## Thursday, June 11, 2015

### How do I Construct these Loci?

If you have two points, A and B,  in the plane and then determine a third point P by measuring the distances to the original two and having the sum of those distances constant, you define an ellipse.

PA + PB = k

Tracing out such a locus is a fairly standard thing that is done in Sketchpad by defining the sum as a segment, creating a point on the segment to partition the length in 2, creating a circle with each partition as a radius, and tracing out the intersections.

What if you have three points, A, B, and C, in the plane and then determine a fourth point P by measuring the distances to the original three and having the sum of those distances be a constant?

PA + PB + PC = k

Here, I have constructed the Fermat Point P[0] which is the point where the sum is a minimum.  Then I traced out three points with the constant sum approximately 10, 12 and 14 units.

I am looking for a more elegant way to trace out the locus corresponding to any given constant sum of distances to the vertices.  I would also like to know what the curve is called and whether there is a way to graph a relation with that shape.

I am calling on the vast readership and commenters to help!

#### 1 comment:

paul martin said...

Not sure if u solved this but saw the word ellipse and referring back umpteen yrs to when I was at Abdn Uni there was work on representing such for various reasons. (I cannot remember just now ...). Anyway a good place to start is the Geometer https://en.wikipedia.org/wiki/Harold_Scott_MacDonald_Coxeter
& before him the papers on the cyclide by James Clerk Maxwell & the literature around that. This may be difficult to access as it is over a hundred years old.

Good luck