Arc Length and Curvature Calc IV Lab Karen Donnelly Saint Joseph's College All rights reserved LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRis= QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiQyQtSSV3aXRoRzYiNiNJKnBsb3R0b29sc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJSEiIg==QyQtSSV3aXRoRzYiNiNJLkxpbmVhckFsZ2VicmFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI=QyQtSSV3aXRoRzYiNiNJL1ZlY3RvckNhbGN1bHVzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi QyQtSSdhc3N1bWVHNiI2IydJInRHRiVJJXJlYWxHRiUiIiI=LUkqaW50ZXJmYWNlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSxzaG93YXNzdW1lZEdGJyIiIQ== Pkklb3B0c0c2IjYkL0klYXhlc0dGJEkmYm94ZWRHRiQvSSdsYWJlbHNHRiQ3JUkieEdGJEkieUdGJEkiekdGJA==

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KS1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy1GLDYlUSJiRidGUkZVLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkmbXNxcnRHRiQ2Iy1GIzYqRistRiM2JC1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYnLUkmbWZyYWNHRiQ2KC1GIzYkLUY1Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjhGO0Y+RkBGQkZERkZGSEZKRk1GOC1GIzYmRistRiM2JUZecC1GLDYlUSJ0RidGUkZVRjhGK0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZdcS8lKWJldmVsbGVkR0Y9LUY1Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y4RjtGPkZARkJGREZGRkhGSkZNLUYjNiYtRiw2JVEieEYnRlJGVS1GNTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y4RjtGPkZARkJGREZGRkhGSkZNLUZlbzYkLUYjNiRGZXBGOEY4RjhGK0Y4RjgtSSNtbkdGJDYkUSIyRidGOC9GZW5GaW5GOC1GNTYtUSIrRidGOEY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmpyLUYjNiQtRmJvNiUtRmVvNiQtRiM2J0Zpb0ZicS1GIzYmLUYsNiVRInlGJ0ZSRlVGanFGXXJGOEYrRjhGOEZhckZlckY4RmZyLUYjNiQtRmJvNiUtRmVvNiQtRiM2J0Zpb0ZicS1GIzYmLUYsNiVRInpGJ0ZSRlVGanFGXXJGOEYrRjhGOEZhckZlckY4RitGOEYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRlt1LyUqbGluZWJyZWFrR1ElYXV0b0YnRl5wRmVwRjhGK0Y4 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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 JSFH

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. PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQiIiUtSSRzaW5HRjA2I0kidEdGJC1GKjYkRjMtSSRjb3NHRjBGNkY3 LUkrU3BhY2VDdXJ2ZUc2IjYlSSJSR0YkL0kidEdGJDsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGLkkoX3N5c2xpYkdGJDYkIiIjSSNQaUdGLkklb3B0c0dGJA== Thus to calculate the arc length of the vector valued function R above we use 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: PkkiVkc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlJHRiRJInRHRiQ= QyUtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiRJIlZHRilGLiIiIi1JKXNpbXBsaWZ5R0YrNiNJIiVHRik= QyUtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JJXNxcnRHRiU2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYmL0krbW9kdWxlbmFtZUdGKEksVHlwZXNldHRpbmdHRiU2JEkiVkdGKEY0L0kidEdGKDsiIiEtSSIqRzYkRiYvRjFJL1ZlY3RvckNhbGN1bHVzR0YlNiQiIiNJI1BpR0YmIiIiLUkmZXZhbGZHRiY2I0kiJUdGKA== Note that we really didnot need to work very hard to integrate this since V has constant length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbXNxcnRHRiQ2JC1GIzYlLUkjbW5HRiQ2JVEjMTdGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGNy8lK2JhY2tncm91bmRHRjYtSSNtb0dGJDYuUSJ+RidGNEY3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlFGPEY0Rjc=so 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: QyQtSSpBcmNMZW5ndGhHNiI2JEkiUkdGJS9JInRHRiU7IiIhLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRi9JKF9zeXNsaWJHRiU2JCIiI0kjUGlHRi8iIiI=LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= JSFH Suppose we make a change in the third coordinate of R ( make it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYoLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUscGxhY2Vob2xkZXJHRjRGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQ0Y+). PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQiIiUtSSRzaW5HRjA2I0kidEdGJC1GKjYkRjMtSSRjb3NHRjBGNiokRjciIiM= LUkrU3BhY2VDdXJ2ZUc2IjYlSSJSR0YkL0kidEdGJDsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGLkkoX3N5c2xpYkdGJDYkIiIjSSNQaUdGLkklb3B0c0dGJA== Now we have a velocity vector with non constant length. PkkiVkc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlJHRiRJInRHRiQ= QyUtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGJi9JK21vZHVsZW5hbWVHRihJLFR5cGVzZXR0aW5nR0YlNiRJIlZHRihGMSIiIi1JKXNpbXBsaWZ5R0YlNiNJIiVHRig= QyUtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JJXNxcnRHRiU2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYmL0krbW9kdWxlbmFtZUdGKEksVHlwZXNldHRpbmdHRiU2JEkiVkdGKEY0L0kidEdGKDsiIiEtSSIqRzYkRiYvRjFJL1ZlY3RvckNhbGN1bHVzR0YlNiQiIiNJI1BpR0YmIiIiLUkmZXZhbGZHRiY2I0kiJUdGKA== or QyQtSSpBcmNMZW5ndGhHNiI2JEkiUkdGJS9JInRHRiU7IiIhLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRi9JKF9zeXNsaWJHRiU2JCIiI0kjUGlHRi8iIiI=LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= JSFH

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.. Theorem 11.9, page 825: Let 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 define a twice differentiable plane curve C. The curvature function K is given by 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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 1/K. 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: LUklcGxvdEc2IjYnPCQqJEkieEdGJCIiIy1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YkNiRGKSokKS1JIitHRiw2JCIiIi1GKzYkIiIlRictRis2JCIiJCNGOUYpISIiL0YoOy1JIi1HRiw2I0Y/Rj8vSSJ5R0YkOyIiISIiJy9JJmNvbG9yR0YkNyRJJmtoYWtpR0YkSSZjb3JhbEdGJC9JKHNjYWxpbmdHRiRJLGNvbnN0cmFpbmVkR0Yk We see that curvature tends to zero as x goes towards either positive or negative infinity by taking JSFH QyUtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkiKkc2JEYmL0krbW9kdWxlbmFtZUdGKEkvVmVjdG9yQ2FsY3VsdXNHRiU2JCIiIyokKS1JIitHRiw2JCIiIi1GKzYkIiIlKiRJInhHRihGMS1GKzYkIiIkI0Y3RjEhIiIvRjxJKWluZmluaXR5R0YmRjctRiQ2JEYqL0Y8LUkiLUdGLDYjRkM= 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. QyQ+SSJwRzYiLUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYnKiRJInhHRiUiIiMvRi07LUkiLUc2JEYpL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHRig2I0YuRi4vSSJ5R0YlOyIiISIiJS9JKHNjYWxpbmdHRiVJLGNvbnN0cmFpbmVkR0YlL0kmY29sb3JHRiVJJHJlZEdGJSEiIg== QyQ+SSJjRzYiLUknY2lyY2xlR0YlNiY3JCIiISQiIiYhIiJGKyIiIi9JJmNvbG9yR0YlSSZncmVlbkdGJUYt LUkoZGlzcGxheUc2IjYjPCRJImNHRiRJInBHRiQ=

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: K = || dT/ds || where T is the unit tangent vector and s is arc length. It is easier to compute K 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 K = ||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 (see text Theorem 4.11, page 289), ds/dt = ||v(t)|| = ||dr/dt|| . Substituting this into the above equation for K, we have K = ||dT/dt|| / ||dr /dt|| which is the first formula in Theorem 12.8, page 871. The second formula for K is more complicated to prove, but follows Properties of the Derivative (Theorem 12.2, page 842): K = ||r'(t) X r''(t)|| / ||r'(t)||^3 Using this second form, we can calculate the curvature K for the helix r(t) = [2*cos(t), 2*sin(t), t], plotted below: PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQiIiUtSSRzaW5HRjA2I0kidEdGJC1GKjYkRjMtSSRjb3NHRjBGNkY3 LUkrU3BhY2VDdXJ2ZUc2IjYlSSJSR0YkL0kidEdGJDsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGLkkoX3N5c2xpYkdGJDYkIiIlSSNQaUdGLkklb3B0c0dGJA== PkkjRFJHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkSSJSR0YkSSJ0R0Yk PkkkRDJSRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkjRFJHRiRJInRHRiQ= PkkjQ1BHNiItSSMmeEdGJDYkSSNEUkdGJEkkRDJSR0Yk QyQ+SSRUT1BHNiItSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGKS9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nR0YoNiRJI0NQR0YlRjMiIiI=PkkkVE9QRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0Yj QyQ+SSdCT1RUT01HNiIqJC1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYqL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHRik2JEkjRFJHRiVGNCIiJCIiIg==PkknQk9UVE9NRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0Yj PkkiS0c2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYoSShfc3lzbGliR0YkNiRJJFRPUEdGJCokSSdCT1RUT01HRiQhIiI= Check our computation against built-in Curvature function in the Vector Calculus package: QyctSSpDdXJ2YXR1cmVHNiI2JEkiUkdGJUkidEdGJSIiIi1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJIiVHRiVGKS1JJmV2YWxmR0YtRi8= LUkqQ3VydmF0dXJlRzYiNiRJIlJHRiRJInRHRiQ= 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. PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQiIiMtSSRjb3NHRjA2I0kidEdGJC1GKjYkRjMtSSRzaW5HRjBGNiokRjdGMw== LUkrU3BhY2VDdXJ2ZUc2IjYlSSJSR0YkL0kidEdGJDsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGLkkoX3N5c2xpYkdGJDYkIiIlSSNQaUdGLkklb3B0c0dGJA== PkkjRFJHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkSSJSR0YkSSJ0R0Yk PkkkRDJSRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkjRFJHRiRJInRHRiQ= PkkjQ1BHNiItSSMmeEdGJDYkSSNEUkdGJEkkRDJSR0Yk QyQ+SSRUT1BHNiItSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGKS9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nR0YoNiRJI0NQR0YlRjMiIiI=PkkkVE9QRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0Yj QyQ+SSdCT1RUT01HNiIqJC1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYqL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHRik2JEkjRFJHRiVGNCIiJCIiIg==PkknQk9UVE9NRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0Yj PkkiS0c2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYoSShfc3lzbGliR0YkNiRJJFRPUEdGJCokSSdCT1RUT01HRiQhIiI= JSFH Curvature at t = 1 and t = 3: JSFHQyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEc2IiQiIiIiIiFJIktHRilGKw==LUklc3Vic0clKnByb3RlY3RlZEc2JC9JInRHNiIkIiIkIiIhSSJLR0Yo Or -- again with the Curvature function PkkiS0c2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSSpDdXJ2YXR1cmVHRiQ2JEkiUkdGJEkidEdGJA==; QyUtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEc2IiQiIiIiIiFJIktHRilGKy1GJDYkL0YoJCIiJEYsRi0=

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

JSFH

JSFH