# Questions tagged [centroid]

"The centroid or geometric center of a plane or solid figure is the arithmetic mean ("average") position of all the points in the shape. " This tag is for questions about the centroid of a geometrical shape, its properties and computation.

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### Petr-Douglas-Neumann theorem - centroids don't match

According to Petr-Douglas-Neumann theorem (Wikipedia article) after all n-2 steps (n-number of sides/vertices in the polygon) are complete, the result is a regular polygon whose centroid coincides ...
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### How would one define the “center” of a closed, non-discrete 1d loop?

Centroids describe the average of all the discrete points on a curve (in 2d space), but why is it that the center of a circle or ellipse is what it is? Moreover, is it possible to determine the center ...
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### Is it possible to express an infinitesimal area as a sum of squares?

I came across a question which was asking the center of mass of an area, the centroid. The proposed solution uses weighted average, the formula is that given a planar region, the x and y coordinates ...
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### Find the center of mass of a quarter circle given by $\sqrt{r^2−x^2}$, $x\in[0,r]$

Q: The center of mass of the quarter circle given by $y=\sqrt{r^2−x^2}$, $x\in[0,r]$ is the point $(x, y)$. My first thought was that the circle was bounded by the points $(0,0)$, $(0,r)$, $(r,0)$ and ...
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### Characterization of the center of a polygon

Let $O$ be the circumcenter of the regular polygon $P_n$. Then for any $A\in P_n$ one has $d(A,O)\leq r$, where $d$ is the usual Euclidean distance and $r$ is the polygon's circumradius. Prove that ...
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### Computing a 3D quadrilateral inscribed sphere at its center?

For my game, I need to know when the player is over a powerup and for this I use a sphere collider: 3rd side view: Legend: orange, the powerup outline, a quad for which I know its centroid green, ...
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### The centroid of the zeros of the kth partial sum of exp(z) is -1?

The question is the next: Let $P_k=1+z+\frac{z^{2}}{2!}+...+\frac{z^{k}}{k!}$, the kth partial sum of $e^{z}$. (a) Show that, for all values of $k\geq 1$, the centroid of the zeros of $P_{k}$ is -1. (...
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### Does a plane passing througth the centroid of two points in euclidean 3D space must contain the two points?

Assume we have two points A and B in euclidean 3D space. The centroid of these two points is C. Now assume we define a plane P by means of the centroid C and an arbitrary normal vector n. Does the ...
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### Solve for the base of an isosceles triangle

The area of an isosceles triangle is $S$ and the angle between the medians to the legs, facing the base, is $\alpha$. Find the base of the triangle. Let $CH$ be the third median of the triangle. The ...
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### Formula for centroid of triangular number figurate

Triangular numbers are of the form T1 = 1 T2 = 1+2 T3 = 1+2+3 these can be represented as figurates. for example, a pool game has 15 balls = 1+2+3+4+5 = T5 ...
I am trying to find the centroid of a convex curve rotated at an angle by $\frac{3}{8}$ Simpson's rule (I have curve points, not the curve equation). The curve is intersected by a line $y=c$ (centroid ...