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Hello, I am trying to solve a geometry problem, which is illustrated in the figure I attached. Two circles and with centers at and are given. Their radii are and respectively. The distance between their center is (not shown in figure). A third circle is to be found such that it satisfies the following constrains: The angles between the tangents at the points of contact should be and as shown in the figure. Also, the area bounded between the three circles, which is shown here in yellow, should be .
I tried solving this problem using the differential equation . This equation needs three boundary conditions: two for the second order ODE and one for constant C. I have three conditions in the form , and angles at the two sides, but this quickly leads to very complicated set of transcendental equations containing several sines and cosines. Maybe my approach is wrong. I would be happy if somebody can come up even with the numerical solution, if not analytical solution. All I need is the profile of the part of the third circle, which is in between the other two. Thanks - DSachan ( talk) 17:14, 6 July 2011 (UTC)
Don't consider differential equations. You already know that the solution is a circle. The circle is specified by three unknown reals, e.g. the center coordinates and the radius. And the constrains must be expressed as equations. Sines and cosines of known angles are just known constants. So write down the equations and solve them. Bo Jacoby ( talk) 17:58, 6 July 2011 (UTC).
The very first tiny simplification to make is to divide all lengths by d and the area by d2. Then the problem is reduced to d=1. When this problem is solved, multiply back by d and d2. Bo Jacoby ( talk) 16:32, 7 July 2011 (UTC).
Let the coordinate system be defined such that
So the area equation is
The pythagorean theorem provides the equations
These are all the equations we need. Unfortunately both the angles x8, x9, x10 and their sines occur. The problem seems not to be algebraic neither in the angles nor in their sines and cosines.
The four variables
are eliminated from the remaining equations, giving
6 equations in 6 unknowns. That looks good. The equation 4 is simplified by substituting eq 2 and 3:
And now good night! Bo Jacoby ( talk) 22:03, 7 July 2011 (UTC).
Mathematics desk | ||
---|---|---|
< July 5 | << Jun | July | Aug >> | July 7 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Hello, I am trying to solve a geometry problem, which is illustrated in the figure I attached. Two circles and with centers at and are given. Their radii are and respectively. The distance between their center is (not shown in figure). A third circle is to be found such that it satisfies the following constrains: The angles between the tangents at the points of contact should be and as shown in the figure. Also, the area bounded between the three circles, which is shown here in yellow, should be .
I tried solving this problem using the differential equation . This equation needs three boundary conditions: two for the second order ODE and one for constant C. I have three conditions in the form , and angles at the two sides, but this quickly leads to very complicated set of transcendental equations containing several sines and cosines. Maybe my approach is wrong. I would be happy if somebody can come up even with the numerical solution, if not analytical solution. All I need is the profile of the part of the third circle, which is in between the other two. Thanks - DSachan ( talk) 17:14, 6 July 2011 (UTC)
Don't consider differential equations. You already know that the solution is a circle. The circle is specified by three unknown reals, e.g. the center coordinates and the radius. And the constrains must be expressed as equations. Sines and cosines of known angles are just known constants. So write down the equations and solve them. Bo Jacoby ( talk) 17:58, 6 July 2011 (UTC).
The very first tiny simplification to make is to divide all lengths by d and the area by d2. Then the problem is reduced to d=1. When this problem is solved, multiply back by d and d2. Bo Jacoby ( talk) 16:32, 7 July 2011 (UTC).
Let the coordinate system be defined such that
So the area equation is
The pythagorean theorem provides the equations
These are all the equations we need. Unfortunately both the angles x8, x9, x10 and their sines occur. The problem seems not to be algebraic neither in the angles nor in their sines and cosines.
The four variables
are eliminated from the remaining equations, giving
6 equations in 6 unknowns. That looks good. The equation 4 is simplified by substituting eq 2 and 3:
And now good night! Bo Jacoby ( talk) 22:03, 7 July 2011 (UTC).