Ch 6, Exercise 3 again

evsikman

Below is an explanation to demonstrate why the circles $\widehat{G}$ and $\widehat{J}$ are orthogonal $\widehat{C}$.

Think of any plane $\mathbb{P}$, which contains the points $v$ and $\widehat{z}$. and the circle $\widehat{J}$.
On the cone with apex $v$ and base $\widehat{C}$, the line from $v$ through any of the two points where $\widehat{J}$ intersects $\widehat{C}$ is orthogonal to $\widehat{C}$ and lies in the plane $\mathbb{P}$.
The fact that the cone only touches $\Sigma$ means that this line must also be tangent to $\widehat{J}$, and so it follows that $\widehat{C}$ and $\widehat{J}$ intersect at right angles.
A similar argument shows that this is also true for $\widehat{C}$ and $\widehat{G}$.

If you have any questions about the above explanation please ask.

Vasco

evsikman

Sorry I didn't make it clear enough in my last post. Here is another attempt:

The points 6a to 6e below are meant to replace point 6 in my original commentary.

6a. Let's define the plane $\mathbb{P}$ as that which is defined by the 3 non-collinear points $v$, $\widehat{z}$ and one of the points of intersection of $\widehat{J}$ with $\widehat{C}$. Let's call this point $\widehat{q}$.

6b. The circle $\widehat{J}$ lies in this plane $\mathbb{P}$, and the line $M$ from $v$ to $\widehat{q}$ also lies in the same plane and is part of the cone $K$.

6c. Because the line $M$ lies on the cone it must be orthogonal to $\widehat{C}$, since this circle is the base of $K$. i.e. the angle between $M$ and the tangent to $\widehat{C}$ at $\widehat{q}$ is $\pi/2$.

6d. The fact that the cone only touches $\Sigma$ together with the result in 6c., means that $M$ must be tangent to $\widehat{J}$, and so it follows that $\widehat{C}$ and $\widehat{J}$ are orthogonal.

6e. A similar argument shows that this is also true for $\widehat{C}$ and $\widehat{G}$.

Vasco

evsikman

In 6c we have that the angle between $M$ and the tangent to $\widehat{C}$ at $\widehat{q}$ is $\pi/2$.
6d tells us that $M$ is tangent to $\widehat{J}$ at the same point $\widehat{q}$
So if $M$ is tangent to $\widehat{J}$ at $\widehat{q}$, and orthogonal to $\widehat{C}$ at $\widehat{q}$, then $\widehat{J}$ and $\widehat{C}$ must be orthogonal.
All this is in 3 dimensions of course.
I'm not saying that the result 6c. affects the angle between C^ and J^ in any way, all I'm saying is that if a line is tangent to a curve and orthogonal to another curve then the two curves must be orthogonal.

Vasco

evsikman

I don't understand your difficulty. M must be tangent to $\widehat{J}$. Please explain in pictures. Thanks

Vasco

Yes, I finally understand it!
I thought about two options, in one of them I forgot about C^ and M lying in one plane, in the other - about only one point of contact between Σ and the cone, as you pointed out. Now all is clear.

Thank you!


evsikman

Would you like to write a more detailed explanation of orthogonality of C^ and J^ in your solution ?
By the way, could you add a link to the forum in chapter 6 , as you did with other chapters? Thanks.


evsikman