Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Brachistochrone"

(Created page with "A brachistochrone is the curve of fastest descent from <math>A</math> to <math>B</math>. It is described by parametric equations which are simple to derive: A cycloid is the...")
 
(spacing)
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
A brachistochrone is the curve of fastest descent from <math>A</math> to <math>B</math>. It is described by parametric equations which are simple to derive:
+
A brachistochrone is the curve of fastest descent from <math>A</math> to <math>B</math>.  
−
 
+
It is described by parametric equations which are simple to derive.
 +
<br />
 +
<br />
 +
==From Circle to Cycloid==
 
A cycloid is the path traced by a point on a rolling circle:
 
A cycloid is the path traced by a point on a rolling circle:
−
[asy]
+
[[File:Rollingcircle.png|How to Create a Cycloid]]
−
draw((0,0)--(9,0));
+
<br />
−
draw(Circle((0,1),1));
+
<br />
−
dot((0,0));
+
<br />
−
draw((0,1)--(3,1),Arrow);
+
<br />
−
[/asy]
+
If the radius of the circle is <math>a</math> and the center of the circle is moving at a speed of <math>a</math> units per second, then it moves <math>2\pi a</math> units, or one revolution, every <math>2\pi</math> seconds (in other words, it revolves 1 radian per 1 second).
 +
<br />
 +
<br />
 +
<br />
 +
<br />
 +
Then
  
−
If the radius of the circle is <math>a</math> and the center of the circle is moving at a speed of <math>a</math> units per second, then it moves <math>2\pi a</math>, or one revolution, every <math>2\pi</math> seconds (in other words, it revolves 1 radian per 1 second). Then
 
 
the x-coordinate of the center relative to the ground is: <math>at</math>
 
the x-coordinate of the center relative to the ground is: <math>at</math>
 +
 
the x-coordinate of the point relative to the center is: <math>a\cos(-\frac{\pi}{2}-t)=-a\sin t</math>
 
the x-coordinate of the point relative to the center is: <math>a\cos(-\frac{\pi}{2}-t)=-a\sin t</math>
 +
 
so the x-coordinate of the point is: <math>a(t-\sin t)</math>
 
so the x-coordinate of the point is: <math>a(t-\sin t)</math>
 +
<br />
 +
<br />
 +
<br />
 +
<br />
 +
Similarly,
  
−
Similarly,
 
 
the y-coordinate of the center relative to the ground is: <math>a</math>
 
the y-coordinate of the center relative to the ground is: <math>a</math>
 +
 
the y-coordinate of the point relative to the center is: <math>a\sin(-\frac{\pi}{2}-t)=-a\cos t</math>
 
the y-coordinate of the point relative to the center is: <math>a\sin(-\frac{\pi}{2}-t)=-a\cos t</math>
 +
 
so the y-coordinate of the point is: <math>a(1-\cos t)</math>
 
so the y-coordinate of the point is: <math>a(1-\cos t)</math>
−
 
+
<br />
 +
<br />
 +
==From Cycloid to Brachistochrone==
 
Since a brachistochrone is an upside-down cycloid, we reverse the sign of y:
 
Since a brachistochrone is an upside-down cycloid, we reverse the sign of y:
 
<math>x=a(t-\sin t)</math>
 
<math>x=a(t-\sin t)</math>
 
<math>y=a(\cos t-1)</math>
 
<math>y=a(\cos t-1)</math>
 
This is a brachistochrone starting at <math>(0,0)</math>.
 
This is a brachistochrone starting at <math>(0,0)</math>.
−
 
+
<br />
 +
<br />
 +
<br />
 +
<br />
 
If you want it to start at <math>(x_0,y_0)</math> you just shift it:
 
If you want it to start at <math>(x_0,y_0)</math> you just shift it:
 
<math>x-x_0=a(t-\sin t)</math>
 
<math>x-x_0=a(t-\sin t)</math>
 
<math>y-y_0=a(\cos t-1)</math>
 
<math>y-y_0=a(\cos t-1)</math>
−
 
+
<br />
 +
<br />
 +
<br />
 +
<br />
 
If you want it to go through <math>(c,d)</math> you need to solve for <math>a</math>:
 
If you want it to go through <math>(c,d)</math> you need to solve for <math>a</math>:
 
<math>c-x_0=a(t-\sin t)</math>
 
<math>c-x_0=a(t-\sin t)</math>
 
<math>d-y_0=a(\cos t-1)</math>
 
<math>d-y_0=a(\cos t-1)</math>
−
 
+
<br />
 +
<br />
 +
<br />
 +
<br />
 
I recommend first solving for <math>t</math> in terms of <math>a</math> using <math>d-y_0=a(\cos t-1)</math>, then substituting into <math>c-x_0=a(t-\sin t)</math>.
 
I recommend first solving for <math>t</math> in terms of <math>a</math> using <math>d-y_0=a(\cos t-1)</math>, then substituting into <math>c-x_0=a(t-\sin t)</math>.

Latest revision as of 02:41, 27 July 2019

A brachistochrone is the curve of fastest descent from $A$ to $B$. It is described by parametric equations which are simple to derive.

From Circle to Cycloid

A cycloid is the path traced by a point on a rolling circle: How to Create a Cycloid



If the radius of the circle is $a$ and the center of the circle is moving at a speed of $a$ units per second, then it moves $2\pi a$ units, or one revolution, every $2\pi$ seconds (in other words, it revolves 1 radian per 1 second).



Then

the x-coordinate of the center relative to the ground is: $at$

the x-coordinate of the point relative to the center is: $a\cos(-\frac{\pi}{2}-t)=-a\sin t$

so the x-coordinate of the point is: $a(t-\sin t)$



Similarly,

the y-coordinate of the center relative to the ground is: $a$

the y-coordinate of the point relative to the center is: $a\sin(-\frac{\pi}{2}-t)=-a\cos t$

so the y-coordinate of the point is: $a(1-\cos t)$

From Cycloid to Brachistochrone

Since a brachistochrone is an upside-down cycloid, we reverse the sign of y: $x=a(t-\sin t)$ $y=a(\cos t-1)$ This is a brachistochrone starting at $(0,0)$.



If you want it to start at $(x_0,y_0)$ you just shift it: $x-x_0=a(t-\sin t)$ $y-y_0=a(\cos t-1)$



If you want it to go through $(c,d)$ you need to solve for $a$: $c-x_0=a(t-\sin t)$ $d-y_0=a(\cos t-1)$



I recommend first solving for $t$ in terms of $a$ using $d-y_0=a(\cos t-1)$, then substituting into $c-x_0=a(t-\sin t)$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Brachistochrone&oldid=107964"