C1 With a Twist
Generating a spline requires some minimum number of points (often referred to as control points) for the curve to pass through in order. Using these control points and some fancy Interpolation maths, a curve can be approximated using very short line segments to any required precision. The curve's smoothness, or Order of Continuity, can also be controlled during approximation. In short, the higher the continuity, the smoother and more accurate the curve.
In most applications, including Coaster Creator, a C1 continuous Cardinal Spline is preferred because it passes through all defined control points (higher order curves usually do not). Lets examine a C1 continuous curve that has control points that are equally spaced apart (as in Coaster Creator):
Yep, it is a curve alright. If it were a Coaster Creator track, it wouldn't be a very good one and our coaster train would get stuck. Let's move the third control point down to give our track a dip instead of a hill. For comparison, we'll make the original track semi-transparent.
Looks good...but wait, there is now a small hill in the first track segment. We weren't changing that part of the track and didn't intend to. A coaster train still wouldn't make it over that bump, which would be very frustrating to the player. Let's take a look at how a Coaster Creator track is built and why it doesn't have this problem.
Notice that the first track segment is completely flat between the two control points with the same height (22.9m). The dip in segment two did not affect our straight piece in segment one, but the track still looks completely smooth. Wonderful, but how? Below is the same image with our usual curved arrow and control point symbols overlaid to show each cardinal spline and control point used to make the track:
Of course! There is more than one spline and only part of each spline is used. The (semi-transparent) parts that are ignored are thrown out. Looking closely, you'll notice that there are also some hidden (semi-transparent) control points...these keep our final spline smooth and unbroken, and give it a tangent of 0 at each non-hidden control point. Calculating the positions of the hidden control points requires some surprisingly simple logic, but I won't get into that here. The portions of each spline that are kept make up our final track spline, which is C1 continuous and hits all shown control points but is influenced by a series of hidden control points...and that's the twist!
To see some traditional C1 splines in action check out this interactive site. You'll notice right away that the spline does not behave as intuitively as the Coaster Creator track spline.