How Do You Convert a Complex Spiral Equation from Polar to Cartesian Form?

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In summary, the conversation revolved around finding parametric and Cartesian equations for the equation |z| = \arg(z). The speaker was preparing to teach this topic to year 12 students and was struggling to explain it. They proposed using the angle \theta = \arg(z) as the parameter for the parametric equations, which simplified the process. They also mentioned that the Cartesian equation they attempted was |z| = \arg(z), but they got stuck and asked for suggestions on how to proceed. The expert provided a solution for the parametric equations and stated that the Cartesian equation was as simple as it could get.
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Rohitpi
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I'm volunteering in a summer school for year 12 students in my area, and have to teach them a few topics. I've been struggling to get the parametric equations from this.

Sketch: [tex]|z| = \arg(z)[/tex]

So I thought that the obvious way to explain it to them would be to say: "that as the magnitude of z increases (ie. distance from the origin) the greater the angle becomes, thus producing a spiral" and I can draw it on the whiteboard.

Upon attempting the Cartesian equation, I got a bit stuck:
[tex]|z| = \arg(z)[/tex]
[tex]\sqrt{x^2 + y^2} = \tan^{-1}(\frac{y}{x})[/tex]
[tex]\tan{(\sqrt{x^2 + y^2)}}=\frac{y}{x}[/tex]
[tex]y=x\tan{({\sqrt{x^2 + y^2}})}[/tex]

And that is where I get stuck unfortunately. Any thoughts/solutions on how to proceed in finding a Cartesian and/or parametric equations?

(sorry if this doesn't make sense, I haven't done any maths since year 12 haha as I study med atm :/ I miss maths!)

Thanks :)
 
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  • #2
You titled this "Cartesian equations" and refer to the Cartesian equation in the body but also ask for parametric equations. Which do you want?

You haven't said what parameter you want to use. I would recommend the angle [itex]\theta= arg(z)[/itex]. Then [itex]x= r cos(\theta)[/itex] and [itex]y= rsin(\theta)[/itex] while [itex]r= |z|= arg(z)= \theta[/itex] so the parametric equations are just [itex]x= \theta cos(\theta)[/itex], [itex]y= \theta sin(\theta)[/itex].

As for the Cartesian equation, I think you have it about a simple as you are going to get it.
 
  • #3
Ah, thanks! Sorry about the title, I should have included both Parametric and Cartesian. The issue I had with the parametric was as you alluded to, I actually didn't know where to start (hence the query was vague).

Thanks a lot! :)
 

FAQ: How Do You Convert a Complex Spiral Equation from Polar to Cartesian Form?

1. What is a Spiral Cartesian equation?

A Spiral Cartesian equation is a mathematical equation that represents a spiral shape on a Cartesian coordinate system. It is a polar equation that can be written in terms of either the radius or angle.

2. How is a Spiral Cartesian equation different from a regular Cartesian equation?

A regular Cartesian equation represents a straight line or a curve on a Cartesian coordinate system, while a Spiral Cartesian equation represents a spiral shape. Additionally, a regular Cartesian equation is written in terms of x and y coordinates, while a Spiral Cartesian equation is written in terms of polar coordinates.

3. What are the key components of a Spiral Cartesian equation?

The key components of a Spiral Cartesian equation are the center point, the starting angle, the winding number, and the radius function. The center point represents the point from which the spiral begins, the starting angle determines the starting position of the spiral, the winding number determines the number of times the spiral will loop around the center point, and the radius function determines the distance between the spiral and the center point at any given angle.

4. How is a Spiral Cartesian equation used in real life?

Spiral Cartesian equations have many real-life applications, such as in architecture, art, and engineering. They can be used to create visually appealing designs and structures, and they also have practical uses in fields such as manufacturing, where they can be used to create spiraled shapes in machines or products.

5. Are there any limitations to using a Spiral Cartesian equation?

One limitation of using a Spiral Cartesian equation is that it can only represent a spiral that follows a specific pattern, such as an Archimedean spiral or a logarithmic spiral. It cannot be used to represent other types of curves or shapes. Additionally, the equation may become more complex and difficult to work with as the winding number increases.

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