# How to draw a curve in polar coordinates?

In summary, the conversation discusses how to draw curves in polar coordinates. The process involves making a rough sketch of the curve in Cartesian coordinates, then transferring the y-values to r-values on a polar coordinate paper. The conversation also touches on using derivatives to find critical points and how one period can give an idea of the behavior of the curve in other points.
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?

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.

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?

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.

## 1. How do I convert cartesian coordinates to polar coordinates to draw a curve?

To convert cartesian coordinates (x,y) to polar coordinates (r,θ), you can use the following equations:
r = √(x² + y²)
θ = tan⁻¹(y/x)
where r is the distance from the origin and θ is the angle with respect to the positive x-axis.

## 2. How do I plot a curve in polar coordinates given an equation?

To plot a curve in polar coordinates, first substitute values for the angle (θ) and solve for the corresponding radius (r) using the given equation. Then, plot the points (r,θ) on a polar coordinate system and connect them to create the curve.

## 3. What is the difference between a polar curve and a cartesian curve?

A polar curve is graphed using polar coordinates, where the distance from the origin and the angle with respect to the positive x-axis determine a point on the curve. A cartesian curve, on the other hand, is graphed using cartesian coordinates (x,y) where the x and y values determine a point on the curve. In polar coordinates, the curve is plotted in a circular or spiral shape, while in cartesian coordinates, the curve is plotted in a straight or curved line.

## 4. Can I draw a straight line in polar coordinates?

Yes, you can draw a straight line in polar coordinates if the equation for the line is in the form of r = a + bθ, where a and b are constants. This type of equation is known as a linear equation in polar coordinates and represents a straight line that passes through the origin.

## 5. How do I draw a polar curve on a graphing calculator?

To draw a polar curve on a graphing calculator, you can use the POLAR function or the POLARGRAPH function, depending on the type of calculator you have. These functions allow you to input the equation for the polar curve and graph it on a polar coordinate system. You can also adjust the window settings to change the range of values for the angles and the radius, which will affect the shape and size of the curve.

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