Prove Analytically: Inversion of a Circle is Also a Circle

In summary, the conversation discusses proving analytically that the inversion of a circle, D, that does not pass through the origin in a unit circle is still a circle. The speaker provides two pdf files and mentions that they have worked out the equation of the image circle, D', but cannot derive it algebraically. They also mention using Geometer's Sketch Pad, which they find useful.
  • #1
zhandele
7
0

Homework Statement



Given the unit circle (in the Euclidean plane) centered at the origin x^2+y^2=1, and a general circle D with equation (x-a)^2+(y-b)^2=c^2 that does not pass through the origin (ie the center of inversion, ie a^2+b^2≠c^2, prove analytically that the inversion of D in the unit cirlce is still a circle.

Homework Equations



See the attached pdf files

The Attempt at a Solution



I can prove this synthetically. I even worked out the equation of the image circle D', but I can't derive it algebraically. I feel I must be missing something very obvious.

I uploaded two pdf files. I was going to upload a GSP file, but I guess this forum can't do that? I'll have to generate pdfs from it or something. Do most of you guys have GSP? It's mind-bogglingly useful.
 

Attachments

  • Circle Inversion Question (1 Algebra).pdf
    34.3 KB · Views: 223
  • Circle Inversion Question (2 Geometric).pdf
    57.5 KB · Views: 267
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  • #2
I'm posting again to upload pdf files I generated from Geometer's Sketch Pad. They may help somebody follow what I write in the other two pdfs. Thanks!
 

Attachments

  • Circle Inversion Question GSP 1.pdf
    8.8 KB · Views: 215
  • Circle Inversion Question GSP 2.pdf
    6.4 KB · Views: 255
  • Circle Inversion Question GSP 3.pdf
    6.7 KB · Views: 265

What is the definition of inversion of a circle?

Inversion of a circle is a geometric transformation that involves flipping a circle inside out by reflecting it through a point called the center of inversion. This results in the creation of a new circle, known as the inverted circle, which intersects the original circle at right angles.

How is the inversion of a circle related to a circle?

The inversion of a circle is related to a circle because it is a special case of inversion where the center of inversion lies on the circle itself. This results in the inverted circle being identical to the original circle, making it a special type of circle.

What are the properties of the inverted circle?

The inverted circle has some unique properties, such as:- It intersects the original circle at right angles.- It shares the same diameter as the original circle.- It is always smaller than the original circle, unless the center of inversion lies on the circle itself.- It is tangent to the original circle at the center of inversion.

How can we prove analytically that the inversion of a circle is also a circle?

To prove analytically that the inversion of a circle is also a circle, we can use the general equation of a circle, (x-a)^2 + (y-b)^2 = r^2, where (a,b) represents the center of the circle and r is the radius. By applying the inversion transformation, we can show that the equation of the inverted circle is also in the form of a circle, thus proving that it is indeed a circle.

What are some real-world applications of the inversion of a circle?

The inversion of a circle has various applications in fields such as engineering, physics, and computer graphics. Some examples include:- Designing optical lenses and mirrors- Studying electromagnetic fields- Creating 3D computer graphics- Solving problems in conformal mapping- Visualizing complex numbers and their transformations.

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