More about ellipse and circle intersection
- To: mathgroup at smc.vnet.net
- Subject: [mg38426] More about ellipse and circle intersection
- From: Tomas Garza <tgarza01 at prodigy.net.mx>
- Date: Sat, 14 Dec 2002 03:20:45 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
In a previous message [mg38381] Re: Ellipse and circle intersection, I
hinted that Solve would not always give the right answer when trying to
obtain the intersections of an ellipse and a circle. Two comments were
received (Tom Burton and Rasmus Debitsch) suggesting that there was
nothing wrong. Still, definitely there is, as shown in the example
below. The situation is as follows:
In[1]:=
ellipse = (x - c)^2/b^2 + (y - d)^2/a^2 == 1;
circ = x^2 + y^2 == 1;
sol = Solve[{ellipse, circ}, {x, y}];
The solution comes out all right in terms of the four parameters a, b,
c, d. The following sets of values for the parametrs are tested:
In[2]:=
example1Tom = {a -> 1.2, b -> 1.3, c -> 0.2, d -> 0.3};
example2Tom = {a -> 1.2, b -> 1.3, c -> 1.2, d -> 1.3};
example3Rasmus = {a -> 2, b -> 1, c -> 1, d -> 1};
example4Tomas = {a -> 0.25, b -> 0.75, c -> 0.5, d -> 0};
(the first two come from Tom Burton, the third one from Rasmus, and the
fourth one is mine). Numerical solutions are then obtained for each set:
In[3]:=
sol1=sol/.example1Tom;
In[4]:=
sol2=sol/.example2Tom;
In[5]:=
sol3=sol/.example3Rasmus//N;
In[6]:=
sol4=sol/.example4Tomas;
In each of the first three cases the correct intersections (as observed
graphically through ImplicitPlot) are found, in addition to some complex
points. However, the fourth case fails to give a correct answer, even if
the two curves intersect very nicely at four different points in the
plane. This points to a weird behavior of Solve. I hope someone will
come forward with some explanation.
Tomas Garza
Mexico City
- Follow-Ups:
- Re: More about ellipse and circle intersection
- From: Daniel Lichtblau <danl@wolfram.com>
- Re: More about ellipse and circle intersection