Archemedian Spiral Flight Path: Calculating Arc Length & Equations

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In summary, the conversation is about developing a flight path for a circular area using an Archimedes spiral. The general equation in polar is r=a(theta)^1/n and the distance between each spiral can be calculated using d = a*(2*pi). There is no Cartesian equation for the spiral. The arc length can be calculated using s=0.5*a[theta*sqrt(1+theta^2)+ln(theta+sqrt(1+theta^2))]. The logic for finding the number of turns needed to cover a specific radius is to add the incremental distances between the spirals until reaching the desired radius. No flaws were mentioned in the conversation.
  • #1
rhimmelblau
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Hi I'm working on a project where I need to develop a flight path to cover a circular area. I was thinking of having the plane follow an archenemies spiral. I found that the general equation in polar is r=a(theta)^1/n
My question is if I have a specific distance I want each spiral to be from the last how do I input that into the equation.
Also is there a Cartesian equation for the spiral?
Edit: Also how does one calculate the arc length of the spiral?
 
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  • #2
The equation for the Archimedes spiral is:
R = a*theta
Each turn is separated from the last (and the next) by a distance (measured radially) of
d = a*(2*pi)
http://mathworld.wolfram.com/ArchimedesSpiral.html
There is no Cartesian Equation because it is not a single-valued function in cartesian space; there is no single value of x (or y) that can be associated with a given value of y (or x).


P.S. Don't tell your archenemies. Make them figure it out for themselves.
 
  • #3
So if I have a radius that I need to search and a distance each successive turn should be from the last, then I can use the arc length equation s=0.5*a[theta*sqrt(1+theta^2)+ln(theta+sqrt(1+theta^2))].
So I would plug in "a" equal to my distance/2*pi,
Then theta would be how many turns I go around the circle, which I can find by adding up the incremental distances between the spirals until I reach the radius of the search area.

Correct me if you see any flaws in my logic.
 
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FAQ: Archemedian Spiral Flight Path: Calculating Arc Length & Equations

What is an Archemedian Spiral Flight Path?

An Archemedian Spiral Flight Path is a type of trajectory in which an object follows a spiral path as it moves through space or air. The path is determined by an equation derived by the mathematician Archimedes, hence the name.

How is the Arc Length of an Archemedian Spiral Flight Path Calculated?

The arc length of an Archemedian Spiral Flight Path can be calculated using the equation S = aθ, where S is the arc length, a is a constant value, and θ is the angle of rotation. Alternatively, it can also be calculated using the formula S = rθ, where r is the distance from the center of the spiral to the point on the path.

What is the Equation for an Archemedian Spiral Flight Path?

The equation for an Archemedian Spiral Flight Path is r = aθ, where r is the distance from the center of the spiral to a point on the path, a is a constant value, and θ is the angle of rotation.

How is an Archemedian Spiral Flight Path Useful?

An Archemedian Spiral Flight Path is useful for a variety of applications, such as in aviation and space travel. It allows for efficient and smooth movement, making it ideal for long-distance travel and navigation.

What Factors Affect an Archemedian Spiral Flight Path?

The main factors that affect an Archemedian Spiral Flight Path are the constant value a and the angle of rotation θ. Changing these values can result in a different spiral trajectory, which can impact the path's arc length and overall efficiency.

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