Smooth curve between 2 points with given gradient at first point? I'm trying to create a smooth curve between 2 given points with a given gradient/tangent at the first point and any gradient at the last. The idea being to be able to join these to create a smooth overall curve. The reason I am not using the standard spline methods is that I do not know any of the future points when drawing each part of the curve.
What is the best/simplest way to achieve this?
 A: Cubic Hermite curve seems to be right for the purpose. A cubic Hermite curve is defined by a series of points and first derivatives at these given points. A cubic Hermite curve is a piecewise continuous curve (in general only C1) and each segment (the portion between two successive points) is in fact a cubic polynomial. See this link for more details. It is easy to implement and the best thing is adding a new point will not affect the shape of the previous segments. If you do not know what first derivative value should be used for any point, you can use finite difference method to infer it.
A: There are quite a few ways it can be done, in fact possibilities are unlimited.
You have defined the stating point,starting slope, ending point and an initial curvature needing to be varied. These four conditions call for a fourth order differential equation  in a Boundary Value Problem.
I chose a relatively simple linear ODE here for y(x) as:
$$ y^{''''}(x) - y^{''}(x) + y(x) = 0. $$
All integrand curves have the same initial slope = 2, pass through the origin (0,0) and a fixed end point (3,10), and prescribed initial point bent curvature $ki$  varies between -60 and + 60 among all the given curves. If there is a particular point you wish the curve to pass through or define slope $k$ anywhere else, one from such a solution set can be chosen.

Also how " simple" or "good " it is, depends on choice of a defined criterion to evaluate the curve. 
