math problem: help me figure this out. (now with paypal reward)

[notfred]

here's a math question for you guys (it's not homework):

You've got two circles with radii x and y, respectively.

these circles both have center points that lie on the X axis. their center points are separated by a distance d. you can assume that the left-most circle's center ponit is at 0,0 if you like.

Using x, y, and d, find the location of points u and v, where u and v are the points at which each circle would touch a straightedge that was set down on top of them so that it was resting on both circles.

I'd prefer to do this with just algebra and trig, and not calculus, if possible. I need to get a computer to do it, and I have trig libraries but not calculus libraries.

Edit: here's a diagram.

Edit2:
The points we are trying to find are not simply the highest points on the circles. They're the points at which a line that is tangent to both circles intersects each circle. I'll draw a more exaggerated diagram if it's still unclear.

Edit 3: diagram with more exaggerated radius differences

Edit 4: $5 via paypal to the first person who comes up with a correct answer with nointegrals or derivatives in it.

 

[Goosemaster]

looks like pythagorean theorem

 

[notfred]

Originally posted by: Goosemaster
looks like pythagorean theorem

I don't want to know the length of the line, I want to know its endpoints.

 

[Goosemaster]

Originally posted by: notfred

Originally posted by: Goosemaster
looks like pythagorean theorem

I don't want to know the length of the line, I want to know its endpoints.

is the straight edge parallel to the x axis?

I am assuming by straighedge you mean a ruler and that it might not be parallel

 

[bobert]

how will you find the endpoints if you don't give any relative location of the circles. do you want your end points in relation to the other circle?

 

[Goosemaster]

Originally posted by: bobert
how will you find the endpoints if you don't give any relative location of the circles. do you want your end points in relation to the other circle?

that is what it will have to be, but if the straightedge is parallel it is much simpler than if it is not

 

[bobert]

if the straight edge is parallel to the X axis and the center of each circle lies on the x-axis, that makes X = Y right?

 

[hypn0tik]

Originally posted by: Goosemaster

Originally posted by: bobert
how will you find the endpoints if you don't give any relative location of the circles. do you want your end points in relation to the other circle?

that is what it will have to be, but if the straightedge is parallel it is much simpler than if it is not

I'm assuming the radii are different, so the straight edge won't be parallel to the axis given that the centres of the circle lie on the axis.

I'm assuming something like this?

u and v are where the line intersects the circles and d is their centre to centre separation.

Edit: I should clarify, that line is tangent to both circles. Correct?

 

[JS80]

Where's Loke? He'll "help" you.

 

[Goosemaster]

Originally posted by: hypn0tik

Originally posted by: Goosemaster

Originally posted by: bobert
how will you find the endpoints if you don't give any relative location of the circles. do you want your end points in relation to the other circle?

that is what it will have to be, but if the straightedge is parallel it is much simpler than if it is not

I'm assuming the radii are different, so the straight edge won't be parallel to the axis given that the centres of the circle lie on the axis.

I'm assuming something like this?

u and v are where the line intersects the circles and d is their centre to centre separation

we are all assuming different thigns:laugh:

 

[hypn0tik]

Originally posted by: Goosemaster

Originally posted by: hypn0tik

Originally posted by: Goosemaster

Originally posted by: bobert
how will you find the endpoints if you don't give any relative location of the circles. do you want your end points in relation to the other circle?

that is what it will have to be, but if the straightedge is parallel it is much simpler than if it is not

I'm assuming the radii are different, so the straight edge won't be parallel to the axis given that the centres of the circle lie on the axis.

I'm assuming something like this?

u and v are where the line intersects the circles and d is their centre to centre separation

we are all assuming different thigns:laugh:

Yeah, OP really needs to clarify and include a picture so we're all on the same page.

 

[Goosemaster]

Originally posted by: JS80
Where's Loke? He'll "help" you.

I could defintiely use a blowjob about now...

 

[Goosemaster]

Originally posted by: hypn0tik

Originally posted by: Goosemaster

Originally posted by: hypn0tik

Originally posted by: Goosemaster

Originally posted by: bobert
how will you find the endpoints if you don't give any relative location of the circles. do you want your end points in relation to the other circle?

that is what it will have to be, but if the straightedge is parallel it is much simpler than if it is not

I'm assuming the radii are different, so the straight edge won't be parallel to the axis given that the centres of the circle lie on the axis.

I'm assuming something like this?

u and v are where the line intersects the circles and d is their centre to centre separation

we are all assuming different thigns:laugh:

Yeah, OP really needs to clarify and include a picture so we're all on the same page.

seriously

 

[notfred]

I just added a diagram to the OP. The center points of the circles are separated by a distance d. You can assume that the left most circle has its center point at 0,0 if that makes things easier.

 

[chuckywang]

The distance from u to v is sqrt(d^2 + (x-y)^2). Where is the origin in this problem? If you know where the origin is, it's a simple trig problem.

 

[notfred]

Originally posted by: chuckywang
The distance from u to v is sqrt(d^2 + (x-y)^2). Where is the origin in this problem? If you know where the origin is, it's a simple trig problem.

Origin is at the center of the left-most circle, and your equation is wrong, and I didn't ask for distance.

 

[bobert]

edit nvm, forgot that he decided the straight edge wasn't parallel

 

[hypn0tik]

Originally posted by: notfred
I just added a diagram to the OP. The center points of the circles are separated by a distance d. You can assume that the left most circle has its center point at 0,0 if that makes things easier.

It isn't too bad of a problem. Can be done with trig only.

 

[RGUN]

If we set the original to point u (0,0), then point v is 'd' across in the x direction and (y-x) in the y direction. Therefore with this point of view, the coordinates are u (0,0) v (d, [y-x])

If you want to use the center of a circle as the origin than you would have to use some calculus to determine where the lines become tangant to your circle.

edit, upon further inspection the d across would actually be [d + cos(theta)*x-cos(theta)*y]

 

[hypn0tik]

Hmmm, never mind. I thought you wanted the distance between u and v. Now that you want the actual locations, it's a bit trickier.

 

[notfred]

Originally posted by: RGUN
If we set the original to point u (0,0), then point v is 'd' across in the x direction and (y-x) in the y direction. Therefore with this point of view, the coordinates are u (0,0) v (d, [y-x])

If you want to use the center of a circle as the origin than you would have to use some calculus to determine where the lines become tangant to your circle.

edit, upon further inspection the d across would actually be [d + cos(theta)*x-cos(theta)*y]

And what's theta?

 

[chuckywang]

Originally posted by: notfred

Originally posted by: chuckywang
The distance from u to v is sqrt(d^2 - (x-y)^2). Where is the origin in this problem? If you know where the origin is, it's a simple trig problem.

Origin is at the center of the left-most circle, and your equation is wrong, and I didn't ask for distance.

Sorry, supposed to be a minus in the sqrt. If you know the distance between u and v, you can find the coordinates pretty easily.

 

[esun]

I can't think of an easy way. There is a brute force way: iterate along the first circle, keeping track of your current coordinates and the slope at the point you're at. Come up with the corresponding equation for a tangent line, then check to see if you can solve that equation simultaneously with the equation of the second circle (to within some certain error). If yes, then you're done. If no, then continue iterating along the first circle.

 

[Goosemaster]

$5 reward?

GOTTA BE HW

 

[notfred]

Originally posted by: Goosemaster
$5 reward?

GOTTA BE HW

Yeah, it's a howework assignment for my Computer Networking class, since that's the only class I haven't already taken the final for. Also, after being here for 5 years and making 36k posts, I have a track record of lieing about homework problems all the time.