A Planar Geometry Problem
 

On Tuesday, May 3, 2011 5:32:27 PM UTC-5, Tim Wescott wrote:


The immediate problem is for making pseudo-ellipsis in a cheap CAD
program (Qcad), so one could even restrict the two 'master' circles to
being the same diameter. But there are times when I'd like to use
different sized circles for this (imagine a triangle with rounded
corners, and 'puffy cheeks').

I can see an algebraic approach for this involving over use of the
Pythagorean theorem, but I'd rather know if there's a solution that can
be done by construction, not by numbers.


I assume your two circles are equidistant from your point, maybe like this if you are trying to construct an ellipse:
http://i.imgur.com/na3nu.png

Point A is what you want the arc to pass through, and Point B is the arc's center found by trial and error until the arc's size allows it to both pass through the point and be tangent to the circles to make the long side of your ellipse, like so:
http://i.imgur.com/g7TUh.png

Ken

A Planar Geometry Problem
 

Ken Grunke wrote:

On Tuesday, May 3, 2011 5:32:27 PM UTC-5, Tim Wescott wrote:


The immediate problem is for making pseudo-ellipsis in a cheap CAD
program (Qcad), so one could even restrict the two 'master' circles to
being the same diameter. But there are times when I'd like to use
different sized circles for this (imagine a triangle with rounded
corners, and 'puffy cheeks').

I can see an algebraic approach for this involving over use of the
Pythagorean theorem, but I'd rather know if there's a solution that can
be done by construction, not by numbers.


I assume your two circles are equidistant from your point, maybe like this if you are trying to construct an ellipse:
http://i.imgur.com/na3nu.png


With that particular set-up its fairly easy to do with compass and
ruler.

Let R equal radius of the 2 circles
Find the distance R from A that is on centerline (call it point P)
draw a line L1 from P to center of one of the circles
Create a perpendicular line L2 from the mid point of L

The point where the line L2 intersects the centerline
is the center of the arc tangent to the two circles through point A


-jim




Point A is what you want the arc to pass through, and Point B is the arc's center found by trial and error until the arc's size allows it to both pass through the point and be tangent to the circles to make the long side of your ellipse, like so:
http://i.imgur.com/g7TUh.png

Ken