# Inverter

"Without a tagline since 2010"

You need a browser which is capable of displaying scalable vector graphics to view this page properly.

Performance settings (Hide)
Number of steps:
Number of points:
Display settings (Show)
Source line colour:
Source fill colour:
Source line opacity:
Source fill opacity:
Curve line colour:
Curve fill colour:
Curve line opacity:
Curve fill opacity:
Axes opacity:
Unit circle opacity:
Canvas color:
Curve settings (Hide)
Input type
r = eg r = sin(4θ)
r = eg r = 1+cos(2*t)
x = eg x = sinh(5*t)
y = eg y = cosh(5*t)
y = eg y = x*x

The inversion transformation takes a point $$(x_0,y_0) = f(r_0,\theta_0)$$ on the plane and inverts it through the origin to give a new point $$(x_1,y_1)$$. The distance of the point $$(x_0,y_0)$$ to the origin is r0 and the distance from the point (x1,y1) to the origin is $$r_1$$. Similarly, the angles made with the $$x$$-axis are $$\theta_0$$ and $$\theta_1$$. The transformation for a point is:

$$(x_0,y_0) = f(r_0,\theta_0)$$
$$r_1 = 1/r_0$$
$$\theta_1 = \theta_0+\pi$$

### Shape definitions

The "source" (by default red) is transformed into a curve (by default green) in steps. The steps are intended to help you to visualise how each point transforms and each step, in isolation, is meaningless. The $$x$$ and $$y$$ axes transform onto themselves, and the unit circle ($$1=x^2+y^2$$) transforms onto itself. Any point inside the unit circle will transform to a point outside the unit circle, and vice versa. No point transforms onto itself.

### Display settings

The opacity and colours of the source, curve, unit circle and axes can be changed using the settings listed below.

### Performance settings

The inverter uses your computer's CPU to perform lots of calculations. By default the settings suit a low end computer, so the animation is not very smooth. If you want a smoother animation you can:

• Increase the number of points to around 5,000, to get a smoother curve.
• Increase the number of steps to get a smoother transition.
• Decrease the delay to speed up the transition.

### Curve definition

There are two ways to define the curve you want to transform. You can either provide a polygon, by specifying the points, or you can provide an equation to define the curve. If you define the curve with an equation and nothing shows, you may have a problem with your equation (eg division by zero, square root of a negative number, natural log of a negative number.)