Difference between revisions of "Geometric warm-up"
(Created page with "Category:Cosmo warm-up") |
|||
| (5 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | [[Category:Cosmo warm-up]] | + | [[Category:Cosmo warm-up|3]] |
| + | |||
| + | __TOC__ | ||
| + | |||
| + | <div id="razm7"></div> | ||
| + | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| + | === Problem 1: triangle on a sphere === | ||
| + | What is maximum sum of angles in a triangle on a sphere? | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;">$540^\circ$.</p> | ||
| + | </div> | ||
| + | </div></div> | ||
| + | |||
| + | |||
| + | |||
| + | <div id="razm8"></div> | ||
| + | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| + | === Problem 2: circle on a sphere === | ||
| + | $^*$ Consider the sphere of radius $R$. A circle is drawn on the sphere which has radius $r$ as measured along the sphere. Find the circumference of the circle as a function of $r$. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;">$\displaystyle L = 2\pi R\sin {r \over R}$.</p> | ||
| + | </div> | ||
| + | </div></div> | ||
| + | |||
| + | |||
| + | |||
| + | <div id="razm9"></div> | ||
| + | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| + | === Problem 3: density on a sphere === | ||
| + | Suppose that galaxies are distributed evenly on a two-dimensional sphere of radius $R$ with number density $n$ per unit area. Determine the total number $N$ of galaxies inside a radius $r$. Do you see more or fewer galaxies out to the same radius, compared to the flat case? | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;">$N = 2\pi nR^2 \left( {1 - \cos {r \over R}} \right) \approx n\pi R^2 \left( {1 - {{r^2 } \over {12R^2 }}} \right) < n\pi R^2 $.</p> | ||
| + | </div> | ||
| + | </div></div> | ||
| + | |||
| + | |||
| + | |||
| + | <div id="razm10"></div> | ||
| + | <div style="border: 1px solid #AAA; padding:5px;"> | ||
| + | === Problem 4: angular sizes in spaces of constant curvature === | ||
| + | An object of size $A$ is situated at distance $B$. Determine the angle at which the object is viewed in flat space and in spaces of constant (positive and negative) curvature. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;">In a flat space we have | ||
| + | \[\alpha = \arccos \left( {1 - \frac{A^2}{2B^2}} \right),\] | ||
| + | while in a space of constant negative curvature with radius of curvature $R$ | ||
| + | \[\alpha = \arccos \left( 1 - \frac{\operatorname{ch} \left( {A/R} \right) - 1}{\operatorname{sh} ^2 (B/R)} \right).\] | ||
| + | When $R \to \infty $ this case turns into the flat one. The expression for constant positive curvate could be obtained by replacement $R \to iR$.</p> | ||
| + | </div> | ||
| + | </div></div> | ||
Latest revision as of 13:17, 11 October 2012
Contents
Problem 1: triangle on a sphere
What is maximum sum of angles in a triangle on a sphere?
$540^\circ$.
Problem 2: circle on a sphere
$^*$ Consider the sphere of radius $R$. A circle is drawn on the sphere which has radius $r$ as measured along the sphere. Find the circumference of the circle as a function of $r$.
$\displaystyle L = 2\pi R\sin {r \over R}$.
Problem 3: density on a sphere
Suppose that galaxies are distributed evenly on a two-dimensional sphere of radius $R$ with number density $n$ per unit area. Determine the total number $N$ of galaxies inside a radius $r$. Do you see more or fewer galaxies out to the same radius, compared to the flat case?
$N = 2\pi nR^2 \left( {1 - \cos {r \over R}} \right) \approx n\pi R^2 \left( {1 - {{r^2 } \over {12R^2 }}} \right) < n\pi R^2 $.
Problem 4: angular sizes in spaces of constant curvature
An object of size $A$ is situated at distance $B$. Determine the angle at which the object is viewed in flat space and in spaces of constant (positive and negative) curvature.
In a flat space we have \[\alpha = \arccos \left( {1 - \frac{A^2}{2B^2}} \right),\] while in a space of constant negative curvature with radius of curvature $R$ \[\alpha = \arccos \left( 1 - \frac{\operatorname{ch} \left( {A/R} \right) - 1}{\operatorname{sh} ^2 (B/R)} \right).\] When $R \to \infty $ this case turns into the flat one. The expression for constant positive curvate could be obtained by replacement $R \to iR$.
