For this study, a roller coaster track will be modeled as a set of elements, aligned such that one begins where the previous one ended. In order to connect these elements to create a track, as well as to accurately model the dynamics, many geometric quantities must be known everywhere along the track. This section will introduce these quantities and explain the general derivation techniques for many of them. It will also explain the techniques used to guarantee curvature continuity throughout the track.
Many of the geometric parameters that need to be known vary from one element to the next, and will be discussed in sections 2.1.1-2.1.5; however, the following quantities must be known for all elements.
Other quantitites that must be determined along the track are functions of the distance into the element, . These include the following.
The curvature at the ends of each element is zero, in order to guarantee curvature continuity throughout the track. Since the middle of an element may have non-zero curvature, a shape must be used to connect the zero curvature region with the non-zero curvature region. This shape is a clothoid,2.1 a geometric entity that has linearly varying curvature with respect to distance along the curve. Clothoids are used in all elements discussed here, except straight elements. All elements at least partially composed of a clothoid will be referred to as clothoidal elements. Clothoids allow curvature continuity to be maintained throughout elements with varying curvature. A clothoidal element is made up of three sections:
A special note should be made about how the geometry of the three-dimensional elements (the curve, loop, and corkscrew) is created. The circular and clothoidal regions of an element are initially created in a plane. (This is the - plane for the curve and the - plane for the loop and corkscrew.) The element is then stretched along the axis perpendicular to the plane of creation, (the direction for curves and the direction for loops and corkscrews), such that the position change in this direction is linear with respect to distance along the element. This results in a constant angle between the plane of creation of the element and the tangent to the element after it has been stretched. This angle will be referred to as the vertical angle, , for the curve, and the shift angle, , for the loop and the corkscrew.
This approach of ``linear stretching'' may be considered somewhat simplistic when compared with the alternate method of allowing any desired curvature variations. However, as will be seen in the following sections, maintaining curvature continuity throughout the elements, even the two-dimensional joint, is a very complex process. This technique of transitioning from two to three-dimensions was used in order to prevent the creation of the these elements from becoming excessively complex. In this way, the goal of curvature continuity is maintained, while enabling the flexibility to create three-dimensional elements.