---- > [!definition] Definition. ([[centroid of a parameterized manifold]]) > For a [[parameterized Euclidean manifold]] $Y _ \alpha$ where $\alpha:A \subset \rrk \to \rrn$, the **centroid** of $Y _\alpha$ is the point $\v c = (c_{1}, \dots, c_{n}),$ where $c_{i}=\frac{1}{v(Y_\alpha)} \int _{Y _\alpha} \pi _{i} \, \d V $ with $\pi_{i}: \rrn \to \rr$ denoting the $i^{th}$ [[projection function]]. > [!basicexample]- > **Question.** let $R > 0$. let $\alpha: (-R,R)\to \rr^{2}$ given by $\alpha(t)=(t,\sqrt{ R^{2}-t^{2} })$. let $Y=\alpha\big( (-R,R) \big)$. Find the [[centroid of a parameterized manifold|centroid]] of $Y _ \alpha$. > > **Solution.** We first want to find $v(Y_{\alpha})$. let > - $A := (-R,R)$, (so $Y:=\alpha(A)$) > - $B:=(0, \pi)$; > - $\beta: B \to \rr^{2}$ be given by $\beta(t)=\big(R\cos t, R\sin t\big)$, so that $Y=\beta(B)$; $\beta$ clearly is $C^\infty$; > - $g: A \to B$ be given by $g(t)=\cos ^{-1} (\frac{t}{R})$. > > (Note that we have seen before that $\alpha, \beta$ both parameterize a hemisphere of radius $R$ in $\rr^{2}$.) > > $g$ is [[continuously differentiable]] with [[continuously differentiable]] inverse $R\cos\left( \frac{t}{R} \right)$, hence a [[Euclidean diffeomorphism|diffeomorphism]]. Moreover, we have $\alpha=\beta \circ g$: observe that $\begin{align} > \beta(g(t))= & \left( R\cos\left( \cos ^{-1}\left( \frac{t}{R} \right) \right), R\sin\left( \cos ^{-1}\left( \frac{t}{R} \right) \right) \right) \\ > = & \left( \frac{Rt}{R}, R\sqrt{ 1-\left( \frac{t^{2}}{R^{2}} \right) } \right) \\ > = & \left( t, R\sqrt{ \frac{R^{2}-t^{2}}{R^{2}} } \right) \\ > = & (t,\sqrt{ R^{2} - t^{2} }) \\ > = & \alpha(t).\end{align}$ > Under these conditions, we have [[invariance under reparameterization of parameterized manifold|invariance of volume under reparameterization of]] $Y_{\alpha}$ by $\beta$, thus it suffices to compute $v(Y_\beta)$. > > Noting that $D\beta=[-R\sin t \ \ R \cos t]^{\top}$ and [[volume of a parameterized manifold|recalling the definition of volume of a parameterized manifold]]: $\begin{align} > v(Y_\alpha) = & v(Y_{\beta}) \\ > = & \int _B V(D \beta(t)) \, \d t \\ > = & \int_{0}^{\pi} V([-R\sin t \ \ R \cos t]^{\top})\,\d t \\ > = & \int_{0}^{\pi}\, \sqrt{ \det ([-R\sin t \ \ R \cos t]^{\top} \cdot [-R\sin t \ \ R \cos t]^{\top}) } \\ > = & \int_{0}^{\pi}\, R \ \d t \\ > = & \pi R. > \end{align}$ > Note that $V(D\beta(t))=R$ for all $t \in B$. > > This accords with our definition of [[arc length of a path]]. Next we want to compute $\int _{Y_{\alpha}} \pi_{i}\, \d V$ for $i \in \{ 1,2\}$; [[invariance under reparameterization of parameterized manifold|again it suffices to]] compute $\int_{Y_{\beta}}^{} \pi_{i} \, \d V$. This is $\begin{align} > \int_{B}^{} ( \pi_{i} \circ \beta ) V(D\beta) = & \int_{0}^{\pi} \pi_{i} (R\cos t, R \sin t) V(D\beta) \, \d t \\ > = & \begin{cases} \int_{0}^{\pi} R^{2} \cos t \, \, \d t & i = 1, \\ \int_{0}^{\pi} R^{2} \sin t\, \ \d t & i=2. \end{cases} \\ > = & \begin{cases} 0 & i = 1, \\ 2R^{2} & i=2. \end{cases} > \end{align}$ > This gives a final answer of $\textcolor{LimeGreen}{\v c = (\frac{0}{\pi R}, \frac{2R^{2}}{\pi R})=(0,2R / \pi)}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```