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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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Calculate coordinates of a centroid of $ABC$

We have a triangle $ABC$, where on the cartesian coordinate system: $A$ lies on $[-3, -2]$, $B$ lies on $[1, 1]$, $C$ lies on $[0, -6]$. How do we calculate coordinates for the centroid of this ...
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2answers
37 views

Proof that the triangle formed by mass centers is equilateral.

Let $ABC$ be a triangle and consider $A_1$, $B_1$, $C_1$ outside the triangle such that triangles $ABC_1$, $BCA_1$ and $ACB_1$ are equilaterals. Consider now $A_2$, $B_2$ and $C_2$ mass centres of $...
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3answers
82 views

Area of the intersection of two triangles.

Let $\triangle{ABC}$ be a triangle with $AB=5$, $BC=7$, and $CA=4$. Define $D$, $E$, and $F$, to be the midpoints of $AB$, $BC$, and $CA$ respectively. Let $G$ the intersection of the medians of $\...
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0answers
16 views

Center of mass for a solid that is symmetrical in one direction?

If I am trying to find the CM of a solid that is symmetrical in one direction (think an unsymmetrical shape that is now extruded from the page in both directions, i.e. into and out of, equally) this ...
2
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1answer
61 views

Intuitive Explanation to Pappus Theorem

Pappus's theorem is as follows: First theorem: The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on ...
3
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1answer
106 views

Centroid within non-convex 2d polygon

The centroid of an object is defined as the arithmetic mean of all points of the object. For non-convex objects, the centroid is often not a part of the object itself: Is there a definition of a ...
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1answer
37 views

centroid algorithm robust to missing poins

I need to find a center point of a person given the coordinates of all the joints. The joints of a person can be represented as a nodes of a graph with a fixed structure. The catch is some of the ...
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2answers
56 views

$z_1^2+z_2^2+z_3^2=3z_0^2$ if $z_1,z_2,z_3$ be the vertices of an equilateral triangle and $z_0$ be the circumcentre

Prove that, if $z_1,z_2,z_3$ be the vertices of an equilateral triangle and $z_0$ be the circumcentre, then $z_1^2+z_2^2+z_3^2=3z_0^2$ My Attempt $$ z_0=\frac{z_1+z_2+z_3}{3}\implies 3z_0=z_1+z_2+z_3\...
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1answer
40 views

How to solve this particular task about centroids

Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there ...
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1answer
44 views

How to find the best fit point inside a cluster?

I have a cluster with many points. Like this: Where I can visually identify a cluster of points and a ...
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0answers
27 views

Find the total mass and COM of a body

So in this exercise I am asked to find the center of mass of a certain solid defined by 2 elipsoids that have the following parametrizations: Elipsoid A: $$x=\cos\alpha \sin\theta\\ y=\frac{1}{2}\...
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2answers
71 views

How to find the barycenter/center of mass of this 2d shape without calculation?

I am given this shape (or any similar shape which is symmetrical). How can I determine the barycenter without calculating anything? My thought intuitively is, that the center should be at C, since it ...
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1answer
87 views

Coordinates of the centroid in a 3D rectangle

I have a 3D rectangle and I have to find the 3D coordinates of its centroid. I tried to take 4 vertices, one let's say the origin $o$ and the three adjacent vertices $a, b, c$. I computed the ...
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0answers
6 views

Nodal Centres in a Finite-Volume Approach

Consider the finite volume element in a cylindrical coordinate system shown in the figure below whose faces are bounded by the radii $r=\bar{r}_{i-1}$ and $r=\bar{r}_{i}$. The issue is how to ...
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1answer
43 views

Finding lengths of sides on triangles with 3 given medians and lengths

The medians of $△TUV$ are $\overline{TX}, \overline{UY},$ and $\overline{VW}$. They meet at a single point $Z$. In other words, $Z$ is the centroid of $△TUV$. Suppose $\overline{UY}=33$, $\overline{TZ}...
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2answers
293 views

Finding the centroid of a tetrahedron

I have four points to form a tetahedron $$A=(0,-\frac{1}{2},-\frac{1}{4}\sqrt{\frac{1}{2}}) \\B=(0,\frac{1}{2},-\frac{1}{4}\sqrt{\frac{1}{2}}) \\C=(-\frac{1}{2},0,\frac{1}{4}\sqrt{\frac{1}{2}}) \\D=(\...
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0answers
18 views

Calculating the centroids of two superposed gaussian functions

I am trying to find a solution to the following problem. I have a set of points which should model a sum of 2 Gaussian functions centered at different points. I need to find these two points. Up to ...
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1answer
77 views

Volumetric center of a polygon?

Is there a quick and efficient algorithm for calculating the volumetric center (probably the wrong term) of a polygon like that shown in the figure below somewhere around the blue dot? I'm not ...
7
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1answer
103 views

Can a spiral have its centroid at the origin?

A spiral is a curve $\gamma$ with the polar equation $r=f(\theta)$ where $f$ is a continuous positive strictly monotone function on some interval $[a, b]$, $-\infty<a<b<\infty$. Best known ...
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1answer
49 views

K-Means equality proof

Is there any geometrical and or short proof for the equality ${\displaystyle {\underset { }{ }}\sum _{i=1}^{k}\sum _{\mathbf {x} \in S_{i}}\left\|\mathbf {x} -{\boldsymbol {\mu}}_{i}\right\|^{2}}={\...
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0answers
67 views

How to find optimum (least square error) “square splitting circular arc” with given two centroids?

I am looking for an analytic solution for the following problem. A unit square is given. Coordinates of two points inside the square are given. What is the best circular arc which splits the square in ...
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3answers
130 views

Convex sum order

If I have a strictly convex function $f(x)$ with $f''(x)>0$ and if I know that for some $a\le b \le c$ and $x \le y \le z$ I have $$a+b+c = x+y+z$$ $$f(a)+f(b)+f(c)=f(x)+f(y)+f(z)$$ can I ...
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3answers
79 views

Centroid of a non-polygonal concave shape determined by an equilateral triangle an its incircle

I have a diagram here of an equilateral triangle ABC, centre O, where circle centre O has tangents which are all three sides of the triangle ABC. M is the midpoint of AB, and F is the intersection of ...
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0answers
119 views

Better “centerpoint” than centroid for placing a map marker inside a concave region (that may have holes)?

I'm using the centroid of polygons to attach a marker in a map application. This works definitely fine for convex polygons and quite good for many concave polygons. However, some polygons (banana, ...
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2answers
72 views

Line through triangle's vertex-centroid midpoint

I've been struggling for a couple of days trying to solve this geometry problem. Here it is: On this triangle, $G$ is its centroid (meaning that $AM = MC$). Also, $BN = NG$. He wants to know the ...
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2answers
215 views

What is the locus of centroid of triangle OAB?

A variable circle having fixed radius $a$, passes through origin and meets the coordinate axes in point $A$ and $B$. What is the locus of centroid of triangle $OAB$, $O$ being the origin?
2
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1answer
226 views

Centroid of quadrilateral on coordinate plane

I'm having trouble understanding the motivation between finding the centroid of a quadrilateral. Q: Find the centroid of a quadrilateral with vertices at (-8,12), (7,15) (13,-9), and (-2,-3). I've ...
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2answers
58 views

find ratio of area of triangle $\Delta{AFG}$ to area of $\Delta {ABC}$

In the figure below, $BD=DE=EC$, $F$ divides $AD$ so that $FA:FD=1:2$ and $G$ divides $AE$ so that $GA:GE=2:1$. Find ratio of area of triangle $\Delta{AFG}$ to area of $\Delta {ABC}$ My Try: I ...
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2answers
44 views

Centre of mass of a rod with infinitesimal width

The following is a thought that I am struggling to comprehend, most likely because of some logical flaw regarding infinitesimal objects. Consider a rod of uniform density, length $L$ and thickness $T$...
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0answers
29 views

Terminology: Point on the edge of convex polygon in the direction of a given point

Consider a convex polygon P with centroid c and a point p. Let q be the closest point to p on the line segment from p to c that is a member of P. What do you call the point q? What do you call ...
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1answer
82 views

How can prove that centroids of three equilateral triangles are collinear

Let $ABC$ be a triangle, $P$ be a point in the plane, $A'B'C'$ be the cevian triangle of $P$. Let point $A_b$ chosen on $CA$, point $A_c$ chosen on $AB$ such that $A'A_bA_c$ be an equilateral triangle ...
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0answers
41 views

Need help with an unusual surface area

I’ve developed a 3D rendering program for bodies of non-spherical revolution, by which I mean that as the curve is rotated about the vertical axis it is modulated by an arbitrary closed curve. ...
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1answer
432 views

Centroid of an Area Between Two Curves by Calculus

This problem is found on Stewart Calculus Metric Version, 7th Edition. It's quite difficult since you can only use integration along $y$. Here are the curve equations: $x = y^2$ and $x = y + 2$ If ...
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0answers
23 views

In a circle $C(O(0,0),1)$ with a polygon inscribed $A_1A_2…A_n$

In a circle $C(O(0,0),1)$ with a polygon inscribed $A_1A_2...A_n$, where $n \in \mathbb{N}, n\ge3$, such that $O$ is situated in the interior of the polygon. Let $G$ be the centroid (mass center) of ...
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1answer
50 views

Will moving towards the centroid of a triangle make us meet?

Suppose we have 3 people standing around and they want to meet. Normally that would be easy but they have quite a lot of restrictions. The only movement thay can make is to move towards the centroid ...
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2answers
276 views

Complex Numbers: Triangle and Centroid [closed]

Show that $(z_1+z_2+z_3)/3$ is the centroid of a triangle whose vertices are $z_1$, $z_2$, $z_3$. (Hint: The centroid divides the median internally in the ratio of 2:1)
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1answer
257 views

Is the Centroid and Circumcenter of a triangle affine invariant?

As the title says, are the centroid and circumcenter of a triangle affine invariant? And how would I go about proving it? Thanks.
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2answers
60 views

Centroid formula ($\bar y$) integral - why difference of squares, rather than squared difference?

When finding the $y$-coordinate of a centroid, for $f$ as a function of $x$, I understand the formula: $\bar y = \frac12 \int f(x)^2 dx$ when the lower function is $y=0$ What I don't understand ...
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1answer
123 views

Center of mass versus center of surface

Let's consider that an object has a uniform mass The center of surface is $$\vec C_s=\frac{\iint_{\mathbb{S}} \vec rdS}{\iint_{\mathbb{S}} dS}$$ And the center of mass is $$\vec C_v=\frac{\iiint_{\...
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1answer
180 views

Centroid in a Poincare disk model

I have $N$ points in a $D$-dimensional Poincare disk model. Ideally, I would like to have a centroid which would be representative of the cluster. To my knowledge, simple centroid calculation doesn't ...
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1answer
85 views

How to find the center of mass for a system of multiple solid spheres?

Assuming spheres have continuous and uniform mass distribution ($\rho$). How can we compute the center of mass (com), e.g. for a system with 3 solid spheres: $(x_1, ...
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1answer
110 views

Finding the centroid of a triangle in hyperspherical polar coordinates

I have three points: $A = (r{_0},\psi{_0},\theta{_0},\phi{_0})$ $B = (r{_1},\psi{_1},\theta{_1},\phi{_1})$ $C = (r{_2},\psi{_2},\theta{_2},\phi{_2})$ Actually, they are orthogonal, so that: $A = {...
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1answer
34 views

What is the proof the centers of mass lie along lines of symmetry and vise versa?

I was reading this and notices multiple references to the fact that lines of symmetry and centera of mass are deeply interconnected, and I was wondering if anyone could give me a proof, or an ...
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1answer
404 views

Center of volume and volume for triangle wedge

Let's say we have a wedge, where the base is a triangle, parallel with the $xy$-plane, and where only one of the sides extends perpendicular to the base: Here is the same wedge, looking down the Z-...
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1answer
90 views

Centroids and Position vectors

I'm currently struggling how to visualise this question. I.e where the vectors fall along the plane and how they are connected. If possible, a diagram would be great. Let K, L, M, N be four points ...
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1answer
315 views

Articles on the “Property I found” and other types of Centers (excluding the Centroid)?

I am a first-year undergraduate student. I came up with a kind of center property which I cannot find in articles online. I found the center on my own but needed help of mathematicians (@Rahul) on ...
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2answers
989 views

Centroid of a Trapezoid using double integrals

So, I need to find the center of mass from that trapezoid using double integration knowing the mass function: $\delta(x, y) = 1 + 2x + y$ What I've found so far is the boundaries are $y = -x + 3$, $y ...
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1answer
99 views

Integrating to find mass and centre of mass

A component of a stone plinth has a square base with corners at coordinates $(0,0,0,)$, $(0,2,0)$, $(2,2,0)$ and $(2,0,0)$. The height of the top surface is defined as $z =(1 + x^2 y)$. (All units ...
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2answers
230 views

I need to find the centre of gravity of a half ball?

Previously I had successfully calculated the C.O.G. of a Tetrahedron but after several attempts, I am not able to do the half ball one. the half ball $H = \{(x, y, z) \in \Bbb R^3| z\ge 0, x^2 + y^2 +...
1
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1answer
32 views

Centroid of a solid with given boundary

Given this solid in the picture, calculate the centroid: First things first: I calculated the filled area: $$A = \int_{0}^af(x_1)dx = b\int_{0}^ax^{\frac{1}{3}}dx = b\cdot\frac{3}{4}\cdot\sqrt[3]...