Hi all,

while helping my daughter with her school algebra problems I thought I might get some benefit for myself and dabble in some CAS. (Back when I studied
math in the 90s I didn't even know such a thing as CAS existed, and I
always found algebra terribly tedious).

Topic of the day, circle equations. So why not try and figure out the intersection points (if any) of two circles.

(%i1) e1: x^2 + y^2 + a*x + b*y + c = 0;
2 2
(%o1) y + b y + x + a x + c = 0
(%i2) e2: x^2 + y^2 + d*x + e*y + f = 0;
2 2
(%o2) y + e y + x + d x + f = 0

X and y in the two equations are the same because we're searching the point(s) both have in common. So here goes:

(%i3) solve([e1, e2], [x, y]);
(%o3) []

That took several seconds and yielded nothing. So let's try a more homeworky approach: subtract the equations to get rid of the square terms, solve for x (or y) and substitute into one of the equations:

(%i5) solve(%, x);
(e - b) y + f - c
(%o5) [x = - -----------------]
d - a

Now I can already see that substituting this into %o1 or %o2 will bring up an idiotically convoluted quadratic equation. Just the right thing for a CAS. Nice. But now I'm stuck. I can't find anything in the docs on how to proceed from here.

Questions:

1) Why didn't Maxima find any solution at %i3?
2) How does one substitute the expression for x at %o5 into %o1?
3) I'm using the Linux console. How can I make Maxima use readline, so that
arrow-up brings up an editable copy of an earlier command line?

Thanks!

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Robert Latest wrote:

[stuff]

My questions from before have largely answered themselves by playing around a bit more with Maxima. Turns out that even the simpler general case of solving the intersection(s) of a circle with a straight line yields unwieldy solutions. When it comes to algebra, I try to teach my kids to "first do it only with letters, then insert numbers at the end," but I had never tried it with circle equations.

I still don't know why Maxima can't solve the general case of two (possibly) intersecting circles, but I have a hunch that it would be a useless jumble of symbols. Much like the surprisingly absurd roots of a cubic equation.

The remaining question is: Is there Readline support for Maxima on the Linux console?

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

"RL" == Robert Latest <boblatest@yahoo.com> writes:

The remaining question is: Is there Readline support for Maxima on the Linux
console?

if running just 'maxima' fails to use readline (whether it does depends
on which lisp system was used to compile maxima), then there will be an 'rmaxima' command you can call instead.

rmaxima is similar to running 'rlwrap maxima'.

-JimC
--
James Cloos <cloos@jhcloos.com> OpenPGP: 0x997A9F17ED7DAEA6

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Robert Latest schrieb:

Hi all,

while helping my daughter with her school algebra problems I thought
I might get some benefit for myself and dabble in some CAS. (Back
when I studied math in the 90s I didn't even know such a thing as CAS existed, and I always found algebra terribly tedious).

Topic of the day, circle equations. So why not try and figure out the intersection points (if any) of two circles.

(%i1) e1: x^2 + y^2 + a*x + b*y + c = 0;
2 2
(%o1) y + b y + x + a x + c = 0
(%i2) e2: x^2 + y^2 + d*x + e*y + f = 0;
2 2
(%o2) y + e y + x + d x + f = 0

X and y in the two equations are the same because we're searching the point(s) both have in common. So here goes:

(%i3) solve([e1, e2], [x, y]);
(%o3) []

That took several seconds and yielded nothing. So let's try a more
homeworky approach: subtract the equations to get rid of the square
terms, solve for x (or y) and substitute into one of the equations:

(%i5) solve(%, x);
(e - b) y + f - c
(%o5) [x = - -----------------]
d - a

Now I can already see that substituting this into %o1 or %o2 will
bring up an idiotically convoluted quadratic equation. Just the right
thing for a CAS. Nice. But now I'm stuck. I can't find anything in
the docs on how to proceed from here.

Questions:

1) Why didn't Maxima find any solution at %i3?
2) How does one substitute the expression for x at %o5 into %o1?
3) I'm using the Linux console. How can I make Maxima use readline,
so that arrow-up brings up an editable copy of an earlier command
line?

Don't know why Maxima fails like this (the problem has been known for a
long time and seems somehow impossible to fix), but feeding your input
into Derive 6.10:

e1 := x^2 + y^2 + a*x + b*y + c = 0

e2 := x^2 + y^2 + d*x + e*y + f = 0

SOLVE([e1, e2], [x, y])

[x = ((b - e)*SQRT(a^2*(e^2 - 4*f) + 2*a*d*(2*(c + f) - b*e)
+ b^2*(d^2 - 4*f) + 4*b*e*(c + f) - 4*(c^2 + c*(d^2 + e^2 - 2*f)
+ f^2))*SIGN(a - d) + a*(b*e - 2*c - e^2 + 2*f) - d*(b^2 - b*e
- 2*(c - f)))/(2*(a^2 - 2*a*d + b^2 - 2*b*e + d^2 + e^2))
AND y = - (SQRT(a^2*(e^2 - 4*f) + 2*a*d*(2*(c + f) - b*e)
+ b^2*(d^2 - 4*f) + 4*b*e*(c + f) - 4*(c^2 + c*(d^2 + e^2 - 2*f)
+ f^2))*ABS(a - d) + a^2*e - a*d*(b + e) + b*(2*c + d^2 - 2*f)
- 2*e*(c - f))/(2*(a^2 - 2*a*d + b^2 - 2*b*e + d^2 + e^2)),
x = - ((b - e)*SQRT(a^2*(e^2 - 4*f) + 2*a*d*(2*(c + f) - b*e)
+ b^2*(d^2 - 4*f) + 4*b*e*(c + f) - 4*(c^2 + c*(d^2 + e^2 - 2*f)
+ f^2))*SIGN(a - d) - a*(b*e - 2*c - e^2 + 2*f) + d*(b^2 - b*e
- 2*(c - f)))/(2*(a^2 - 2*a*d + b^2 - 2*b*e + d^2 + e^2))
AND y = (SQRT(a^2*(e^2 - 4*f) + 2*a*d*(2*(c + f) - b*e)
+ b^2*(d^2 - 4*f) + 4*b*e*(c + f) - 4*(c^2 + c*(d^2 + e^2 - 2*f)
+ f^2))*ABS(a - d) - a^2*e + a*d*(b + e) - b*(2*c + d^2 - 2*f)
+ 2*e*(c - f))/(2*(a^2 - 2*a*d + b^2 - 2*b*e + d^2 + e^2))]

yields an answer within seconds. Derive has been discontinued, however,
and was only available for Windows anyway. For your school algebra
problems on Linux, maybe you should try Giac?

Martin.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Robert Latest <boblatest@yahoo.com> wrote:

Hi all,

while helping my daughter with her school algebra problems I thought I might get some benefit for myself and dabble in some CAS. (Back when I studied
math in the 90s I didn't even know such a thing as CAS existed, and I
always found algebra terribly tedious).

Topic of the day, circle equations. So why not try and figure out the intersection points (if any) of two circles.

(%i1) e1: x^2 + y^2 + a*x + b*y + c = 0;
2 2
(%o1) y + b y + x + a x + c = 0
(%i2) e2: x^2 + y^2 + d*x + e*y + f = 0;
2 2
(%o2) y + e y + x + d x + f = 0

X and y in the two equations are the same because we're searching the point(s)
both have in common. So here goes:

(%i3) solve([e1, e2], [x, y]);
(%o3) []

Using FriCAS:

(37) -> eq1 := x^2 + y^2 + a*x + b*y + c

2 2
(37) y + b y + x + a x + c
Type: Polynomial(Integer) (38) -> eq2 := x^2 + y^2 + d*x + e*y + f

2 2
(38) y + e y + x + d x + f
Type: Polynomial(Integer) (39) -> res := solve([eq1, eq2], [x, y])

(39)
[
(- e + b)y - f + c
[x = ------------------,
d - a

2 2 2 2 2
(e - 2 b e + d - 2 a d + b + a )y
+
2 2 2
((2 e - 2 b)f + (- a d - 2 c + a )e + b d - a b d + 2 b c)y + f
+
2 2 2
(- a d - 2 c + a )f + c d - a c d + c
=
0
]
]
Type: List(List(Equation(Fraction(Polynomial(Integer)))))

This is not an explicit solution, but arguably is more useful: intead of
two equations of order 2 in two variables you get so called triangular
system, where last equation contains only single variable and other
equations reduce to equation in single variable when you plug in
solutions of later equation. In fact, here first equation gives
you x in terms of y.

If you want it more explicit we can solve quadratic equation:
(40) -> radicalSolve(res(1)(2), y)
(40)
[
y
=
(- d + a)
*
ROOT
2 2 2 2 2
- 4 f + (4 b e + 4 a d + 8 c - 4 b - 4 a )f + (- 4 c + a )e
+
2 2 2
(- 2 a b d + 4 b c)e + (- 4 c + b )d + 4 a c d - 4 c
+
2 2
(- 2 e + 2 b)f + (a d + 2 c - a )e - b d + a b d - 2 b c
/
2 2 2 2
2 e - 4 b e + 2 d - 4 a d + 2 b + 2 a
,

y
=
(d - a)
*
ROOT
2 2 2 2 2
- 4 f + (4 b e + 4 a d + 8 c - 4 b - 4 a )f + (- 4 c + a )e
+
2 2 2
(- 2 a b d + 4 b c)e + (- 4 c + b )d + 4 a c d - 4 c
+
2 2
(- 2 e + 2 b)f + (a d + 2 c - a )e - b d + a b d - 2 b c
/
2 2 2 2
2 e - 4 b e + 2 d - 4 a d + 2 b + 2 a
]
Type: List(Equation(Expression(Integer)))

This is solved using usual quadratic formula, but since coefficients
of the quadratic are complicated the result is large. One could
plug in this y into formula for x, but result would be equally large.

Note: when you look at "general solution" there are subtleties.
You can see this already from the first solution: there is
division by d - a. And indeed, depending on the other parameters
beside normal case of two solutions there may be one solution,
no solutions or infinitely many solutions. Similarely, quadratic
formula gives "generic solution" and ignores special cases.

Normally a CAS will give you "generic solution", that is solution
which is valid as long as there are no divisions and which may
miss some special cases. Some teachers insist of analysiong
all cases, that quickly gets tedious both for people and at
somewhat larger scale for computers. If you have numbers in
the problem, than it is possible to see during computation
if there are special cases and handle such cases apropriately,
this avoids purely theoretical difficulties. OTOH if you
really need general solution, then there is some extra work
to check usual CAS solution and write proper conditions.

--
Waldek Hebisch

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

antispam@math.uni.wroc.pl wrote:

This is solved using usual quadratic formula, but since coefficients
of the quadratic are complicated the result is large. One could
plug in this y into formula for x, but result would be equally large.

Note: when you look at "general solution" there are subtleties.
You can see this already from the first solution: there is
division by d - a. And indeed, depending on the other parameters
beside normal case of two solutions there may be one solution,
no solutions or infinitely many solutions. Similarely, quadratic
formula gives "generic solution" and ignores special cases.

I know all that. I didn't really *need* to find a general solution for the intersections of two circles. I just took my daughter's homework assignment as inspiration to play around with a CAS (which I always wanted but didn't have any reason to). I expected a screenful of messy solutions, but not an empty result.

Normally a CAS will give you "generic solution", that is solution
which is valid as long as there are no divisions and which may
miss some special cases. Some teachers insist of analysiong
all cases, that quickly gets tedious both for people and at
somewhat larger scale for computers.

Again, the homework assignment wasn't the general solution.

If you have numbers in
the problem, than it is possible to see during computation
if there are special cases and handle such cases apropriately,

That's of course more along school homework lines.

this avoids purely theoretical difficulties. OTOH if you
really need general solution, then there is some extra work
to check usual CAS solution and write proper conditions.

Like I said, I don't. I was just curious.


--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Robert Latest <boblatest@yahoo.com> wrote:

antispam@math.uni.wroc.pl wrote:

This is solved using usual quadratic formula, but since coefficients
of the quadratic are complicated the result is large. One could
plug in this y into formula for x, but result would be equally large.

Note: when you look at "general solution" there are subtleties.
You can see this already from the first solution: there is
division by d - a. And indeed, depending on the other parameters
beside normal case of two solutions there may be one solution,
no solutions or infinitely many solutions. Similarely, quadratic
formula gives "generic solution" and ignores special cases.

I know all that. I didn't really *need* to find a general solution for the intersections of two circles. I just took my daughter's homework assignment as
inspiration to play around with a CAS (which I always wanted but didn't have any reason to). I expected a screenful of messy solutions, but not an empty result.

OK. So use a CAS with reasonable equation solver (and not Maxima
which is (in)famous for failures in its solver).

--
Waldek Hebisch

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

antispam@math.uni.wroc.pl wrote:

OK. So use a CAS with reasonable equation solver (and not Maxima
which is (in)famous for failures in its solver).

Thanks. Didn't know there were big differences.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 16 Oct 2022 17:55:14 GMT, Robert Latest wrote:

Hi all,

while helping my daughter with her school algebra problems I thought I
might get some benefit for myself and dabble in some CAS. (Back when I studied math in the 90s I didn't even know such a thing as CAS existed,
and I always found algebra terribly tedious).

Topic of the day, circle equations. So why not try and figure out the intersection points (if any) of two circles.

(%i1) e1: x^2 + y^2 + a*x + b*y + c = 0;
2 2
(%o1) y + b y + x + a x + c = 0 (%i2) e2: x^2 +
y^2 + d*x + e*y + f = 0;
2 2
(%o2) y + e y + x + d x + f = 0

X and y in the two equations are the same because we're searching the point(s)
both have in common. So here goes:

(%i3) solve([e1, e2], [x, y]);
(%o3) []

That took several seconds and yielded nothing. So let's try a more
homeworky approach: subtract the equations to get rid of the square
terms, solve for x (or y) and substitute into one of the equations:

(%i5) solve(%, x);
(e - b) y + f - c
(%o5) [x = - -----------------]
d - a

Now I can already see that substituting this into %o1 or %o2 will bring
up an idiotically convoluted quadratic equation. Just the right thing
for a CAS. Nice. But now I'm stuck. I can't find anything in the docs on
how to proceed from here.

Questions:

1) Why didn't Maxima find any solution at %i3?
2) How does one substitute the expression for x at %o5 into %o1?
3) I'm using the Linux console. How can I make Maxima use readline, so
that
arrow-up brings up an editable copy of an earlier command line?

Thanks!

No wonder it won't give "solutions". In general, there are none.

For instance
e1: x^2+y^2 + 1 =0
is not even a circle, yet obeys to your e1.

First of all, use wxmaxima. It provides menus, help and many other things
that simplify life to a beginner.

Then, perhaps you could find a more recent computer.

(%i14) solve([e1,e2],[x,y]);
Evaluation took 2.6020 seconds (1.6910 elapsed)
(%o14) []

Finally, you should use a correct approach, as solve is right returning an empty set. For instance:

(%i10) eliminate([e1,e2],[x,y]);
(%o10) [((d-a)*sqrt(-4*f^2+(4*b*e+4*a*d+8*c-4*b^2-4*a^2)*f+ (a^2-4*c)*e^2+(4*b*c-2*a*b*d)*e+(b^2-4*c)*d^2+4*a*c*d-4*c^2)+(2*b-2*e)*f+ (a*d+2*c-a^2)*e-b*
d^2+a*b*d-2*b*c)/(2*e^2-4*b*e+2*d^2-4*a*d+2*b^2+2*a^2)]

(%i11) eliminate([e1,e2],[x]);
(%o11) [(e^2-2*b*e+d^2-2*a*d+b^2+a^2)*y^2+(2*e*f+b*(-2*f+d^2-a*d)- a*d*e+a^2*e+c*(2*b-2*e))*y+f^2-a*d*f+a^2*f+c*(-2*f+d^2-a*d)+c^2]

(%i12) eliminate([e1,e2],[y]);
(%o12) [(e^2-2*b*e+d^2-2*a*d+b^2+a^2)*x^2+(2*d*f+a*(e^2-2*f)+b*(-d*e-a*e) +b^2*d+c*(2*a-2*d))*x+f^2-b*e*f+b^2*f+c*(-2*f+e^2-b*e)+c^2]

Here are your quadratic equations. Maybe they have real solutions, maybe
not. Good luck exploring your 6 dimensional parameter space.

There are plenty of tutorials around for wxmaxima/maxima that could be
helpful.

HTH.

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