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Math Help - two radii with a common tangent

  1. #1
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    two radii with a common tangent

    In the attached jpg, I need to specify x1, y1, x2 and y2 in terms of r1, r2, H and W. I'm stuck!

    My friend was trying to help me, and he found this information helped him visualize the problem, and why its stated the way it is.

    This is acutally a real-world application... not homework! What you are looking at is a tool path for a computer operated lathe. The cutting tool starts in the bottom right, and the user wants move to a new cutting diameter using a tapered step with two radii. The solution is in terms of variables because this is a very common operation and I am trying to create a subroutine where the user will simply enter the height and width of the step as shown, and the two radii.

    In order to do this, my subroutine specifies coordinates for the center of radius 1 (i.e. (0,r1)), and the end point of radius 1 (i.e. (x1, y1)). The subroutine then specifies a taper by moving the tool to the coordinates (W-x2,H-y2). Finally, the subroutine specifies the center of radius 2 (W,H-r2), and the end point (W,H).

    Good luck, and thanks!
    Attached Thumbnails Attached Thumbnails two radii with a common tangent-radtaper-1-.jpg  
    Last edited by ParadiseIsle; April 20th 2006 at 02:04 PM. Reason: Additional Info
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ParadiseIsle
    In the attached jpg, I need to specify x1, y1, x2 and y2 in terms of r1, r2, H and W. I'm stuck!

    My friend was trying to help me, and he found this information helped him visualize the problem, and why its stated the way it is.

    This is acutally a real-world application... not homework! What you are looking at is a tool path for a computer operated lathe. The cutting tool starts in the bottom right, and the user wants move to a new cutting diameter using a tapered step with two radii. The solution is in terms of variables because this is a very common operation and I am trying to create a subroutine where the user will simply enter the height and width of the step as shown, and the two radii.

    In order to do this, my subroutine specifies coordinates for the center of radius 1 (i.e. (0,r1)), and the end point of radius 1 (i.e. (x1, y1)). The subroutine then specifies a taper by moving the tool to the coordinates (W-x2,H-y2). Finally, the subroutine specifies the center of radius 2 (W,H-r2), and the end point (W,H).

    Good luck, and thanks!
    Label the two angles in the diagram \theta_1 and \theta_2. Then \theta_1=\theta_2=\theta and the
    slope of the common tangent is -\tan(\theta).

    Now write x_1,\ y_1,\ x_2,\ y_2 in terms of r_1,\ r_2,\ W,\ H,\ \theta.

    Now write the equation of lines of slope -\tan(\theta) through
    (x_1,\ y_1), and (x_2,\ y_2) (in terms of r_1,\ r_2,\ W,\ H,\ \theta). As these are two
    ways of writing the equation of the common tangent equate the constant
    terms and solve for \theta, which will allow you to express the
    common tangent in terms of what you know.

    It may in practice be easiest to solve for \theta numerically.

    RonL
    Attached Thumbnails Attached Thumbnails two radii with a common tangent-new-picture-2-.jpg  
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  3. #3
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    Thank you

    Thanks for your help. I'm delighted to report that I received a very nice solution by posting it on a forum at cnczone.com (a machinists forum). For those still interested, his simple solution is at http://www.cnczone.com/forums/showthread.php?t=19771
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