Inversion
Circle Inversion |
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Circle Inversion
- This image is an example of a fractal pattern that can be created with repeated inversion in circles.
Contents
Basic Description
Inversion is a type of transformation that moves points from the inside of a circle to the outside and from the outside of a circle to the inside using a specific rule. It can be used to create images such as the one on this page, but more importantly it can be used to simplify certain proofs. Inversion is also a method for understanding and solving the Problem of Apollonius
Inversion of a Point
The inversion transformation is defined by the rule that the distance from the center of the circle, O, to the original point, P', times the distance from O to the inverse point, P, equals the length of the radius of the circle, k, squared. In mathematical notation: .
This applet can be seen on its own page at: Inversion of A Point
To the left you can see exactly how inversion works by moving the red point which you can think of as P' and watching the blue inverse point , P, move depending on the rule:.
A More Mathematical Explanation
Important Properties
In the diagram we can see other properties of inversion as well. For example:
- Q is a point on the circle such that OQ is perpendicular to PQ
- P' is the foot of the altitude of the triangle OQP where QP' is perpendicular to OP
- We can now see that triangle OP'Q is similar to triangle OQP
From these properties we are able to deduce the equation which is equivalent to our first equation: .
Notice therefore that a point on the inside of the circle is inverted to a point outside the circle and vice versa, a point on the circle is inverted to itself, and the point at the center of the circle is inverted to the point at infinity and vice versa.
This concept of a point at infinity becomes clear when we observe using the point at the center of the circle as P. OP is therefore equal to zero so it follows that as well. Since , we now have . Solving the equation where k is a constant, it follows that OP' must be infinitely large and P' must be the point at infinity.
The point at infinity can be thought of as very close to points extremely far out in all directions. So because lines go out to infinity in both directions you can think of their end points meeting at the point at infinity. If then this line's end points meet then you can think of it as a circle.
Inversion of Curves
This applet can be seen on its own page at: Inversion of A ShapeThis inversion transformation can also be used to invert entire curves like the ones in the main image of this page. This is done by inverting all of the points along the original curve to get the inverted curve. The applet to the left shows how the red circle is inverted about the black circle to get the blue inversion curve. Let's call the center of the black circle 'O' for clarity's sake.
By moving around the red curve in the applet on the left, we notice that inversion of a circle that passes through the O will always be a line that does not pass through O. The inversion of a line not passing through O is always a circle that does pass through O.
Based on the equations and what we observe about the point at the center of the circle being inverted to the point at infinity and vice versa, we can think of a line as a curve where the two ends meet at the point at infinity.
Circles that do not pass through O invert to another circle that does not pass through O. Circles and lines invert to themselves if and only if they are orthogonal to the circle of inversion.
3D Inversion
Teaching Materials
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Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
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|AuthorName=Xah Lee
|SiteName=Circle Inversion Gallery
|SiteURL=http://www.xahlee.org/SpecialPlaneCurves_dir/InversionGallery_dir/inversionGallery.html
|Field=Geometry
|Field2=Fractals
|ToDo=Maybe we could have the diagram be interactive. Also maybe something that showed a circle or a line being inverted.
|InProgress=Yes