Re: Geometric Transformation Question

Norm Dresner wrote :
<markh@xxxxxxxxxxx> wrote in message
news:1140128693.664388.80920@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Given a point P in the plane, and lines l and m not passing through P,
is there some simple way of constructing points Q and R (on l and m,
respectively) such that P, Q and R are the vertices of an equilateral
triangle?

Geometrically, take a compass and draw a circle centered at P of radius r.
This circle will intersect
(a) neither lines l nor m if the radius r is smaller than the distance
from P to these two lines
(b) one of the lines, say l, if the r is larger than the distance from P
to l but smaller than distance P to m
(c) both lines if r is larger than the larger of the distances P to l
and P to m. Call the minimum such radius r_0.
Any circle of radius > r_0 centered at P intersects the lines l and m at two
points each. Choose one of each pair and you have your three points P, Q,
and R.

Of course not.
Your method gives just *isosceles* triangles PQ=PR (infinitely many)
Only two specific radii will result into equilateral triangles
PQ=PR=QR.

Analytically, you can compute d_l and d_m as the distances from the point P
to the two lines l and m respectively. Chose any r > max( d_l , d_m ) and
you have a circle radius that satisfies the geometric conditions above. You
can then form the quadratic equations that define the 2 points on each line
that are the distance r from the point P.

Norm

That is analytically you must go on with computing the distance between
these two points F(r, data) and solve equation F(r, data) = r.
(two tedious for me, without symbolic math software)

Another method would be to draw any auxiliary line from P, intersecting
the given lines in B and C to build a (scalene) triangle ABC, A being
intersection point of the two given lines.

Then write in the complex plane
P,Q,R on the three lines :
P = u*B + (1-u)*C
Q = v*A + (1-v)*B
R = w*A + (1-w)*C

and PQR equilateral :
P + j*Q + j^2*R = 0 (direct) or P + j*R + j^2*Q = 0 (retrograde)
with j = e^(i*pi/3)

Regards.

--
philippe
mail : chephip at free dot fr
site : http://chephip.free.fr/


.

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