# cartesian coordinates / two circles

• Apr 8th 2010, 02:06 AM
redbaseone
cartesian coordinates / two circles
hello,

I have two circles with known r1 & r2 radii.
I know the centers cartesian coordinates C0(a0,b0) & C1(a1,b1).
I know the distance "d" between centers of Circle 1 & Circle 2
(circles never overlap in my problem)

Then, i have (the two blue lines) like in this picture :

[IMG] http://img180.imageshack.us/img180/6710/twocircles.png [/IMG]

These 2 blue lines are common tangent lines
(interior tangent & exterior tangent) to those two circles.

I really need to find out the cartesian coordinates of :

case A (exterior tangent):

T0(x0,y0) = ?
T1(x1,y1) = ?

case B (interior tangent):
T2(x2,y2) = ?
T3(x3,y3) = ?

i.e , find for example for T0 point , x0 = ??? and y0 = ???

Please help, im not so good at math , need this in a very little game im trying to make.

note: this is use in a very little game im trying to make :)
• Apr 8th 2010, 02:58 AM
Sudharaka
Quote:

Originally Posted by redbaseone
hello,

I have two circles with known r1 & r2 radii.
I know the centers cartesian coordinates C0(a0,b0) & C1(a1,b1).
I know the distance "d" between centers of Circle 1 & Circle 2
(circles never overlap in my problem)

Then, i have (the two blue lines) like in this picture :

[IMG] http://img180.imageshack.us/img180/6710/twocircles.png [/IMG]

These 2 blue lines are common tangent lines
(interior tangent & exterior tangent) to those two circles.

I really need to find out the cartesian coordinates of :

case A (exterior tangent):

T0(x0,y0) = ?
T1(x1,y1) = ?

case B (interior tangent):
T2(x2,y2) = ?
T3(x3,y3) = ?

i.e , find for example for T0 point , x0 = ??? and y0 = ???

Please help, im not so good at math , need this in a very little game im trying to make.

note: this is use in a very little game im trying to make :)

Dear redbaseone,

This is a general method of finding the cartesian coordinates, and it could be used in both exterior and interior cases.

First you have to find the intersection point of the tangent and the line that connects the two origins.(i.e: the intersection point of $\displaystyle C_{0}C_{1}~and~T_{2}T_{3}$ in the case of the the interior tangent.)

Now let me take this point as D={x,y},

Then $\displaystyle \frac{C_{1}D}{C_{o}D}=\frac{r_{2}}{r_{1}}$ ; Since the two tringles are similar.

Now by using the ratio theorem you could find the intersection point.

$\displaystyle x=\frac{r_{2}a_{1}+r_{1}a_{0}}{r_{1}+r_{2}}$

$\displaystyle y=\frac{r_{2}b_{1}+r_{1}b_{0}}{r_{1}+r_{2}}$

Also,

$\displaystyle \sqrt{(y_{2}-b_{0})^2+(x_{2}-a_{0})^2}=r_{1}$-----------1

$\displaystyle (b_{0}-y)^2+(a_{0}-x)^2-r_{1}^{2}=(y-y_{2})^2+(x-x_{2})^2$------2

Substituting the values obtained for x and y in equations 1 and 2 and simplifing them you could obtain $\displaystyle x_{2} ~and~ y_{2}$.

Similarly you could obtain $\displaystyle x_{3}~ and~ y_{3}.$

You could extend $\displaystyle C_{0}C_{1}~and~T_{0}T_{1}$ to obtain a set of two similar traingles and get along with a similar method for obtiaining $\displaystyle T_{0}~and~T_{1}$

• Apr 8th 2010, 03:13 AM
redbaseone
i think i got it (must work it on paper)

1) did you popped this below line from Thales Theorem in triangles (C0,T2,D) & (C1,T3,D) ? :

" Then http://www.mathhelpforum.com/math-he...e3babc21-1.gif ; Since the two tringles are similar. "

2)

"and

http://www.mathhelpforum.com/math-he...e845f69e-1.gif-----------1

http://www.mathhelpforum.com/math-he...02e89124-1.gif------2 "

equation 1 = is pithagorean theorem in triangle (C0,T2,D) ?
equation 2 = is this circle equation in point T2 wich equals line (T2,D)
equation in same T2 point ?

thank you !! (Happy)
• Apr 8th 2010, 06:00 AM
Sudharaka
Quote:

Originally Posted by redbaseone
i think i got it (must work it on paper)

1) did you popped this below line from Thales Theorem in triangles (C0,T2,D) & (C1,T3,D) ? :

" Then http://www.mathhelpforum.com/math-he...e3babc21-1.gif ; Since the two tringles are similar. "

2)

"and

http://www.mathhelpforum.com/math-he...e845f69e-1.gif-----------1

http://www.mathhelpforum.com/math-he...02e89124-1.gif------2 "

equation 1 = is pithagorean theorem in triangle (C0,T2,D) ?
equation 2 = is this circle equation in point T2 wich equals line (T2,D)
equation in same T2 point ?

thank you !! (Happy)

Dear redbaseone,

For the Thales' theorem all three points of the traingle must be on a circumference of a circle (Refer:Thales' theorem - Wikipedia, the free encyclopedia). I did'nt use the Thales theorem. Instead this is a theorem that is valid for all similar traingles (Refer:Triangle similarity, ratios of parts - Math Open Reference).

Equation 1) and 2) are obtained from the Pythogorean Theorem.