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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How to find distance between center of rectangle and bound with degree?

Given width and height of rectangle, coordinates of centroid and degree, how to find coordinates of a point extended from centroid with degree? In the below image, I want to calculate (??, ??). Any ...
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How to calculate the centroid of a Polytope?

Given a polytope is divided into simplexes, is it correct to calculate the centroid of the polytope as the average sum of its simplex centroid coordinates
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Prove that $AP \cdot AQ+CP \cdot CQ=BP\cdot BQ$

Let $G$ be the centroid of $\triangle ABC$. A line $M$ through $G$ intersects the circumcircle of $\triangle ABC$ at $P$ and $Q$, where $A$ and $C$ lie on same side of $M$. Prove that $AP \cdot AQ +CP ...
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Is this function monotonically increasing as $x_2$ increases?

Suppose I have a differentiable and continuous function $f(x)>0$, the monotonicity of $f(x)$ is unknown. Assume that $x_1< x_2< x_3 \in \mathcal{S}$, $\mathcal{S}$ is the domain of $f(x)$. ...
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How to construct the centroid of a quadrilateral

I know how to construct the centroid of a quadrilateral as mentioned here. But my question is different from that. We know that if points B,C,G are given in geometry plane and for locating point A ...
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Intersection of half angle with centroid

I am in calculus 2 and I recently came up with a question. With the formulas f(x)=n and g(x)=x^2 is there a value for n in which the half angle between the intersection of the parabola and the ...
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Moment Question

I am looking for feedback on some points actually. I am looking at the centroid equation $\frac{\int x f(x) dx}{\int f(x) dx}$, assuming that the integrals are bounded, will not consider constants. I ...
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Find list of closest points around a point in 3D space.

I have a list of points (x1,y1,z1) ...(xn,yn,zn). I have another point in this space, (X,Y,Z). I want to find a subset of the original points that form a boundary around (X,Y,Z). Or in other words, a ...
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In a triangle ABC, if certain areas are equal then P is its centroid

Let $P$ be a point in the interior of $\triangle ABC$. Extend $AP$, $BP$, and $CP$ to meet $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. If $\triangle APF$, $\triangle BPD$, and $\triangle ...
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n-fold centered tensor analog to double centered matrix?

A double centered matrix has the properties, that: the mean over all components the mean of each column the mean of each row is 0 (in every component). To double-center any given matrix $M$ with ...
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Pappus centroid theorem and Hypercones.

The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have: $$ V_{C3}=\int_0^h \pi r^...
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A system of particles distributed on the surface of a ball, what is the "center of mass" of them on the surface?

Suppose a system of $n$ particles distributed on the surface of a ball, what is the "center of mass" of them on the surface? Does the following optimization problem have an analytical ...
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How does one derive the centroid formula for multiple shapes?

I know how to derive the formula for the centroid of n sets of finite points, each with $m_k$ points and centroid $C_k$. The formula is: $$C=\frac{\sum_{k=1}^{n}C_km_k}{\sum_{k=1}^{n}m_k}$$ However, ...
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k-means clustering identity

Suppose $x_1, x_2, \dots, x_n$ are distinct points in $\mathbb{R}^d$, $\mathcal{S} = \{ S_1, S_2, \dots, S_k \}$ is a partition of $\{ x_i : i \in [n] \}$ (with $k \leq n$), and $\mu_i = \frac{\sum_{x ...
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For natural numbers $n$ and $k$, with $n \ge k$, if $k$ can be expressed as a sum of prime factors of $n$, then can $n-k$ not to be? (centre of mass)

Example I: $n=12$ and $k=5$, note that $5$ can be expressed as $2+3$ (which is a sum of prime factors of $12$), similarly $n-k=12-5=7$ can be expressed as $2+2+3$ (which is a sum of prime factors of $...
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Any interpretation for the fact that centroid = optimal point for maximising volume of such cuboid?

(Visualising a sample case via the image at the bottom) Consider a plane $\frac xa+\frac yb+\frac zc=1$ so that it intercepts with the axis at $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$, $a,b,c >0$. Now ...
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Thermal center of a set

The current Wikipedia article on the heat equation includes a gif which shows how the solution to the heat equation evolves over time if the initial data is the indicator function of a domain. ...
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Every $2$-colouring of the lattice points of $\mathbb R^m$ has $n$ monochromatic points whose centroid is a lattice point of the same colour

I was asked the question Prove that every $2$-colouring of the lattice points of $\mathbb R^m$ has a collection of $n$ monochromatic points whose centroid is a lattice point of the same colour Now, ...
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Lines from vertices through centroid of a triangle bisect opposite sides.

It is well known that the lines from the vertices through the centroid of a general triangle bisect the sides opposite to each vertex. Is there a simple geometrical proof for this? I've managed to ...
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Is it guaranteed that the centroid of a convex polygon will be the intersection points of lines connecting opposite vertices?

From Wikipedia, the centroid $\mathbf{C}$ of a finite set of points $\mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_k$ in $\mathbb{R}^n$ is: $$ \mathbf{C} = \frac{1}{k} \left( \mathbf{x}_1 + \cdots + \...
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Prove that the area centroid of a planar region in $\mathbb{R}^3$ projects onto that of its projection onto the $xy$-plane

Consider an arbitrary convex bounded region $\Omega'\in\mathbb{R}^2$ which is the projection onto the $xy$-plane of a region $\Omega$ that lies on a plane in $\mathbb{R}^3$ (that is not parallel to ...
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Centroid of connected 3D shapes

This might be a dumb question, but I tried to search for answers online and couldn't find any. I couldn't find too much information about centre of mass of irregular 3D objects in general. So, I have ...
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Explanation for the formula for the centroid $\bar{x}$

First, can someone provide a simple explanation for the $\bar{x}$ formula: $$\frac{\iint xdA}{area}$$ My understanding of the formula is as follows: we let $z=x$, calculate the volume, and divide by ...
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Under what conditions does every line through the centroid divide in half?

Let $G$ be an enclosed region of the complex plane, such that the property $\iint_G z dA=0$ holds; in other words, such that the center of mass is the origin. Draw an arbitrarily line $C$ passing ...
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The point of intersection of the four space diagonals of a general parallelepiped, the centroid

I know that the centroid of a parallelogram is the intersection of the diagonals, but is it true that the centroid of a parallelepiped is the intersection of the space diagonals? I'm talking about the ...
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Volume of a rotated ellipse by x=y

I'm calculating a volume of a rotated ellipse by the line $x=y$ using Pappus Theorem, the ellipse has an equation of : $$(9x^2/16)+(36y^2/25)=1$$ Using Pappus Theorem, I can just plug in the length of ...
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Centroid of the region bounded by y = f(x) and x = f(y) curve [closed]

How do i find the centroid of the region if $x = f(y)$ is involved ? For example : find the centroid of the region bounded by $y = x^2$ and $x = y^2$ And how do i find it if there is curve ...
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2 votes
1 answer
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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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Finding value of a when the centroid of a region (p,0) is bounded by $y^2 = 4px$ and x = a

The problem is basically the title, however I just got stuck when I did two solutions. Here is one of them: $$y^2 = 4px\\ f(x) = \sqrt{4px}\\ g(x) = -\sqrt{4px}\\ \bar{x} = p = \frac{M_y}{M}\\ M_y = \...
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Measuring symmetry / Is there a way to estimate the "centroid of an ideal shape" if I measure a skewed one?

How to measure asymmetry? Is there a way to estimate the "centroid of an ideal shape" if I measure a skewed one? I.e. I produce a measure of the centroid and it's "off", since the ...
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3 votes
4 answers
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In a triangle prove that $IG \perp BC \iff b + c = 3 a$

Consider the triangle $ABC$. Denote the incenter with $I$, the centroid with $G$ and $AB = c$, $AC = b$, $BC = a$. Prove that $IG \perp BC \iff b + c = 3 a$. I have tried using centroid's and incenter'...
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Condition for a rigid body in equilibrium on a horizontal surface

The following question is from a past paper from Further Mathematics and it has been bothering me immensely I have spent hours cracking at a way to make sense of it, the question is in two parts ...
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If we transform a set of unit vectors in $R^n$ using some $n \times n$ permutation matrix, is the centroid also transformed by the permutation matrix?

Given a set of unit vectors $V$ in $R^n$, the set's centroid $c^V$, and some $n \times n$ permutation matrix $P$, we define $V'=\{Pv~|~\forall v \in V\}$. Is it true that $c^{V'} = Pc^V$? If so, can ...
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How to find angle at centroid of triangle by its edge lengths?

I need to write a program that takes length of a triangle's edges and calculates the angle $\angle APB$ ($P$ is the centroid of the triangle). Thanks for any help or clue.
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How would I find the limit of this recursive sequence?

The Question $$x_n(c)=\frac{\int_{1}^{c} x_{n-1}(b)y_{n-1}(b)db}{\int_{1}^{c} y_{n-1}(b)x_{n-1}(b)'db}, x_1(c)=\frac{\int_{1}^{c} xf(x)dx}{\int_{1}^{c} f(x)dx} $$ $$y_n(c)=\frac{\int_{1}^{c} y_{n-1}(...
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3 answers
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Area of a triangle formed by mid-points of two sides and the other vertex

Let the medians $AK$ and $BM$ of a triangle $ABC$ intersect at $O$, $AB=13,BC=14,CA=15$. Find the area of $\triangle AOM$. The lengths of the medians are given by:$$m_a=AK=\dfrac12\sqrt{2b^2+2c^2-a^2}...
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Show for the centroid $M$ of a triangle $ABC$ that $\vec{OM}=\frac13(\vec{OA}+\vec{OB}+\vec{OC})$

Show for the centroid $M$ of a triangle $ABC$ that $$\vec{OM}=\dfrac13\left(\vec{OA}+\vec{OB}+\vec{OC}\right)$$ where $O$ is an arbitrary point. I haven't studied position vectors and am very new to ...
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2 votes
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Twisted ribbons and centre of gravity

NOTE After posting this question, I realised I had not defined the overall shape of the form. The final note at the bottom was intended to correct that omission. I am no longer certain how this ...
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Finding the centre of mass of two right cones joined combined

The following question is given in my textbook Two uniform cones with base radius $r$ are joined together by their plane faces.Their lines of symmetry are aligned, the height of one cone is $6r$ and ...
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Centroid of the Region bounded by the functions: $y = x, x = \frac{64}{y^2}$, and $y = 8$.

Can someone help and teach me how to solve this problem? Find the centroid of the region bounded by the graphs of $$\begin{align} &y = x\\ &x = \frac{64}{y^2}\\ &y = 8 \end{align}$$ ...
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2 answers
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Finding the center of mass of a composite lamina

The following lamina is given and I have to find the distance from center of mass from AB and AC I started by splitting the lamina into two shapes, one rectangle of width $4a$ height $5a$ and one of ...
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Centroid of semi-circular arc vs Average of coordinates

Equation for the centroid a semi circular arc is x=2*r/(pi). If I have a circle with 1 cm of radius then its centroid is 2r/(pi)= 0.637 cm. Now I use the equation of this circle x^2+y^2=1^2 in Matlab ...
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Centroid and "Centre of mass"

I thought that the centroid was also the point at which the area in all directions was net 0. Aka area above the cenrroid was the same area as below. But for a triangle everything I can see seems to ...
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Given x, y coordinates of a path, find the point that minimizes the interquartile range of radii away from that point.

I have a list of x, y ∈ [-100, 100] coordinates that form a path. Example path I wish to find the point x,y ∈ [-100, 100] that minimizes the interquartile range (IQR) of radii away from the point. I ...
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1 vote
2 answers
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Computing the centroid of some $n$ points in 3-dimensions

Solving for the centroid of some set of points in 2-dimensions is trivial. I'd like, however, to determine the centroid in 3-dimensions, given an arbitrary set of $n$ points. How can this most easily ...
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Can irregular polygons with unique vertices be guaranteed of a unique center point? Formula?

To visualize this, I've created the following diagram. I have sets of 80 numbers each. Here, just to draw it easily, I'm showing them as sets of 12 numbers each. The sets ...
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Calculating the centroid of a tetrahedron

Hope doing well and being healthy. I have a basic question on centroid of tetrahedrons. Are the coordinate of the centroid always the averages of $x$ and $y$ and $z$, by which I mean $$\frac14(x_1+...
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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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1 answer
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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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