Position on the plane of a man

[hlin818]

Homework Statement

An man starts on 0+0i of the complex plane. During the first walk step he moves a distance 1 to the right and lands on 1+0i. At each walk phase after this initial one the walk length decreases by a factor f<1 and he changes direction by an angle theta (measured from the horizontal and going counterclockwise). On what point does the man end up after N iterations of this? After an infinite number? What are all possible N=infinity locations?

Homework Equations

None that I know of. The path seems to form a spiral like pattern from the origin that curves counterclockwise inwards towards a center point.

The Attempt at a Solution

I tried to calculate the the position of the man by adding to the previous steps at each step by using trigonometry and adding the components to get the position at each step but that got very messy fast since the orientation of the angle is always changing and thus I would have to alternate between sines for the x values and cosines for the y values at seemingly irregular places. It looks like a formula is possible using this method but it would be incredibly messy and I doubt this is the way to do it.

 

[Dick]

Represent the step vector in complex polar form. Like f*exp(i*theta).

 

[LCKurtz]

Homework Statement

An man starts on 0+0i of the complex plane. During the first walk step he moves a distance 1 to the right and lands on 1+0i. At each walk phase after this initial one the walk length decreases by a factor f<1 and he changes direction by an angle theta (measured from the vertical and going counterclockwise). On what point does the man end up after N iterations of this? After an infinite number? What are all possible N=infinity locations?

Homework Equations

None that I know of. The path seems to form a spiral like pattern from the origin that curves counterclockwise inwards towards a center point.

The Attempt at a Solution

I tried to calculate the the position of the man by adding to the previous steps at each step by using trigonometry and adding the components to get the position at each step but that got very messy fast since the orientation of the angle is always changing and thus I would have to alternate between sines for the x values and cosines for the y values at seemingly irregular places. It looks like a formula is possible using this method but it would be incredibly messy and I doubt this is the way to do it.

Try writing the complex numbers in polar form, so the first two steps would be$$
z_2 = 1e^{i0}+fe^{i\theta}$$

 

[Simon Bridge]

Perhaps you can express each successive position in polar form? [edit]yes - I too will says it!
$$\vec{z}=Re^{j\theta}$$

Is the change in direction always the same angle to the vertical?
If that angle were 0 radians, the at the second step he'd go in the imaginary direction ... but successive steps would not change direction quite as much. If he'd started his first step to 0+i he'd just have kept going in that direction wouldn't he?

 

[hlin818]

Thank you guys, that takes care of the angle problem.

So would the general nth position be Zn=1*e^(i0)+f*e^(i*theta)+...+f^n*e^(n*i*theta)? Somehow I doubt this is correct - shouldn't the radius continually be decreasing?Oops - I meant to the *horizontal*. After the initial step theta goes counterclockwise like the unit circle does.

 

[LCKurtz]

Thank you guys, that takes care of the angle problem.

So would the general n+1th term be Zn+1=1*e^(i0)+f*e^(i*theta)+...+f^n*e^(n*i*theta)?

Yes, that is correct.

Somehow I doubt this is correct - shouldn't the radius continually be decreasing?

Isn't $f<1$ given?

 

[hlin818]

Bah. Never mind, I was looking at something wrong.

So would the set of all n=infinity locations be the whole argand plane? It seems that varying values the angle theta and the step function could eventually allow the sum of values to "converge" to any point on the plane.

 

[HallsofIvy]

Looks to me like you are going to end up with a geometric series.

 

[Dick]

Bah. Never mind, I was looking at something wrong.

So would the set of all n=infinity locations be the whole argand plane? It seems that varying values the angle theta and the step function could eventually allow the sum of values to "converge" to any point on the plane.

Since your series only converges for f<1, if you look carefully I don't think you'll find that the whole plane is covered.