Without further assumptions of your used constants the solution is quite lengthy
eqs = {
u == c1 + (x - c1) (1 +
k1 ((x - c1) (x - c1) + (y - c2) (y - c2))),
v == c2 + (y - c2) (1 + k2 ((x - c1) (x - c1) + (y - c2) (y - c2)))
};
Reduce[eqs, {x, y}]
If you can provide numerical values for your constants $C_n$ and $k_n$ it is probably possible to shorten the solution.
Please have a look on your system and notice, how the number of possible (real) solutions varies when the variables change
Manipulate[
ContourPlot[{u ==
c1 + (x - c1) (1 + k1 ((x - c1) (x - c1) + (y - c2) (y - c2))),
v == c2 + (y - c2) (1 +
k2 ((x - c1) (x - c1) + (y - c2) (y - c2)))}, {x, -5,
5}, {y, -5, 5}, PlotPoints -> ControlActive[10, 40],
MaxRecursion -> ControlActive[1, 5]],
{u, -1, 1},
{v, -1, 1},
{c1, -1, 1},
{c2, -1, 1},
{k1, -1, 1},
{k2, -1, 1}
]

Addionally, lets investigate in the first solution you get from the Reduce
call
k1 == 0 && x == u && k2 != 0 &&
(y == Root[(-c1^2)*c2*k2 - c2^3*k2 + 2*c1*c2*k2*u - c2*k2*u^2 - v +
(1 + c1^2*k2 + 3*c2^2*k2 - 2*c1*k2*u + k2*u^2)*#1 - 3*c2*k2*#1^2 + k2*#1^3 & ,1] ||
y == Root[(-c1^2)*c2*k2 - c2^3*k2 + 2*c1*c2*k2*u - c2*k2*u^2 - v +
(1 + c1^2*k2 + 3*c2^2*k2 - 2*c1*k2*u + k2*u^2)*#1 - 3*c2*k2*#1^2 + k2*#1^3 & ,2] ||
y == Root[(-c1^2)*c2*k2 - c2^3*k2 + 2*c1*c2*k2*u - c2*k2*u^2 - v +
(1 + c1^2*k2 + 3*c2^2*k2 - 2*c1*k2*u + k2*u^2)*#1 - 3*c2*k2*#1^2 + k2*#1^3 & ,3])
What I want to show you is that you can hack the output of Reduce
directly into C++. The only thing you need is a if/else
way through all the possible forms your solution can have.
Looking at the output above, you see that when k1==0
and k2!=0
your solution is that x=u
and y
can take 3 values. These three values are the roots of a polynomial of third order. Therefore, your three points are {x,y1}, {x,y2}, {x,y3}
. Using the Manipulate
and set k1
to zero shows, that this is correct:

The points where the red and the blue lines cross have indeed the same x
and 3 different y
.
Therefore, the only thing required for your C++ code are basic arithmetic operations and a root-solver for polynomials of third order.
Solve[]
? $\endgroup$Solve
but it gives me large and not optimised output,I want to place output in my c++ aplication, even usingCForm
not help, how to better optimise it? $\endgroup$Solve
is almost useless. If you substitute parameters for certain values you'll get more useful results, e.g.Solve[{eq1, eq2} /. {C1 -> 1, C2 -> -1, k1 -> 1, k2 -> 1, u -> 3, v -> 0}, {x, y}]
$\endgroup$FullSimplify[]
as well? $\endgroup$FullSimplify[]
,another question is how to get rid ofSlot
functions in output? $\endgroup$