# two radii with a common tangent

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• Apr 18th 2006, 01:24 PM
ParadiseIsle
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!
• Apr 21st 2006, 02:59 AM
CaptainBlack
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
• Apr 21st 2006, 11:36 AM
ParadiseIsle
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