The angle between two tilted circles
Two-Part Question:
1.) Two circles (C1 and C2) sit tangent to both each other, and to the space between an inner circle C3 and an outer circle C4. If r1 and r2 are known, how do we find theta?
2.) Can this problem be "generalized" so that instead of the smaller circles occupying the space between C3 and C4, they lie in the space between two concentric ellipses? Or, is that a different problem entirely?
Thank you!
geometry
asked Dec 3 '18 at 20:56
kreeser1
10110
add a comment |
Two-Part Question:
1.) Two circles (C1 and C2) sit tangent to both each other, and to the space between an inner circle C3 and an outer circle C4. If r1 and r2 are known, how do we find theta?
2.) Can this problem be "generalized" so that instead of the smaller circles occupying the space between C3 and C4, they lie in the space between two concentric ellipses? Or, is that a different problem entirely?
Thank you!
geometry
asked Dec 3 '18 at 20:56
kreeser1
10110
add a comment |
Two-Part Question:
1.) Two circles (C1 and C2) sit tangent to both each other, and to the space between an inner circle C3 and an outer circle C4. If r1 and r2 are known, how do we find theta?
2.) Can this problem be "generalized" so that instead of the smaller circles occupying the space between C3 and C4, they lie in the space between two concentric ellipses? Or, is that a different problem entirely?
Thank you!
geometry
asked Dec 3 '18 at 20:56
kreeser1
10110
Two-Part Question:
1.) Two circles (C1 and C2) sit tangent to both each other, and to the space between an inner circle C3 and an outer circle C4. If r1 and r2 are known, how do we find theta?
2.) Can this problem be "generalized" so that instead of the smaller circles occupying the space between C3 and C4, they lie in the space between two concentric ellipses? Or, is that a different problem entirely?
Thank you!
geometry
geometry
asked Dec 3 '18 at 20:56
kreeser1
10110
asked Dec 3 '18 at 20:56
kreeser1
10110
asked Dec 3 '18 at 20:56
kreeser1
10110
asked Dec 3 '18 at 20:56
kreeser1
10110
asked Dec 3 '18 at 20:56
kreeser1
10110
10110
add a comment |
add a comment |
1 Answer
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Hint:
the centers of $C_1$ and $C_2$ are at a distance from the center of the great circle :
$$
r_3=r_1+left(frac{r_2-r_1}{2} right)=r_2-left(frac{r_2-r_1}{2} right)
$$
$APE$ is a rectangular triangle with $angle EAP=theta/2$ and we know $AE=r_3$ and $EP=r_3-r_1$. And we have: $sin (theta/2)=frac{EP}{EA}$
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
Thank you for the comment. Should your comment be that the centers of C1 and C2 are at a distance from the center of the great circle: r3 = r2 - ((r2 - r1)/2)?
– kreeser1
Dec 3 '18 at 21:26
Yes, sorry for the typo. I edit.:)
– Emilio Novati
Dec 3 '18 at 21:59
Thank you Emilio. I still do not understand how to solve the problem, but I will keep your comment in mind. I appreciate your input!
– kreeser1
Dec 3 '18 at 22:01
I added a figure.
– Emilio Novati
Dec 3 '18 at 22:18
Thank you Emilio, I understand now. Not sure why I couldn't see that before when you added your first hint.
– kreeser1
Dec 3 '18 at 22:26
add a comment |
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1 Answer
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1 Answer
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oldest
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Hint:
the centers of $C_1$ and $C_2$ are at a distance from the center of the great circle :
$$
r_3=r_1+left(frac{r_2-r_1}{2} right)=r_2-left(frac{r_2-r_1}{2} right)
$$
$APE$ is a rectangular triangle with $angle EAP=theta/2$ and we know $AE=r_3$ and $EP=r_3-r_1$. And we have: $sin (theta/2)=frac{EP}{EA}$
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
Thank you for the comment. Should your comment be that the centers of C1 and C2 are at a distance from the center of the great circle: r3 = r2 - ((r2 - r1)/2)?
– kreeser1
Dec 3 '18 at 21:26
Yes, sorry for the typo. I edit.:)
– Emilio Novati
Dec 3 '18 at 21:59
Thank you Emilio. I still do not understand how to solve the problem, but I will keep your comment in mind. I appreciate your input!
– kreeser1
Dec 3 '18 at 22:01
I added a figure.
– Emilio Novati
Dec 3 '18 at 22:18
Thank you Emilio, I understand now. Not sure why I couldn't see that before when you added your first hint.
– kreeser1
Dec 3 '18 at 22:26
add a comment |
Hint:
the centers of $C_1$ and $C_2$ are at a distance from the center of the great circle :
$$
r_3=r_1+left(frac{r_2-r_1}{2} right)=r_2-left(frac{r_2-r_1}{2} right)
$$
$APE$ is a rectangular triangle with $angle EAP=theta/2$ and we know $AE=r_3$ and $EP=r_3-r_1$. And we have: $sin (theta/2)=frac{EP}{EA}$
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
Thank you for the comment. Should your comment be that the centers of C1 and C2 are at a distance from the center of the great circle: r3 = r2 - ((r2 - r1)/2)?
– kreeser1
Dec 3 '18 at 21:26
Yes, sorry for the typo. I edit.:)
– Emilio Novati
Dec 3 '18 at 21:59
Thank you Emilio. I still do not understand how to solve the problem, but I will keep your comment in mind. I appreciate your input!
– kreeser1
Dec 3 '18 at 22:01
I added a figure.
– Emilio Novati
Dec 3 '18 at 22:18
Thank you Emilio, I understand now. Not sure why I couldn't see that before when you added your first hint.
– kreeser1
Dec 3 '18 at 22:26
add a comment |
Hint:
the centers of $C_1$ and $C_2$ are at a distance from the center of the great circle :
$$
r_3=r_1+left(frac{r_2-r_1}{2} right)=r_2-left(frac{r_2-r_1}{2} right)
$$
$APE$ is a rectangular triangle with $angle EAP=theta/2$ and we know $AE=r_3$ and $EP=r_3-r_1$. And we have: $sin (theta/2)=frac{EP}{EA}$
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
Hint:
the centers of $C_1$ and $C_2$ are at a distance from the center of the great circle :
$$
r_3=r_1+left(frac{r_2-r_1}{2} right)=r_2-left(frac{r_2-r_1}{2} right)
$$
$APE$ is a rectangular triangle with $angle EAP=theta/2$ and we know $AE=r_3$ and $EP=r_3-r_1$. And we have: $sin (theta/2)=frac{EP}{EA}$
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
edited Dec 3 '18 at 22:24
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
answered Dec 3 '18 at 21:00
Emilio Novati
51.5k43472
51.5k43472
Thank you for the comment. Should your comment be that the centers of C1 and C2 are at a distance from the center of the great circle: r3 = r2 - ((r2 - r1)/2)?
– kreeser1
Dec 3 '18 at 21:26
Yes, sorry for the typo. I edit.:)
– Emilio Novati
Dec 3 '18 at 21:59
Thank you Emilio. I still do not understand how to solve the problem, but I will keep your comment in mind. I appreciate your input!
– kreeser1
Dec 3 '18 at 22:01
I added a figure.
– Emilio Novati
Dec 3 '18 at 22:18
Thank you Emilio, I understand now. Not sure why I couldn't see that before when you added your first hint.
– kreeser1
Dec 3 '18 at 22:26
add a comment |
Thank you for the comment. Should your comment be that the centers of C1 and C2 are at a distance from the center of the great circle: r3 = r2 - ((r2 - r1)/2)?
– kreeser1
Dec 3 '18 at 21:26
Yes, sorry for the typo. I edit.:)
– Emilio Novati
Dec 3 '18 at 21:59
Thank you Emilio. I still do not understand how to solve the problem, but I will keep your comment in mind. I appreciate your input!
– kreeser1
Dec 3 '18 at 22:01
I added a figure.
– Emilio Novati
Dec 3 '18 at 22:18
Thank you Emilio, I understand now. Not sure why I couldn't see that before when you added your first hint.
– kreeser1
Dec 3 '18 at 22:26
Thank you for the comment. Should your comment be that the centers of C1 and C2 are at a distance from the center of the great circle: r3 = r2 - ((r2 - r1)/2)?
– kreeser1
Dec 3 '18 at 21:26
Thank you for the comment. Should your comment be that the centers of C1 and C2 are at a distance from the center of the great circle: r3 = r2 - ((r2 - r1)/2)?
– kreeser1
Dec 3 '18 at 21:26
Yes, sorry for the typo. I edit.:)
– Emilio Novati
Dec 3 '18 at 21:59
Yes, sorry for the typo. I edit.:)
– Emilio Novati
Dec 3 '18 at 21:59
Thank you Emilio. I still do not understand how to solve the problem, but I will keep your comment in mind. I appreciate your input!
– kreeser1
Dec 3 '18 at 22:01
Thank you Emilio. I still do not understand how to solve the problem, but I will keep your comment in mind. I appreciate your input!
– kreeser1
Dec 3 '18 at 22:01
I added a figure.
– Emilio Novati
Dec 3 '18 at 22:18
I added a figure.
– Emilio Novati
Dec 3 '18 at 22:18
Thank you Emilio, I understand now. Not sure why I couldn't see that before when you added your first hint.
– kreeser1
Dec 3 '18 at 22:26
Thank you Emilio, I understand now. Not sure why I couldn't see that before when you added your first hint.
– kreeser1
Dec 3 '18 at 22:26
add a comment |
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