does not allow for this. --Tarnjp 05:57, 16 November 2006 (UTC)

That is correct, however, is it a practical method? DMZ 13:36, 17 November 2006 (UTC)

The coordinate of the centroid along the line on which the base of the triangle lies is simply ${\frac {a+b}{3}}$, where b is the length of the base of the triangle, and a is the distance between the upper vertex and one of the lower vertices, along a line parallel to the base. The number so obtained is relative to the lower vertex used. In the simplest case, the base of the triangle lies on the x axis, and one of the vertices lies at the origin (as in the previously linked illustration). One then uses the length of the base and the x coordinate of the upper vertex for a and b. Given that both of the required quantities can be easily measured, it is a practical method. --Tarnjp 01:35, 18 November 2006 (UTC)

Fair enough. Feel free to add it, just make sure it's illustrated correctly in the figure.. DMZ 19:02, 18 November 2006 (UTC)

"Sector Area" vs. "Circular Sector"

Based on the (slightly ambiguous) description of "Sector Area", it seems that "Sector Area" is a duplicate of "Circular Sector." The y coordinate of the centroid being zero suggests that the area described is symmetric about the origin, however the other formulae do not agree. In fact, with the given formula for area of a sector ($A={\frac {2\pi ^{2}r^{2}}{\alpha }}$), area varies inversely as $\alpha$. Furthermore, the formula for the x coordinate of the centroid is negative for values of $\alpha >\pi$. Both of these discrepancies need justification. --Tarnjp 04:43, 21 November 2006 (UTC)

Consistency and Usability

In order to be really usable, there need to be figures for all of these, not just some. You could tell from a figure what a parabolic spandrel is, for example.

Also, there is no consistency between labeling and shading of shapes. Some areas are not shaded, some are shaded grey, and one is shaded yellow. Sometimes the radius of a circle is R, sometimes r, and sometimes ${\rho }$. And for the circular arcs, the quantities listed under "Area" are actually lengths. —Preceding unsigned comment added by Hermanoere (talk • contribs) 19:09, 22 April 2009 (UTC)

Centroids and Bisectors of 3d Objects

The main article starts:

The centroid of an object X in n-dimensional space is the intersection of all hyperplanes that divide X into two parts of equal moment about the hyperplane.

And goes on:

For an object of uniform composition, the centroid of a body is also its center of mass.

Other sources state that it is possible for the 'centroid' of an object to be located outside of its geometric boundaries.

This may erroneously or otherwise facilitate a deduction allowing an object hyperplane bisector that does not pass through the object's centroid. This may erroneously or otherwise facilitate a deduction allowing a hyperplane on which the centroid lies not bisecting the object to which the centroid belongs. — Preceding unsigned comment added by 82.29.184.92 (talk) 12:42, 29 September 2020 (UTC)

The first line should make clear whether or not "all hyperplanes that divide" means 'every hyperplane that divides'.

i.e. Could it read:

The centroid of an object X in n-dimensional space is an intersection of, some/the majority of, hyperplanes that divide X into two parts of equal moment about the hyperplane.

Or is the intention to mean that the Centroid, be it inside or outside the object's geometric boundaries, is such that:

Any hyperplane on which an object's centroid lies bisects the object and that there are no other bisecting hyperplanes?

My interest concerns the Ham Sandwich Theorem and whether each of three objects can be considered as centroids as per the line immediately above. After finding out what is meant (consensually); it only remains to find out if it is true.

82.25.128.13 (talk) 23:13, 14 September 2020 (UTC)

Unambiguous Definition of Centroid

It is not clear that Centroid is taken universally by the mathematical community to be the point, such that only hyperplanes it lies on, can be equal volume bisectors of a compound object and no hyperplane it lies on cannot be. 82.29.184.92 (talk) 12:16, 15 October 2020 (UTC)

It is not clear:-

https://byjus.com/maths/centroid/ defines a property of a centroid thusly: It should always lie inside the object.

Wikipedia states: For an object of uniform composition, the centroid of a body is also its center of mass. https://en.wikipedia.org/wiki/List_of_centroid

            and: The center of mass may be located outside the physical body. https://en.wikipedia.org/wiki/Center_of_mass

orbital1337 YouTube as of 29/09/20, writes: " For a convex body of uniform density you can get up to 1 - 1/e on one side of a hyperplane through the center of mass (in the limit as the dimension goes to infinity)".

We need mathematicians to state explicitly what a centroid is and to then furnish a proof.

My interest concerns the Ham Sandwich Theorem and whether each of three objects can be considered as centroids (points) for the purpose of equal volume bisection. After finding out what is meant (consensually); it only remains to find out if it is true.

82.29.184.92 (talk) 13:02, 15 November 2020 (UTC)

82.29.184.92 (talk) 13:56, 29 September 2020 (UTC) 82.29.184.92 (talk) 12:16, 15 October 2020 (UTC)