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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3answers
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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
14 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
55 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 ...
2
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1answer
66 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
33 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
52 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
$$
z_0=\frac{z_1+z_2+z_3}{3}\implies 3z_0=z_1+z_2+z_3\...
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1answer
39 views
How to solve this particular task about centroids
Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there ...
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1answer
43 views
How to find the best fit point inside a cluster?
I have a cluster with many points.
Like this:
Where I can visually identify a cluster of points and a ...
0
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0answers
25 views
Find the total mass and COM of a body
So in this exercise I am asked to find the center of mass of a certain solid defined by 2 elipsoids that have the following parametrizations:
Elipsoid A:
$$x=\cos\alpha \sin\theta\\
y=\frac{1}{2}\...
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2answers
58 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
58 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
33 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
197 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
17 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
67 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
100 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 ...
1
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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}}={\...
1
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0answers
62 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 ...
4
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3answers
129 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 ...
2
votes
3answers
76 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
101 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
65 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
194 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
173 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 ...
3
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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
43 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
28 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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0answers
29 views
Why centroid of triangle is point of median?
I know what is centroid, but I don't know why intersection of median is centroid.
Would you explain about this logic?
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1answer
69 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. ...
1
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1answer
381 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
49 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
229 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
242 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
53 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 ...
2
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1answer
99 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_{\...
3
votes
1answer
167 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 ...
1
vote
1answer
71 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
97 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
32 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 ...
1
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1answer
320 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-...
1
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1answer
72 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 ...
6
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1answer
313 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 ...
2
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2answers
836 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
97 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
187 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
31 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]...
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1answer
3k views
Finding third vertex given 2 points and centroid
I am given the following two vertices: $A=(0,0)$, $B=(6,0)$ and centroid at $P=(4,2)$. Instead of finding midpoints such as $\overline{AC}$ and $\overline{BC}$ (because I end up getting really ...
