A circle is easy to make:
For loudness N 1 sone: loudness level L N in phons = 40 × (N + 0.0005) 0.35 N in sones According to Stanley Smith Stevens' definition, 1 sone is equivalent to 40 phons, which is defined as the loudness level of a pure 1 kHz tone at L N = 40 dBSPL, but only (!) for a sine wave of 1 kHz and not for broadband noise. Downloads for each release of.NET Core 1.0; Release information Build apps - SDK Run apps - Runtime v 1.0.16. Security patch Release notes. Released 2019-05-14: SDK 1.1.14 Included in Visual Studio 15.0 Included runtimes.NET Core Runtime 1.1.13 ASP.NET Core Runtime 1.1.13. HueShifter is a standalone Mac app that works right alongside Adobe Photoshop and many other image editing applications. HueShifter supports Drag & Drop and Copy & Paste, so it's easy to get image data in and out. Drag or paste images into HueShifter, then drag or copy the results back to another app.
Draw a curve that is 'radius' away
from a central point.
from a central point.
Hueshifter 1 2 0 1
And so:
The current version of HueShifter can't do this, but the feature is coming. When we developed HueShifter for our own use, it frankly never occurred to us to put in the ability to colorize grayscale pixels. We always had the 'shift' function in our heads, which meant that there had to be some color in a pixel before it could be shifted. This expression checks the validity of a date (US, but it is easily editable for other format's). Year's 1990-9999, Month's 1 or 01 to 12, Day's 1 or 01 to 31.
All points are the same distance
from the center.
from the center.
In fact the definition of a circle is
Circle: The set of all points on a plane that are a fixed distance from a center.
Circle on a Graph
Let us put a circle of radius 5 on a graph:
Now let's work out exactly where all the points are.
Hueshifter 1 2 0 3
Mweb 3 3 7 equals. We make a right-angled triangle:
And then use Pythagoras:
x2 + y2 = 52
There are an infinite number of those points, here are some examples:
x | y | x2 + y2 |
---|---|---|
5 | 0 | 52 + 02 = 25 + 0 = 25 |
3 | 4 | 32 + 42 = 9 + 16 = 25 |
0 | 5 | 02 + 52 = 0 + 25 = 25 |
−4 | −3 | (−4)2 + (−3)2 = 16 + 9 = 25 |
0 | −5 | 02 + (−5)2 = 0 + 25 = 25 |
In all cases a point on the circle follows the rule x2 + y2 = radius2
We can use that idea to find a missing value
Example: x value of 2, and a radius of 5
Values we know:22 + y2 = 52
Square root both sides: y = ±√(52 − 22)
y ≈ ±4.58..
(The ± means there are two possible values: one with + the other with −)
And here are the two points:
More General Case
Now let us put the center at (a,b)
So the circle is all the points (x,y) that are 'r' away from the center (a,b).
Now lets work out where the points are (using a right-angled triangle and Pythagoras):
It is the same idea as before, but we need to subtract a and b:
(x−a)2 + (y−b)2 = r2
And that is the 'Standard Form' for the equation of a circle!
It shows all the important information at a glance: the center (a,b) and the radius r.
Example: A circle with center at (3,4) and a radius of 6:
Start with:
(x−a)2 + (y−b)2 = r2
Put in (a,b) and r:
(x−3)2 + (y−4)2 = 62
We can then use our algebra skills to simplify and rearrange that equation, depending on what we need it for.
Try it Yourself
'General Form'
But you may see a circle equation and not know it! Draft control 1 5 2.
Because it may not be in the neat 'Standard Form' above.
As an example, let us put some values to a, b and r and then expand it
Example: a=1, b=2, r=3:(x−1)2 + (y−2)2 = 32
Gather like terms:x2 + y2 − 2x − 4y + 1 + 4 − 9 = 0
And we end up with this:
x2 + y2 − 2x − 4y − 4 = 0
It is a circle equation, but 'in disguise'!
So when you see something like that think 'hmm .. that might be a circle!'
In fact we can write it in 'General Form' by putting constants instead of the numbers:
Note: General Form always has x2 + y2 for the first two terms.
Going From General Form to Standard Form
Now imagine we have an equation in General Form:
x2 + y2 + Ax + By + C = 0
How can we get it into Standard Form like this?
(x−a)2 + (y−b)2 = r2
The answer is to Complete the Square (read about that) twice .. once for x and once for y:
Example: x2 + y2 − 2x − 4y − 4 = 0
Put xs and ys together:(x2 − 2x) + (y2 − 4y) − 4 = 0
Now complete the square for x (take half of the −2, square it, and add to both sides):
And complete the square for y (take half of the −4, square it, and add to both sides):
(x2 − 2x + (−1)2) + (y2 − 4y + (−2)2) = 4 + (−1)2 + (−2)2
Tidy up:
Finally:(x − 1)2 + (y − 2)2 = 32
And we have it in Standard Form!
(Note: this used the a=1, b=2, r=3 example from before, so we got it right!)
Unit Circle
If we place the circle center at (0,0) and set the radius to 1 we get:
(x−a)2 + (y−b)2 = r2 (x−0)2 + (y−0)2 = 12 x2 + y2 = 1 Which is the equation of the Unit Circle |
How to Plot a Circle by Hand
1. Plot the center (a,b)
2. Plot 4 points 'radius' away from the center in the up, down, left and right direction
3. Sketch it in!
Example: Plot (x−4)2 + (y−2)2 = 25
The formula for a circle is (x−a)2 + (y−b)2 = r2
So the center is at (4,2)
And r2 is 25, so the radius is √25 = 5
So we can plot:
- The Center: (4,2)
- Up: (4,2+5) = (4,7)
- Down: (4,2−5) = (4,−3)
- Left: (4−5,2) = (−1,2)
- Right: (4+5,2) = (9,2)
Now, just sketch in the circle the best we can!
How to Plot a Circle on the Computer
We need to rearrange the formula so we get 'y='.
We should end up with two equations (top and bottom of circle) that can then be plotted.
Example: Plot (x−4)2 + (y−2)2 = 25
So the center is at (4,2), and the radius is √25 = 5
Rearrange to get 'y=':
Move (x−4)2 to the right: (y−2)2 = 25 − (x−4)2
(notice the ± 'plus/minus' ..
there can be two square roots!)
there can be two square roots!)
So when we plot these two equations we should have a circle:
- y = 2 + √[25 − (x−4)2]
- y = 2 − √[25 − (x−4)2]
Try plotting those functions on the Function Grapher.
It is also possible to use the Equation Grapher to do it all in one go.