Post by Admin on May 31, 2018 4:17:48 GMT
May 31, 2018 0:05:01 GMT Gary said:
Gary
No, because although in figure 4 on page 128 the arc $[bc]$ is mapped to the arc $[\tilde{b}\tilde{c}]$, the line $[bc]$ is not mapped to the line $[\tilde{b}\tilde{c}]$, but to an arc through $\tilde{b}$ and $\tilde{c}$, which is not shown in figure 4. The angle $abc$ will be equal to the angle between the line $q\tilde{a}$ and the tangent to the arc which is not shown.
Don't forget that in figure 4 only the circles are mapped by the inversion to the other circle. The lines are drawn in afterwards and are not part of the mapping.
Vasco
Yes, it is as you say. By (7), if an extended chord of C does not pass through q, it must map to an arc of a circle that does pass through q. In the attached plot, the arc $\tilde{b}\tilde{c}$ is the inversion of the chord $bc$ in K. It is not apparent from the first plot, but when sufficiently extended, it must pass through q, as it would do in the second plot if the extension of chord bc were sufficiently long.
Gary
Gary
Yes and the angle between $[bc]$ and $[ba]$ is equal to that between the tangent to the red circle at $\tilde{b}$ and the line from $\tilde{b}$ to $\tilde{a}$.
"sufficiently long" being in this case infinity. The chord $[bc]$ would have to be extended to infinity. The point at infinity is mapped to $q$.
When I wrote in my previous answer that the lines are not mapped to each other I should have excluded the lines through $q$, which are mapped to themselves. I think its best to think of these lines as circles of infinite radius.
Vasco