Using the information you have supplied above I think you are saying you have this situation:
You know the coordinates of both point A, point B and the angle that the curve makes with the horizontal at point A (in other words the gradient at A).
That is 3 pieces of information. If you are using a cubic spline you need one more piece of information, the angle the curve makes at point B as well. But you have not specified that so I assume you have only 3 pieces of information which means we can use a quadratic rather than a cubic spline.
So we set up the following:
y = y (x) - ie our y coordinate is some function of x.
we know the following:
Coordinates of A we shall write as (0,A) - ie x=0, y=A
Coordinates of B we shall write as (L,B) - ie X=L, y=B
Angle curve makes at A is Theta => Gradient at this point = Tan(Theta).
So:
Y(0)=A
Y(L)=B
Y'(0)=Tan(Theta) - which we will just call "C"
so Y'(0)=C
then, since we have 3 points and are using a quadratic to fit the curve we do this:
Y(x) = ax^2+bx+c (a,b,c not the same as A,B,C)
Y(0)=A = a*0^2 + b*0 + c, therefore c=A
Y(L)=B = a*L^2 + b*L + A
Y'(0) = C = b, therefore b=C
then solving for "a" we get this:
a = (B-CL-A)/(L^2)
Okay so our equation for your curve that passes through the points and with the initial angle as specified is:
Y(x)=((B-CL-A)/(L^2))*x^2 + Cx + A
Now use this equation for Y and calculate the length of the line using the method I outlined above.
I will write it anyway:
Okay - you now want to find the length of the curve I specified:
Length = Integrate Between X=0 and X=L the following function:
(Square Root of (1 + Y'(X)^2)) dX
You could do this analytically but it would probably be just as easy and not too inaccurate to do so with a numerical technique ranging from something simple like the midpoint method to something more complex like the Runge Kutta method
You know the coordinates of both point A, point B and the angle that the curve makes with the horizontal at point A (in other words the gradient at A).
That is 3 pieces of information. If you are using a cubic spline you need one more piece of information, the angle the curve makes at point B as well. But you have not specified that so I assume you have only 3 pieces of information which means we can use a quadratic rather than a cubic spline.
So we set up the following:
y = y (x) - ie our y coordinate is some function of x.
we know the following:
Coordinates of A we shall write as (0,A) - ie x=0, y=A
Coordinates of B we shall write as (L,B) - ie X=L, y=B
Angle curve makes at A is Theta => Gradient at this point = Tan(Theta).
So:
Y(0)=A
Y(L)=B
Y'(0)=Tan(Theta) - which we will just call "C"
so Y'(0)=C
then, since we have 3 points and are using a quadratic to fit the curve we do this:
Y(x) = ax^2+bx+c (a,b,c not the same as A,B,C)
Y(0)=A = a*0^2 + b*0 + c, therefore c=A
Y(L)=B = a*L^2 + b*L + A
Y'(0) = C = b, therefore b=C
then solving for "a" we get this:
a = (B-CL-A)/(L^2)
Okay so our equation for your curve that passes through the points and with the initial angle as specified is:
Y(x)=((B-CL-A)/(L^2))*x^2 + Cx + A
Now use this equation for Y and calculate the length of the line using the method I outlined above.
I will write it anyway:
Okay - you now want to find the length of the curve I specified:
Length = Integrate Between X=0 and X=L the following function:
(Square Root of (1 + Y'(X)^2)) dX
You could do this analytically but it would probably be just as easy and not too inaccurate to do so with a numerical technique ranging from something simple like the midpoint method to something more complex like the Runge Kutta method