Geometric warm-up - Revision history

Igor at 13:17, 11 October 2012

2012-10-11T13:17:17Z

Cosmo All at 14:40, 20 August 2012

2012-08-20T14:40:37Z

Cosmo All: /* Problem 1 */

2012-08-20T14:40:11Z

‎Problem 1

Cosmo All: /* Problem 1 */

2012-08-20T14:39:35Z

‎Problem 1

Cosmo All at 14:37, 20 August 2012

2012-08-20T14:37:42Z

Cosmo All at 21:56, 23 July 2012

2012-07-23T21:56:54Z

Cosmo All: Created page with "Category:Cosmo warm-up"

2012-06-17T16:12:48Z

Created page with "Category:Cosmo warm-up"

New page

[[Category:Cosmo warm-up]]

@@ Line 1: / Line 1: @@
 [[Category:Cosmo warm-up|3]]
-__NOTOC__
+__TOC__
 <div id="razm7"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 1: triangle on a sphere ===
 What is maximum sum of angles in a triangle on a sphere?
 <div class="NavFrame collapsed">
@@ Line 18: / Line 18: @@
 <div id="razm8"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 2 ===
+=== 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 $r$.
+$^*$ 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>
@@ Line 31: / Line 31: @@
 <div id="razm9"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 3 ===
+=== 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">
@@ Line 44: / Line 44: @@
 <div id="razm10"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 4 ===
+=== 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">

@@ Line 18: / Line 18: @@
 <div id="razm8"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 2 ===
 * 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">
@@ Line 31: / Line 31: @@
 <div id="razm9"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 3 ===
 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">
@@ Line 44: / Line 44: @@
 <div id="razm10"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 4 ===
 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">

@@ Line 52: / Line 52: @@
 \[\alpha  = \arccos \left( {1 - \frac{A^2}{2B^2}} \right),\]
 while in a space of constant negative curvature with radius of curvature $R$
-\[\alpha  = \arccos \left( {1 - \frac{\operatorname{ch} \left( {A/R} \right) - 1}{\operatorname{sh} ^2 (B/R)} \right).\]
+\[\alpha  = \arccos \left( 1 - \frac{\operatorname{ch} \left( {A/R} \right) - 1}{\operatorname{sh} ^2 (B/R)} \right).\]
 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>

@@ Line 50: / Line 50: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">In a flat space we have
-\[
+\[\alpha  = \arccos \left( {1 - \frac{A^2}{2B^2}} \right),\]
-\alpha  = \arccos \left( {1 - \frac{{A^2 }} {{2B^2 }}} \right),
 \]
 while in a space of constant negative curvature with radius of curvature $R$
-\[
+\[\alpha  = \arccos \left( {1 - \frac{\operatorname{ch} \left( {A/R} \right) - 1}{\operatorname{sh} ^2 (B/R)} \right).\]
-\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>

← Older revision		Revision as of 14:37, 20 August 2012
Line 1:		Line 1:
	[[Category:Cosmo warm-up\|3]]		[[Category:Cosmo warm-up\|3]]
		+
		+	__NOTOC__
		+
		+	<div id="razm7"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	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 1 ===
		+	* 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 1 ===
		+	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 1 ===
		+	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>

← Older revision		Revision as of 21:56, 23 July 2012
Line 1:		Line 1:
−	[[Category:Cosmo warm-up]]	+	[[Category:Cosmo warm-up\|3]]