How to simplify output from solve

I execute the following commands:

% Define symbolic variables
syms a b c d e f g h i
syms x1 y1 x2 y2 x3 y3 x4 y4
syms x y

% Define the formula
denom = g*x + h*y + i;
f1 = (a*x + b*y + c)/denom;
f2 = (d*x + e*y + f)/denom;

% Define the constraint
constraint = [subs(f1, [x, y], [1, 1]) == x1,
subs(f1, [x, y], [-1, 1]) == x2,
subs(f1, [x, y], [1, -1]) == x3,
subs(f1, [x, y], [-1, -1]) == x4,
subs(f2, [x, y], [1, 1]) == y1,
subs(f2, [x, y], [-1, 1]) == y2,
subs(f2, [x, y], [1, -1]) == y3,
subs(f2, [x, y], [-1, -1]) == y4];


% Solve for a to i
vars = [a, b, c, d, e, f, g, h, i];
solution = solve(constraint, vars);

% Display the solution
disp('Solution:');
disp(solution);

Which outputs:

a: (x1*x3*y2 - x2*x3*y1 - x1*x4*y2 + x2*x4*y1 - x1*x3*y4 + x1*x4*y3 + x2*x3*y4 - x2*x4*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

b: -(x1*x2*y3 - x2*x3*y1 - x1*x2*y4 + x1*x4*y2 - x1*x4*y3 + x3*x4*y1 + x2*x3*y4 - x3*x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

c: -(x1*x2*y3 - x1*x3*y2 - x1*x2*y4 + x2*x4*y1 + x1*x3*y4 - x3*x4*y1 - x2*x4*y3 + x3*x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

d: (x1*y2*y3 - x2*y1*y3 - x1*y2*y4 + x2*y1*y4 - x3*y1*y4 + x4*y1*y3 + x3*y2*y4 - x4*y2*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

e: -(x1*y2*y3 - x3*y1*y2 - x2*y1*y4 + x4*y1*y2 - x1*y3*y4 + x3*y1*y4 + x2*y3*y4 - x4*y2*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

f: -(x2*y1*y3 - x3*y1*y2 - x1*y2*y4 + x4*y1*y2 + x1*y3*y4 - x4*y1*y3 - x2*y3*y4 + x3*y2*y4)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

g: (x1*y3 - x3*y1 - x1*y4 - x2*y3 + x3*y2 + x4*y1 + x2*y4 - x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

h: -(x1*y2 - x2*y1 - x1*y4 + x2*y3 - x3*y2 + x4*y1 + x3*y4 - x4*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

i: 1

What commands should I run in order to simplify these expressions and find common factors between these formulas?

Thanks!

[John D'Errico]

Why do you think the result is not about as simple as it can be? Compuations as you have done tend to create a massive mess of terms. That seems about normal to me.

[Torsten]

You should normalize variable i to 1 right at the beginning so that you have 8 equations in 8, not 9 unknowns.

% Define symbolic variables
syms a b c d e f g h 
syms x1 y1 x2 y2 x3 y3 x4 y4
syms x y
% Define the formula
denom = g*x + h*y + 1;
f1 = (a*x + b*y + c)/denom;
f2 = (d*x + e*y + f)/denom;
% Define the constraint
constraint = [subs(f1, [x, y], [1, 1]) == x1,
subs(f1, [x, y], [-1, 1]) == x2,
subs(f1, [x, y], [1, -1]) == x3,
subs(f1, [x, y], [-1, -1]) == x4,
subs(f2, [x, y], [1, 1]) == y1,
subs(f2, [x, y], [-1, 1]) == y2,
subs(f2, [x, y], [1, -1]) == y3,
subs(f2, [x, y], [-1, -1]) == y4];
% Solve for a to i
vars = [a, b, c, d, e, f, g, h];
solution = solve(constraint, vars);
% Display the solution
disp('Solution:');

Solution:

disp(solution);

    a: (x1*x3*y2 - x2*x3*y1 - x1*x4*y2 + x2*x4*y1 - x1*x3*y4 + x1*x4*y3 + x2*x3*y4 - x2*x4*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)
    b: -(x1*x2*y3 - x2*x3*y1 - x1*x2*y4 + x1*x4*y2 - x1*x4*y3 + x3*x4*y1 + x2*x3*y4 - x3*x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)
    c: -(x1*x2*y3 - x1*x3*y2 - x1*x2*y4 + x2*x4*y1 + x1*x3*y4 - x3*x4*y1 - x2*x4*y3 + x3*x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)
    d: (x1*y2*y3 - x2*y1*y3 - x1*y2*y4 + x2*y1*y4 - x3*y1*y4 + x4*y1*y3 + x3*y2*y4 - x4*y2*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)
    e: -(x1*y2*y3 - x3*y1*y2 - x2*y1*y4 + x4*y1*y2 - x1*y3*y4 + x3*y1*y4 + x2*y3*y4 - x4*y2*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)
    f: -(x2*y1*y3 - x3*y1*y2 - x1*y2*y4 + x4*y1*y2 + x1*y3*y4 - x4*y1*y3 - x2*y3*y4 + x3*y2*y4)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)
    g: (x1*y3 - x3*y1 - x1*y4 - x2*y3 + x3*y2 + x4*y1 + x2*y4 - x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)
    h: -(x1*y2 - x2*y1 - x1*y4 + x2*y3 - x3*y2 + x4*y1 + x3*y4 - x4*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)

[Simon]

@John D'Errico I know from this thread that it can be simplified https://math.stackexchange.com/questions/186286/get-transformation-matrix-from-points. However as I want to extend the original formulas, having such simplifications automated would be helpful (and useful to know about in general).

Answer 1

[Star Strider]

I can’t be certain that this is a significant improvement, however it is the best I can do with your data —

% Define symbolic variables
syms a b c d e f g h i
syms x1 y1 x2 y2 x3 y3 x4 y4
syms x y

% Define the formula
denom = g*x + h*y + i;
f1 = (a*x + b*y + c)/denom;
f2 = (d*x + e*y + f)/denom;

% Define the constraint
constraint = [subs(f1, [x, y], [1, 1]) == x1,
subs(f1, [x, y], [-1, 1]) == x2,
subs(f1, [x, y], [1, -1]) == x3,
subs(f1, [x, y], [-1, -1]) == x4,
subs(f2, [x, y], [1, 1]) == y1,
subs(f2, [x, y], [-1, 1]) == y2,
subs(f2, [x, y], [1, -1]) == y3,
subs(f2, [x, y], [-1, -1]) == y4];

% Solve for a to i
vars = [a, b, c, d, e, f, g, h, i];
solution = solve(constraint, vars);

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

sc = struct2cell(solution);
solutions = cell(size(sc));
for k = 1:numel(sc)
    solutions{k,:} = [vars(k)  simplify(sc{k}, 500)];
end

% Display the solution
disp('Solution:');

Solution:

disp(solutions);

    {[a    (x1*x3*y2 - x2*x3*y1 - x1*x4*y2 + x2*x4*y1 - x1*x3*y4 + x1*x4*y3 + x2*x3*y4 - x2*x4*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3) ]}
    {[b    -(x1*x2*y3 - x2*x3*y1 - x1*x2*y4 + x1*x4*y2 - x1*x4*y3 + x3*x4*y1 + x2*x3*y4 - x3*x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)]}
    {[c    -(x1*x2*y3 - x1*x3*y2 - x1*x2*y4 + x2*x4*y1 + x1*x3*y4 - x3*x4*y1 - x2*x4*y3 + x3*x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)]}
    {[d    (x1*y2*y3 - x2*y1*y3 - x1*y2*y4 + x2*y1*y4 - x3*y1*y4 + x4*y1*y3 + x3*y2*y4 - x4*y2*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3) ]}
    {[e    -(x1*y2*y3 - x3*y1*y2 - x2*y1*y4 + x4*y1*y2 - x1*y3*y4 + x3*y1*y4 + x2*y3*y4 - x4*y2*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)]}
    {[f    -(x2*y1*y3 - x3*y1*y2 - x1*y2*y4 + x4*y1*y2 + x1*y3*y4 - x4*y1*y3 - x2*y3*y4 + x3*y2*y4)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)]}
    {[g    (x1*y3 - x3*y1 - x1*y4 - x2*y3 + x3*y2 + x4*y1 + x2*y4 - x4*y2)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)                         ]}
    {[h    -(x1*y2 - x2*y1 - x1*y4 + x2*y3 - x3*y2 + x4*y1 + x3*y4 - x4*y3)/(x1*y2 - x2*y1 - x1*y3 + x3*y1 + x2*y4 - x4*y2 - x3*y4 + x4*y3)                        ]}
    {[i    1                                                                                                                                                       ]}

.

[Simon]

Thanks! I did try similar things already without seeing too much difference. I was hoping there was a way for matlab to identify common factors between the outputted formulas, like here https://math.stackexchange.com/questions/186286/get-transformation-matrix-from-points but maybe it's not capable of doing that (as I was going to extend the original equations).

[Star Strider]

My pleasure!

In my experience, simplify uses every method (such as collect) at its disposal to provide the best result.

You could use coeffs to see if you can isolate common factors, however I doubt that the result would be in any way illuminating.