Help describing a region in polar coordinates

[BillyC]

Homework Statement

If (r, θ) are the polar coordinates of a point then describe the region defined by the restrictions
-1 < r < 0, π/2 < θ < 3π/2

Homework Equations

No clue

The Attempt at a Solution

I tried drawing the curve in a polar grid by starting at π/2 and finishing at 3π/2. I was careful to draw it according to the negative values of r. I ended up getting a shape that looked like half of a heart. So my answer was that it was a cardioid.

 

[BvU]

Hi,

Very unusual definition of polar coordinates (negative distance ? )

looked like half of a heart

Strange. Can you post it ?

 

[Mark44]

Hi,

Very unusual definition of polar coordinates (negative distance ? )
Strange. Can you post it ?

The r coordinate can be negative. For example, $(2, 5\pi/4)$ and $(-2, \pi/4)$ identify the same location.

I ended up getting a shape that looked like half of a heart. So my answer was that it was a cardioid.

No, that isn't right. The shape you get should be half a circular disk.

@Nidum, in answer to the question you posted, but deleted, yes, that is what is meant.

 

[Nidum]

Here it is again :

I think though that using ( r , theta + pi ) would be preferable to using ( - r , theta ) for the coordinates . It's difficult to visualise what - r means in polar coordinates .

 

[Nidum]

 

[Mark44]

Here it is again :

View attachment 196254

I think though that using ( r , theta + pi ) would be preferable to using ( - r , theta ) for the coordinates . It's difficult to visualise what - r means in polar coordinates .

It's really not that difficult to visualize. The ray in the first quadrant has r > 0. For a negative r with theta being the same, just go out the opposite direction as r is in. This works the same way as it does for vectors, where u and -u point in opposite directions.

 

[Buffu]

Homework Statement

If (r, θ) are the polar coordinates of a point then describe the region defined by the restrictions
-1 < r < 0, π/2 < θ < 3π/2

Homework Equations

No clue

The Attempt at a Solution

I tried drawing the curve in a polar grid by starting at π/2 and finishing at 3π/2. I was careful to draw it according to the negative values of r. I ended up getting a shape that looked like half of a heart. So my answer was that it was a cardioid.

With that negative $r$, I don't how you will plot the curve easily since most graphing calculator assume r to be positive.

Also it is impossible to get -ve $r$ from cartesian coordinates since $\sqrt{x^2 + y^2}= r$.

 

[Nidum]

No calculator needed . In fact no calculation needed .

Once you work out what is actually meant by the problem statement you can draw the required shape using simple straight lines and arcs .

 

[Mark44]

With that negative $r$, I don't how you will plot the curve easily since most graphing calculator assume r to be positive.

Also it is impossible to get -ve $r$ from cartesian coordinates since $\sqrt{x^2 + y^2}= r$.

But if the relationship is instead $r^2 = x^2 + y^2$, then you can get negative values of r.

From wikipedia, https://en.wikipedia.org/wiki/Polar_coordinate_system:
Emphasis added

Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates (r, ϕ ± n×360°) or (−r, ϕ ± (2n + 1)180°), where n is any integer. Moreover, the pole itself can be expressed as (0, ϕ) for any angle ϕ.

Where a unique representation is needed for any point, it is usual to limit r to non-negative numbers (r ≥ 0) and ϕ to the interval [0, 360°) or (−180°, 180°] (in radians, [0, 2π) or (−π, π]). One must also choose a unique azimuth for the pole, e.g., ϕ = 0.