How to draw a curve in polar coordinates?

[AdrianZ]

Hi all

I'm trying to find out how to draw a curve in polar coordinates. Can anyone help me with a book or something and help me find out how to draw curves in polar coordinates?

 

[LCKurtz]

Hi all

I'm trying to find out how to draw a curve in polar coordinates. Can anyone help me with a book or something and help me find out how to draw curves in polar coordinates?

I presume you want to sketch it by hand. Suppose you are trying to draw the graph in polar coordinates of r = sin(θ) + 1/2. First make a freehand sketch of y = sin(x) + 1/2, which is just an ordinary sine curve moved up by 1/2. You don't need to plug in any values; just make a nice looking sine curve drawn approximately to scale. You are going to use this sketch as a free-hand "table of values". Now draw a piece of polar coordinate paper which is just an xy axis with lines radiating from the origin every 30°.

Next, you visually transfer the y values from your xy graph to r values on your polar graph. So, for θ = 0 you notice your y value is positive so go the same distance in the r direction on the line θ = 0 and mark a point. Do the same thing for θ = 30°, 60°, and so on. You will see how the arches on the xy graph determine loops on the polar graph. You will have some negative values which plot in the negative r direction giving an inside loop. Once you have done a couple of these you will have the idea and will be ready to try other examples in your book.

Finally, if you need a really accurate graphs, you can use the exact values of the common angles instead of eyeballing it.

 

[AdrianZ]

Actually I was thinking about doing the same. Can I use derivatives as well? like I take dr/dθ and see where it's positive or negative and find the critical points? I think when dr/dθ is positive that means that when I rotate counter-clockwise r is increasing and when It's negative it means the opposite. in critical points when the derivative becomes zero it means that the behavior of r is changing, like it stops increasing and starts decreasing instead or the opposite. if the derivative didn't exist that means the curve at that point won't be smooth, it'll be something like a curvy V. am I right?
and can I say that if I draw the curve for one period, I'll have a clue of how the curve behaves in other points?

 

[LCKurtz]

Actually I was thinking about doing the same. Can I use derivatives as well? like I take dr/dθ and see where it's positive or negative and find the critical points? I think when dr/dθ is positive that means that when I rotate counter-clockwise r is increasing and when It's negative it means the opposite. in critical points when the derivative becomes zero it means that the behavior of r is changing, like it stops increasing and starts decreasing instead or the opposite. if the derivative didn't exist that means the curve at that point won't be smooth, it'll be something like a curvy V. am I right?
and can I say that if I draw the curve for one period, I'll have a clue of how the curve behaves in other points?

Yes, I pretty much agree with all of that. Of course, you will need more than one period for fiumctions like sin(4θ) but, yes, one loop of a multi-leafed rose is much like any other.