How can I make a quadratic approximation given two points and incidence angle at the second point?

I'm trying to make a parabolic equation that will run through two given points, and with given incidence angle at the 2nd point.

Answer 1

[James Tursa]

Maybe something like this is what you need? (assuming by "incidence angle" you mean slope)

% Construct some sample data for y = a*x^2 + b*x + c
a = 2;
b = 3;
c = -4;
p = [a b c];
point1 = [-2 polyval(p,-2)];    % (x1,y1)
point2 = [ 3 polyval(p, 3)];    % (x2,y2)
slope2 = polyval(polyder(p),3); % slope at x2

% Form the Ax=B from these equations:
%
%   a*x1^2 + b*x1 + c = y1
%   a*x2^2 + b*x2 + c = y2
%   2*a*x2 + b        = slope2
%
% Yields this 3x3 * 3x1 = 3x1 linear set of equations in a,b,c
%
%   [ x1^2 x1  1 ]   [a]   [y1]
%   [ x2^2 x2  1 ] * [b] = [y2]
%   [ 2*x2  1  0 ]   [c]   [slope2]
%
% Then just use backslash to solve for the coefficients

A = [point1(1)^2 point1(1) 1;
     point2(1)^2 point2(1) 1;
     2*point2(1) 1         0];
B = [point1(2);
     point2(2);
     slope2];

% Solve for the coefficients of the polynomial
coeff = A\B;

The values in coeff should match the original a, b, c coefficients.