# Polar and logarithmic images

## Polar coordinate image:

If you have studied advanced mathematics, you will definitely not be unfamiliar with the calculus inside, and the images inside will not be unfamiliar. Recall that when we draw these strange images with our hands, we can actually `MATLAB`

draw them easily. Let's take a look at the polar coordinate image. Its parameters include polar diameter `r`

and polar angle. `theta`

As the first example, we draw a simple curve: Archimedes spiral,

which `a`

is a constant. We draw a polar coordinate image between `a = 2`

and angle. In the `0～2*pi`

first statement, we define a constant:

```
a = 2;
```

Then we define the function r, which needs to be completed in two steps. In the first step, we treat the angle as a variable and define its interval length and increment:

```
>> theta = [0:pi/90;2*pi];
```

Then we write the function:

```
>> r = a*theta;
```

Then we draw the image:

In the second example, suppose we draw the following polar coordinate image:

the angle range here is `0～6*pi`

drawn with a dotted line.

First we define the angle interval:

```
>> theta = [0:pi/90:6*pi];
```

Input function:

```
>> r = 1 + 2*cos(theta);
```

We tell `MATLAB`

to draw with a red dotted line:

```
>> polar(theta,r,'r-.')
```

image:

## Logarithmic image:

Now let's see `MATLAB`

how to draw a logarithmic image. This used to give me a headache. If you are an electronic engineer, you will find this feature very useful. The first logarithmic image we can use is an `log-log`

image. We use a very classic example in an electronic circuit to see how to use it. This circuit contains a voltage source, capacitance and resistance. Since many readers are not electronic engineers, we do not need to discuss the source of this formula. Our purpose is only one: How to get a logarithmic image

It turns out that if the input voltage satisfies the sine relationship:

then the output voltage will be another sine function:

where the response frequency, which is the ratio of output to input (magnification), they have the following relationship:

general The frequency response tells us how much the output signal enhances the input signal at different frequencies. We can use it `拉普拉斯变换`

, so we can make

We take `w`

the range `1～100`

between, the unit of the product of resistance `R`

and capacitance is seconds. For our example, let seconds. These quantities are defined below :`C`

`RC`

`RC = 0.25`

`MATLAB`

```
>> RC = 0.25;
>>s = [1:100]*i;
```

Note that in the second line, we `s`

define complex variables. Frequency response is a logarithmic relationship between output / input and frequency, so we define:

So we pass the function to:

```
>> F = abs(1./(1+RC*s));
```

Now that all the conditions are there, the same `plot`

command can be `loglog`

used in:

```
>> loglog(imag(s),F),grid on,xlabel('频率 (rad/s)'),ylabel('输出/输入比'),title('频率响应')
```

This produces a very beautiful image:

## Another case of using logarithmic images:

`log-log`

Another type of image that uses images is when the given function transforms very quickly within a small range of the domain.

Let's look at an example of a function, `x`

the scope of which is `0～20`

between:

we first use ordinary methods to draw images:

```
>> x = [0:0.01:20];
>> y = exp(-10*x.^2);
>> plot(x,y),grid on
```

We can see that everything happened within a small range of the data set, let's try logarithmic images:

```
>> loglog(x,y)
```

Our `x`

axes and `y`

axes are in logarithmic form, we can also let them use `1，2，3`

such direct values directly, then we need the following two commands

`semilogx(x,y)`

: The image`x`

axis it produces uses logarithmic values, while the`y`

axis uses direct values`semilogy(x,y)`

: The image`y`

axis it produces uses logarithmic values, while the`x`

axis uses direct values