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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Centroids of a polygon
Obviously, for any polygon we can define at least $3$ different centroids:
$C1:\;$ mass center of the lamina;
$C2:\;$ mass center of vertices with equal masses;
$C3:\;$ mass center of the perimeter.
...
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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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Centroid and circumcenter -- how close?
Suppose $R$ is some planar region, bounded by a curve. Let $C_1$ be the centroid of $R$, and let $C_2$ be the center of the "circumcircle" (the smallest circle enclosing $R$). Intuitively, it seems ...
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Proof-validation of centroid's existence
So, a friend of mine came up with this unorthodox proof of the centroid's existence so I figured I could share it here so that someone can confirm that it's a fine one. I think it is correct, but I ...
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On the shoelace formulas
I've stumbled on this nice formula to compute the barycenter $\bar{c}$ of an arbitrary (but not self-intersecting) polygon
\begin{equation}
\bar{c} \triangleq \sum_{i=1}^n \frac{A_i}{A} \bar{c}_i
\...
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Geometry Question on the Circumcentre, Incentre and Centroid of a Triangle
Question
If there is a triangle $ABC$ where its circumcentre, incentre and centroid are named $O, I, G$ respectively, $AB=c$, $BC=a$, $CA=b$, and the radii of the circumcircle and incircle of the ...
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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
$$\frac14(x_1+...
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Finding centroid's coordinates using Pappus theorem
The task is to find the centroid of the given triangle (see the image above). We also should use the fact that the volume of a cone of radius $r$ and height $h$ is $V = \frac{1}{3}\Pi r^2h$. My ...
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How to Prove Triangle Centers in Tetrahedra
How would you prove the existence of triangle centers in tetrahedra, for example, the incenter, circumcenter, or centroid?
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Weighted center of mass by number of neighbors
As indicated by the tag, this is a reference request of a notion I thought of, but have not been able to find any literature on it. So, maybe someone here can link some resource.
Suppose there are $n$ ...
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k-means clustering identity
Suppose $x_1, x_2, \dots, x_n$ are distinct points in $\mathbb{R}^d$, $\mathcal{S} = \{ S_1, S_2, \dots, S_k \}$ is a partition of $\{ x_i : i \in [n] \}$ (with $k \leq n$), and $\mu_i = \frac{\sum_{x ...
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Every $2$-colouring of the lattice points of $\mathbb R^m$ has $n$ monochromatic points whose centroid is a lattice point of the same colour
I was asked the question
Prove that every $2$-colouring of the lattice points of $\mathbb R^m$ has a collection of $n$ monochromatic points whose centroid is a lattice point of the same colour
Now, ...
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Volume of a rotated ellipse by x=y
I'm calculating a volume of a rotated ellipse by the line $x=y$ using Pappus Theorem, the ellipse has an equation of :
$$(9x^2/16)+(36y^2/25)=1$$
Using Pappus Theorem, I can just plug in the length of ...
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Why is k proportional to centroid linkage distance mean and variance in k-means?
I've noticed that if I'm doing k-means clustering (in MATLAB) on any set of data, the mean and variance in centroid linkage distance appears to always be proportional to k.
Is k always proportional ...
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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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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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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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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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Calculating the center of mass of a solid
Let T be the solid consisting of the cylinder $x^2+y^2=1$, $z=cos(x^2+y^2)$ and the surface $z=0$, and let $F(x,y,z)=(-xz-2y)i+(x^2-yz)j+(z^2+1)k$
A) What is the center of mass of T?
B) (Irrelevant ...
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Find the centroid of the triangle
I'm having trouble computing the following exercise concerning the centroid of a general triangle, be it scalene, isosceles or equilateral. It goes like this:
Point A is the origin, point B belongs to ...
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Centroid of circle intersection
A region $R$ is bounded by the circle radius $4$ centred at $(5,0)$, the circle radius $2$ centred at $(0,0)$ and the x-axis, as shown.
What is the centroid of this region?
Finding the top point ...
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Calculate the area of a solid of revolution
So the subject title is self-descriptive. My question is how can I calculate the area of a solid of revolution with the information below:
...
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Barycenters and convexity
Background
Let $c(t):[0,1]\mapsto\mathbb{R}^2$ be a curve with the following properties
$c(t)$ is closed;
$c(t)$ is symmetric with respect to the horizontal axis;
$c(t)$ is contained in the rectangle ...
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Create a 2D plane tessellation starting from given centroids
I have a problem: I want to find a method which, given a set of centroids as x,y coordinates (the number of the centroids does not matter, more than one) it returns ...
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Unusual centroid of a figure
One day the question arose: how to determine centroid of annulus to be in the specified place?
Of course, it can't be usual centroid as a mass center.
We can use symmetry, radus and concentric circle ...
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Average of a function and center of mass
The geometric center of an object is the center of mass of an object with uniform density.
The average height of a semicircular wire is $\pi R/4$ where R is the radius. But the actual y coordinate of ...
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Why does the centroid of a triangle stay in place when one side of it is rotated?
I was experimenting with GeoGebra and I found something that I don't quite understand.
I wanted to see what would happen to the centroid of a triangle if one of its sides rotated inside a circle ($\...
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Circumscribed Circle And Inscribed Circle As Minima
Let $X\subseteq\mathbf{R}^2$ be open, bounded, non-empty and define $c_p$ as the set of points minimizing $\int_{x\in X} \|x-c\|^p$, so e.g. $c_2$ contains only the centroid of $X$.
I have the ...
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The gravity center
Good morning,
I want to prove that all points (A, B, C, D), as shown in the figure Image, compute the same center of gravity of the square.
Is there a way to prove that by using the local coordinates ...
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Vertices of a regular pentagon from center and side length?
I was wondering if it was possible to calculate the coordinates of every vertex in a regular pentagon using only its side length and its center?
The pentagon's center will not be fixed at the origin, ...
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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 ...
user999691
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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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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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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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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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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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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 of volume of revolution
Consider a solid generated by the curve $y^2 =ax^2+2bx+c$,rotated about the $x$-axis, and two plane surfaces at right angles to the latter, distance $h$ apart, and with areas $A$ and $B$. To prove ...
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Coordinates of centroid of a triangle
If $A(x_1,y_1) , B(x_2,y_2) , C(x_3,y_3)$ be the vertices of a triangle then prove that coordinates of centroid are given by $(\frac{(x_1 + x_2 + x_3)}3 , \frac{(y_1 + y_2 + y_3)}3)$
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find centroid of hyperpyramid
I'm having trouble computing centroid of hyperpyramid (assume we have n points in n dimension). I already search a lot and I find how can calculate triangle and pyramid centroid, but I don't know how ...
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What is the centroid of the area bounded by x^3 and 10-x?
I understand what the problem is asking, but I don't fully understand the concept behind it.
I know the formula for x bar is 1/A times the integral of x(f(x)-g(x)),
and that the formula for y bar is ...
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Area moments of a Freeman chain contour
The area moments of a polygon can be computed by generalizations of the shoelace formula for area.
In particular, the first and second order moments are given by
https://en.wikipedia.org/wiki/...
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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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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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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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How to understand a mysterious method to calculate the Centroid of a Quadrilateral?
We are trying to find the centroid of a quadrilateral by using a method that is different from Wiki: Centroid of a polygon.
For a convex (maybe applicable for an arbitrary) 3D quadrilateral has the ...
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Finding the Centroid of Solid G?
I saw this problem on one of my assignments and had no idea how to do it, mostly because I missed the section where it was covered. Anyways it states: Find the centroid of solid G defined by the ...
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Finding the center of mass
How can I find the coordinate $\bar{x}$ of centre of mass of the solid with the bounds $z=1-x^{2}$, $x=0$, $y=0$, $y=2$ and $z=0?$. Assuming a constant density.
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Find the exact coordinates of the centroid for the region bounded by the curves $y=\frac 1x$, $y=x$, $y=0$ and $x=2$
Im not sure how to set up these integrals. I tried using $y=x$ as the upper curve and subtract $y=\frac 1x$ in my integrals but I still am not getting the correct answer. If anyone knows how to do ...