Mathematics for simulations

Lots of calculations are needed, both to arrive at the parameters needed in a simulation program, and to make proper use of those parameters. We will here show what is required in this regard. As of this writing, the text is not completely worked out. This page will only deal with the FLYWAY® system. We have a page titled "Computer Simulation of the Network" for those who want more general information.		List of contents: Calculating guideway coordinates

Coordinates for Beam Sections How would one go about simulating a desired network of beams? Well, one way would be to imagine a 3-dimensional cube that covers the area to be served. 3-dimensional, because the terrain usually undulates, and the beams have to adapt to that to some extent. Moreover, there might be sloping beams at stops, and so forth. To calculate coordinates and angular orientation of curved, constant-speed guideways in a series of segments, containing curves, slopes, off-line stations and straight runs, we will start at a given set of coordinates for X, Y, Z. The Z-coordinate is of course the height above a "reference" ground. And then, as we add new sections to this gradually growing network, we will need to designate the X, Y and Z coordinates for the end-point of straight section, as well as angular coordinates for curved sections. As might be realized, a beam section that alternates between straight and curved beams might have to be sub-divided into smaller sections, separated by "artificial nodes", in order to handle this section mathematically.	Figure 1:1 Figure 1:2	Using the The Azimuth angle The azimuth angle is, in astronomy, the angle on the horizon, between due south and the point towards the western horizon where a straight line from zenith through a certain star hits the horizon, as illustrated in figure 1:1. Thus, the position of a star in heaven, at a certain time, could always be determined by the 2 angles: the aimuth the angular hight from that point on the horizon towards zenith. Generally speaking, all curved beams could be regarded as part of a circle, with a specific radius of curvature. We then need to determine the center of this imagined circle, the angle at the start of the segment and the angle at the end of the segment, counted from a reference point in the azimuth plane. The advantage of this method is that if we build this simulated network in real life, we will get beam sections with constant radiuses of curvature. This means that we could calculate with a certain speed and a certain acceleration for this section. Care must of course be taken that the beams are not joined in such a way so as to create sudden changes in this radius of curvature.

Calculating Curved Beams Figure 1:3 Look at figure 1:3. Two straight beam elements, L1 and L2, are joined with a curved element, with radius of curvature = r. We will assume level beams, and forget about the Z-coordinate in this illustration. The points where the beams join, x1y1 and x2y2 respectively, could both be on the curving part of the curved beam, or one or both of them could be on a straight part of the curved beam segment. In the figure, we have chosen to let point x2y2 be a bit away from the circle that designates the curvature. How would we calculate then? Extending the straight beams from points x1y1 and x2y2, we find that those extensions come together at point P. These extensions can be regarded as y being a funcion of x. Since they are straight, they have constant slopes, k1 and k2. We designate these slopes as: k1 = y/x and k2 = y/x. Then; For L1: y = k1*x + y1 For L2: y = k2*x + y2 At point xp, ypthe lines cross each other. Thus: y on L1 = y on L2 = yp and: k1*xp + y1 = k2*xp + y2 ==> k1*xp = k2*xp + y2 - y1 ==> (k1 - k2)*xp = y2 - y1 And we arrive at the coordinates of point P: xp = (y2 - y1)/(k1 - k2) yp = k1*Xp + y1.

Let us define the bisector that goes through the center of the circle and through point P: kb is thus the slope of the bisector, and we find that: y - yp = kb(x - xp). We will then define the point P3 as having the coordinates x3 and y3. We can now designate the endpoint coordinates of the radius r on the perifery as S1 and S3. Then: The bisector will then have the slope k3. The equation for the radius r to point P3 could then be expressed as: kr = -1/k3 y - y3 = kr*(x- x3) In this manner, we have found a way to convert between the circular coordinates of curved beams and the "straight" coordinates of the XY-system. Sloping beams can then be treated in a similar manner, and be mapped onto the XYZ-coordinate system. Sloping beams are in themselves straight, but the junctures between horizontal and sloping beams can be regarded as vertical curves. The most complicated features to map onto a coordinate system would be those beams that bend horizontally and vertically at the same time. But these formulas make it posible to tackle those situations as well.