Finding points on parallel arcs

[John_Waggs]

I am not good at math whatsoever and would appreciate some help with this. Hopefully it's a simple problem. I need to have three arcs run parallel to each other (1 1/2" apart). As the top arc gets wider or taller, the two lines below it that were parallel are no longer. I need to figure out a way to calculate where the end points of the middle line (see attached image) will be to keep lines parallel no matter what the width or height of the top arc will be.

Any suggestions?

Thanks in advance.

John W.

 

[HallsofIvy]

I don't understand your question. You say "As the top arc gets wider or taller, the two lines below it that were parallel are no longer." That's not true if the arcs are circular.

For example, the arcs given by (a) \(\displaystyle x= 8.5 cos(t)\), \(\displaystyle y= 8.5 sin(t)\); (b) \(\displaystyle x= 10cos(t)\), \(\displaystyle y= 10sin(t)\); (c) \(\displaystyle x= 11.5 cos(t), y= 11.5 sin(t)\), with t from 0 to \(\displaystyle \frac{\pi}{4}\), are "parallel" and all the same distance, 1.5, apart for their full lengths.

 

[John_Waggs]

Sorry if I'm not clear. lets try this. If I have an arc and I know the two end points and also know the height of the spring and width. Now, if I wish to create another arc will be 1 1/2" parallel from that arc how can I find the two end points to start that new arc? I am not using any radius center point for any of these arcs.

 

[HallsofIvy]

Sorry if I'm not clear. lets try this. If I have an arc and I know the two end points and also know the height of the spring and width. Now, if I wish to create another arc will be 1 1/2" parallel from that arc how can I find the two end points to start that new arc? I am not using any radius center point for any of these arcs.

Lines at the two endpoints if the first arc, perpendicular to the arc (perpendicular to the tangent lines there) intersect at the center point and is the center point for the other two arcs.