Arc Length and Curvature Calc IV Lab Karen Donnelly Saint Joseph's College

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Let r(t) = <x(t) , y(t) , z(t) > be a differentiable vector valued function on [a,b]. Then the arc length s is defined by 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 If r(t) = <x(t) , y(t) > is a vector-valued function in the plane then the formula becomes 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 In either case we can remember this formula simply as the integral of the "speed" -- magnitude of the velocity vector over the interval (a,b). 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Consider the following helix: (Note we define r as a list for the space curve command and make R the vector version of it so we may use in VectorCalculus package commands. 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 Thus to calculate the arc length of the vector valued function R above we use 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: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiktRiw2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSZtZnJhY0dGJDYoLUY9Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRkAvRkNRJnVuc2V0RicvRkZGZ24vRkhGZ24vRkpGZ24vRkxGZ24vRk5GZ24vRlBGZ24vRlJRJjAuMGVtRicvRlVGX28tRiM2JkZZLUY9Ni1RIn5GJ0ZARkJGRUZHRklGS0ZNRk9GXm9GYG8tRiw2JVEidEYnRjZGOUZALyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZecC8lKWJldmVsbGVkR0ZERmNvLUYsNiVRIlJGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0ZmcEZARitGZnBGQA== 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 Note that we really didnot need to work very hard to integrate this since V has constant length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbXNxcnRHRiQ2JC1GIzYlLUkjbW5HRiQ2JVEjMTdGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGN0Y0LUkjbW9HRiQ2LlEifkYnRjRGNy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZPRjpGNEY3so the integral is just the length of the interval of integration times this constant. (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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIzE3RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSkYy ). Maple has a built in command ArcLength as part of the VectorCalculus package for computing this arclength: 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 Suppose we make a change in the third coordinate of R ( make it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYoLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUscGxhY2Vob2xkZXJHRjRGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQ0Y+). 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 Now we have a velocity vector with non constant length. 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 or 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Curvature at a point on a curve gives a measure of how sharply the curve bends there. Curves which turn tightly through a short arc length have a large curvature, whereas curves which turn through a large arc length have a smaller curvature.. Formula for Curvature for rectangular equations in the plane: Let 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 define a twice differentiable plane curve C. The curvature function \316\272 can be computed as 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 If the curve 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 has curvature K at point P, then the circle of curvature is that circle which passes through P on the concave side with radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJm1mcmFjR0YkNigtRiM2JS1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjotRiM2JS1GLDYlUSJLRicvJSdpdGFsaWNHUSV0cnVlRicvRjtRJ2l0YWxpY0YnRj1GOi8lLmxpbmV0aGlja25lc3NHRjkvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGTi8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y9RjpGK0Y9Rjo=. The center of the circle is called the center of curvature at P. The radius of the circle is called the radius of curvature at P.

The curvature of the parabola 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 is 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. (Verify this using the above formula by hand.) Below we plot both the parabola (khaki) and its curvature: 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 We see that curvature tends to zero as x goes towards either positive or negative infinity by taking LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 At the origin the curvature is 2. ( Verify). Hence the circle of curvature at P = (0,0) has radius 1/2 and and center (0,0.5). The graphs of the parabola and its circle of curvature are plotted belowat the origin are both shown in the figure below. 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

The curvature K, at a given point, of a space curve is defined as the magnitude of rate of change of the unit tangent vector T with respect to arc length s, or more formally: \316\272 = || dT/ds || where T is the unit tangent vector and s is arc length. It is easier to compute \316\272 in terms of the parameter t instead of s. We can derive a formula for this as follows: By the chain rule: dT/dt = (dT/ds) (ds/dt) so dT/ds = (dT/dt) / (ds/dt) and \316\272 = ||dT/ds|| = ||dT/dt|| / |ds/dt| But by the definition of arc length function s(t) = 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 and the second Fundamental Theorem of Calculus, ds/dt = ||v(t)|| = ||dr/dt|| . Substituting this into the above equation for K, we have \316\272 = ||dT/dt|| / ||dr /dt|| A second formula for K is more complicated to prove, but follows from properties of the derivative.: \316\272 = ||r'(t) X r''(t)|| / ||r'(t)||3 Using this second form, we can calculate the curvature \316\272 for the helix 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, plotted below: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqQ3VydmF0dXJlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEiUkYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidEYnRi9GMi8lK2V4ZWN1dGFibGVHRkVGQUZBRmVuRkE= Not surprisingly, the curvature for this helix is constant. If we again make a "minor" change to the helix: R = < 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 > we will see curvature is not constant. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEkVE9QRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JJm1zcXJ0R0YkNiMtRiM2Jy1GLDYlUSNDUEYnRjZGOS1GPTYtUSIuRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRlxvRmVuLyUrZXhlY3V0YWJsZUdGREZARl5vRkBGK0Zeb0ZALUY9Ni1RIjtGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GW29GVEZeb0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEkVE9QRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYnLUYsNiVRKXNpbXBsaWZ5RidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYlRjMvJStleGVjdXRhYmxlR0ZERkBGQEZgb0ZARitGYG9GQEYrRmBvRkBGK0Zgb0ZA LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEnQk9UVE9NRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYkLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUkmbXNxcnRHRiQ2Iy1GIzYnLUYsNiVRI0RSRidGNkY5LUY9Ni1RIi5GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGZG9GXW8vJStleGVjdXRhYmxlR0ZERkBGQC1JI21uR0YkNiRRIjNGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZARitGZm9GQEYrRmZvRkAtRj02LVEiO0YnRkBGQi9GRkY4RkdGSUZLRk1GT0Zjb0ZURmZvRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEnQk9UVE9NRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYnLUYsNiVRKXNpbXBsaWZ5RidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYlRjMvJStleGVjdXRhYmxlR0ZERkBGQEZgb0ZARitGYG9GQEYrRmBvRkBGK0Zgb0ZA LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEkVE9QRidGNkY5LyUrZXhlY3V0YWJsZUdGREZALUYjNiUtRiw2JVEnQk9UVE9NRidGNkY5RmhuRkAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmRvLyUpYmV2ZWxsZWRHRkRGaG5GQEYrRmhuRkBGK0ZobkZA LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Curvature at t = 1 and t = 3: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 Or -- again with the Curvature function 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; 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