Clothoids: The Supreme Expression of Equality By: Kyle Rush Calculus 4 Project Dr. Karen Donelley April 8, 2008 restart; with(plots):

Clothoids, also known as the Spiral of Cornu or the Euler Spirals, are a family of curves in which the curvature of the function is directly proportional to its arc length(or proportional to a polynomic equation of). This means that in the simplest case, as you move further away from the origin the curve becomes "curvier". The basic set of parametric equations of a clothoid is: 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

Clothoids are based on the work of Augustin-Jean Fresnel who created the Fresnel integrals to explain certain forms of light diffraction. The clothoids were first studied in the 1740\342\200\231s by Leonhard Euler. However, he did not study them analytically and merely included them in his exploration of curves. The first analytical study of clothoids occurred in the 1870\342\200\231s with the work of a French scientist Marie Alfred Cornu. Cornu studied the curve in connection with the diffraction of light. Thus a clothoid can also be called a Cornu Spiral or Euler\342\200\231s Spiral.

Here we plot the basic clothoid 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And now we can animate the curve and see it being drawn out 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Here are some variations of clothoids 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In order to understand what a clothoid is, one must first understand the concepts of curvature LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSgma2FwcGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y3RjctSSNtb0dGJDYtUSJ+RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk5GNw== and arc length (s).

Arc length measures the length of a segment of a curve (Usually measured from the origin). In order to find the arc length of our function, R(t) we: find its derivative, R\342\200\262(t), then find the magnitude of R\342\200\262(t), and finally, integrate it over the desired interval.

An informal definition of curvature is a measure of how "curvy" a function is. Curvature is the rate of change of the angle \316\270 that a curve's tangent (line) makes with some fixed line (often the x-axis). Saying rate of change is equivalent to taking the derivative of a function, so \316\272 = \316\270\342\200\262(t)

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 Now, we must recognize the trigonometric identity 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 JSFH

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therefore, we know that, for a clothoid: 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 t can be function of t. Since we want 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we simply take the derivative of both sides of the equation and arrive at: 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 order to evaluate this, we will use equations that have been determined by previous mathematicians, as it is not easy to find the derivative of an inverse trigonometric function. The math is a little long, but in the end we arrive at 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjQTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjQTJGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdGMUYy 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJGRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSV0cmlnRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ== And so we see that the \316\272LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUSJ+RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSYwLjBlbUYnL0ZFRlNGLw==and sLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0Yv, therefore curvature equals arc length.

JSFH