Solve polynomial functions with four unknown variables [duplicate]

Question

I want to make curve with polynominal function like this figure

but only 2 known point to solve this one. I try to used solve function for this one. but the result is not the same like the figure.

here is my code

h1 = 50;
h2 = 80;

\[Theta]1 = 5 \[Pi];
\[Theta]2 = 9 \[Pi];


eq1 = a0 + (a1*\[Theta]1) + (a2*\[Theta]1^2) + (a3*\[Theta]1^3) == 
   h1;
eq2 = a1 + (a2*\[Theta]1) + (a3*\[Theta]1^2) == 0;
eq3 = a0 + (a1*\[Theta]2) + (a2*\[Theta]2^2) + (a3*\[Theta]2^3) == 
   h2;
eq4 = a1 + (a2*\[Theta]2) + (a3*\[Theta]2^2) == 0;
va = NSolve[{eq1, eq2, eq3, eq4}, {a0, a1, a2, a3}, Reals, 
   WorkingPrecision -> 7];
eq = (a0 + a1 # + a2 #^2 + a3 #^3) &;
P = eq[t] /. va[[1]] // N
Plot[P, {t, \[Theta]1, \[Theta]2}]

this is my curve result with that code

Thank you everyone

Answer 1

I'm assuming what the OP wants is the lowest order polynomial that goes through two given points and has 0 gradient at those two points. The latter isn't explicitly stated, but I implied it from the diagram and the use of a cubic polynomial.

For this problem, we have 4 constraints, and therefore require a polynomial with 4 coefficients, aka, a cubic. This can then be passed to solved along with the constraints and plotted, giving a graph similar to what was in the question:

h1 = 50;
h2 = 80;
\[Theta]1 = 5 \[Pi];
\[Theta]2 = 9 \[Pi];

f[\[Theta]_]:= a \[Theta]^3+b \[Theta]^2 + c \[Theta] + d
sols = Solve[{f[\[Theta]1] == h1, f[\[Theta]2]==h2, f'[\[Theta]1]==f'[\[Theta]2]==0},
             {a, b, c, d}];

fSolve[\[Theta]_] = f[\[Theta]] /. sols\[[1]]

Show[
  Plot[fSolve[\[Theta]], {\[Theta], \[Theta]1, \[Theta]2},
      PlotRange->{{\[Theta]1-\[Pi], \[Theta]2+\[Pi]}, {h1-10, h2+10}}],
    Graphics[{PointSize[Large], Red, Point[{\[Theta]1, h1}]}],
    Graphics[{PointSize[Large], Red, Point[{\[Theta]2, h2}]}]
]

out = 4925/16-(2025 \[Theta])/(16 \[Pi])+(315 \[Theta]^2)/(16 \[Pi]^2)-(15 \[Theta]^3)/(16 \[Pi]^3)