if you have two circles, how do you define the two tangents which are common to both?

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February 16th 2011, 06:15 AM
Alphecca

if you have two circles, how do you define the two tangents which are common to both?

To all,

I am new and am studying Aircraft Engineering at university.

I have a problem i am trying to work out for a project I’m working on currently.

As in the title; if you have two circles, how do you define the two tangents which are common to both?

The situation is relative to an aircraft flying full 360 degree circles. So i have calculated the co-ordinates and headings of the aircraft every second through two circles.

I know i can draw out the circles and simply draw both tangents of the circles with a ruler and measure it like that. But i need something i can put into excel or a formula i can show will work for any 2 circles of the same size give the common tangents.

I think that’s everything. Both circles are the same size but can be anywhere relative to each other. I know the tangents have a common link that is the heading out of one circle will equal the reciprocal heading of the second circle.

Thanks in advance

Alphecca
February 16th 2011, 06:17 AM
Alphecca

Apologies in advance i have posted this in the wrong area, i looked at each and was unsure so i posted here. i can move if requested. thank you
February 16th 2011, 07:05 AM
Wilmer

Seems to me there are 4 tangents common to both circles.

Code:

aA bB O O cC dD

4 tangents: lines ab, cd, AD and BC

My labelling confusing? Like, aA = 2 different points on circle...
February 16th 2011, 07:49 AM
earboth

1 Attachment(s)

Quote:

Originally Posted by Alphecca

...

As in the title; if you have two circles, how do you define the two tangents which are common to both?
...
I know i can draw out the circles and simply draw both tangents of the circles with a ruler and measure it like that. But i need something i can put into excel or a formula i can show will work for any 2 circles of the same size give the common tangents.

I think that’s everything. Both circles are the same size but can be anywhere relative to each other. I know the tangents have a common link that is the heading out of one circle will equal the reciprocal heading of the second circle.

Thanks in advance

Alphecca

1. Two circles with the same radius can have 2, 3 or 4 common tangents.

2. The number of tangents depends on the distance between the two centers of the circles.
February 16th 2011, 09:29 AM
Alphecca

Earboth,

I found your diagrams very interesting!

Wilmer i think i understand you’re labelling.

So it seems there can be four tangents! Thank you for your assistance.

Now i see how it is possible to work them out by drawing them, is there a way i can do it with a formula which will hold for any situation??

Thank you again
February 17th 2011, 12:21 PM
Alphecca

Progress but not complete

I have made some progress into this problem. It seems the key lies in knowing the bearing of angle between the centres of the two circles!

This bearing compared with the headings will give only two matches. They are your tangents

However; this doesn’t solve the tangents labelled in red which cross over, only the ones labelled in blue.

Is anyone willing to help me out in this problem knowing this new information?

I’m you if i ponder for more hours over diagrams i might get it but some help would be greatly appreciated.

Thank you
February 17th 2011, 10:47 PM
earboth

1 Attachment(s)

Quote:

Originally Posted by Alphecca

I have made some progress into this problem. It seems the key lies in knowing the bearing of angle between the centres of the two circles!

This bearing compared with the headings will give only two matches. They are your tangents

However; this doesn’t solve the tangents labelled in red which cross over, only the ones labelled in blue.

Is anyone willing to help me out in this problem knowing this new information?

I’m you if i ponder for more hours over diagrams i might get it but some help would be greatly appreciated.

Thank you

1. I've made an exact construction of the interior tangents in the more general case of two circles with different radii. (see attachment) (In your case the lengthes of $r_1$ and $r_2$ are equal)

2. The value of angle $\alpha$ is determined by the dimensions of the greyed right triangle $\Delta(M_1M_2T_a)$ :

$\cos(\alpha)=\dfrac{r_1+r_2}{|\overline{M_1M_2}|}$

3. In your case $r_1 = r_2$ this formula is reduced to:

$\cos(\alpha)=\dfrac{2r}{|\overline{M_1M_2}|}$

c_1, c_2 := circles
c_a := auxiliar circle to get the tangent point T_a
c_T := circle of Thales to obtain a right angle at T_a
February 17th 2011, 10:52 PM
earboth

1 Attachment(s)

Quote:

Originally Posted by Alphecca

I have made some progress into this problem. It seems the key lies in knowing the bearing of angle between the centres of the two circles!

This bearing compared with the headings will give only two matches. They are your tangents

However; this doesn’t solve the tangents labelled in red which cross over, only the ones labelled in blue.

Is anyone willing to help me out in this problem knowing this new information?

I’m you if i ponder for more hours over diagrams i might get it but some help would be greatly appreciated.

Thank you

1. I've made an exact construction of the exterior tangents in the more general case of two circles with different radii. (see attachment) (In your case the lengthes of $r_1$ and $r_2$ are equal)

2. The value of angle $\alpha$ is determined by the dimensions of the greyed right triangle $\Delta(M_1M_2T_a)$ :

$\cos(\alpha)=\dfrac{r_2-r_1}{|\overline{M_1M_2}|}$

3. In your case $r_1 = r_2$ this formula is reduced to:

$\cos(\alpha)=\dfrac{0}{|\overline{M_1M_2}|}~\impli es~\alpha = 90^\circ$

c_1, c_2 := circles
c_a := auxiliar circle to get the tangent point T_a
c_T := circle of Thales to obtain a right angle at T_a