Reflection: Line about a Circle
- Author:
- Ryan Hirst
Theorem without proof:
The image of a line reflected across the arc of a circle, is a circle.
The image of a ray or segment is a circular arc.
I used algebra for the proof. Having guessed (observation) that the reflection was a circle, I set up the vector problem:
There is a single point M and a corresponding constant c which satisfy, for any point X on the line,
X' = M + c (cos θ, sin θ)
The crux is this. Drop an altitude from A to the line, and draw the intersection P.
Reflect P about α:
On the reflection, P' is the closest point to the given line.
Now consider a second point Z. very far from A. The further Z gets, the closer Z' is to A:
has constant length, and r is a constant. The denominator increases without bound.
In the limit as Z gets infinitely far away, Z' = A.
So both ends of the line project to point A. The figure is closed. A line doesn't make any weird deviations, it just keeps on keepin' on. The reflection should have constant curvature. A closed figure with constant curvature is a circle, and if this is all true, AP' should be a diameter.
Test the midpoint M of AP' directly.
Success.
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The Tangent Circle Problem:
- 1. Tangent along the rim: solve for k
- 2a. Initial position: http://www.geogebratube.org/material/show/id/58360
- 2b. Tangent to equal circles: http://www.geogebratube.org/material/show/id/58455
- 3a. Four mutually tangent & exterior circles (Apollonius): http://www.geogebratube.org/material/show/id/58189
- 3b. Vector reduction: http://www.geogebratube.org/material/show/id/58461
- Affine Transformation http://www.geogebratube.org/material/show/id/58177
- →Reflection: Line about a Circle
- Reflection: Circle about a Circle http://www.geogebratube.org/material/show/id/58185
- Circle Inversion: The Metric Space http://www.geogebratube.org/material/show/id/60132
- Sequences 1: Formation http://www.geogebratube.org/material/show/id/58896
- Sequence 1: Formation http://www.geogebratube.org/material/show/id/59816
- Sequence 1: Iteration 1 http://www.geogebratube.org/material/show/id/59828
- Example of equivalent projections: http://www.geogebratube.org/material/show/id/65754
- Final Diagram: http://www.geogebratube.org/material/show/id/65755