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### Clothoid 1:

The variable is used for clothoid calculations; however, is actually a function of the projected arclength, . In this way, is used both as a function and as a variable. Since is a function of , which is itself a function of , which is a function of , is ultimately the independent variable of all functions, as discussed in section 1.2.

The equation of arclength of a clothoid is used to determine .

The azimuth angle along the curve is found from the value of , using equation A.7.

The projection of the curve in the - plane is determined by the Fresnel Integrals, which are described in appendix A. These integrals, when multiplied by the scaling factor , will produce the coordinates of the points on a clothoid. The coordinates of the curve vary linearly with the arclength along the curve.

 (2.21) (2.22) (2.23)

Derivatives with respect to arclength will need to be taken to determine the remaining functions. Since the position vector is expressed in terms of , the chain rule is applied. The chain rule will also be applied to quantities expressed in terms of , so the required relationships are developed here.

The forward vector is the derivative of the position vector with respect to arclength.

 (2.24) (2.25) (2.26)

The radial vector is the derivative of the forward vector with respect to arclength.

The unit vector in the direction of the radial vector must be determined.
 (2.27) (2.28) (2.29)

The curvature is the magnitude of the (non-unit) radial vector.
 (2.30)

For a curve with no elevation change, equation 2.30 reduces to , which is the equation for the curvature of a planar clothoid.2.4 The only difference in curvature between a two-dimensional curve and a three-dimensional curve is the term, which has a constant value. This simple result is due to the method of linearly stretching the two-dimensional curve to create the three-dimensional element, as explained in section 1.4.

2.4 This is shown toward the beginning of this section and in appendix A.

Next: Circular Region: Up: Curve Elements Previous: Curve Elements   Contents
Darla Weiss 2000-02-13