, the formula for the curvature (and radius of curvature) is stated in all calculus textbooks
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1 7. Curvature Given the function, the formula for the curvature (and radius of curvature) is stated in all calculus textbooks Definition (Curvature), Definition (Radius of Curvature). Definition (Osculating Circle) At the point {x,f[x]} on the curve y = f[x], the osculating circle is tangent to the curve and has radius r[x]. Example 1. Consider the parabola Solution 1. and the point {0,f[0]} = (0,0) on the curve. Find the radius of curvature and the circle of curvature. Finding Curvature at Any Point For the above example the circle of curvature was easy to locate because it's center lies on the y-axis. How do you locate the center if the point of tangency is not the origin? To begin, we need the concepts of tangent and normal vectors. Tangent and Normal Vectors Given the graph, a vector tangent to the curve at the point (x,f(x)) is. The unit tangent vector is, which can be written as Definition (Unit Tangent) For vectors in, a corresponding perpendicular vector called the unit normal vector is given by Lemma (Unit Normal). Example 2. Consider the parabola and the point (1,f(1)) = (1,1) on the curve. Find the unit tangent and unit normal at point (1,1). Solution 2. Example 3. Consider the parabola and the point (1,f(1)) = (1,1) on the curve. Find the radius of curvature and the circle of curvature. Solution 3. A new construction of the Circle of Curvature file:///c /numerical_analysis/chap04/curvaturemod.html (1 of 6) :24:08
![and the point {0,f[0]} = (0,0) on the curve. Find the radius of curvature and the circle of curvature.](/docs-images/44/21069410/images/page_1.jpg "Finding Curvature at Any Point For the above example the circle of curvature was easy to locate because it's center lies on the y-axis.")
2 What determines a circle? A center and a radius. The formula for the radius of curvature is well established. What idea could we use to help understand the situation. We could use the fact that three points determine a circle and see where this leads. Example 4. Consider the parabola and the point (0,f(0)) = (0,0) on the curve. Find collocation circle to go through the three points,, and, and explore the situation for h = 1,.1,.01. Solution 4., Derivation of the Radius of Curvature The standard derivation of the formula for radius curvature involves the rate of change of the unit tangent vector. This new derivation starts with the collocation the collocation circle to go through the three points,, and on the curve. The limit as is the osculating circle to the curve at the point. The radius of curvature and formulas for the location of its center are simultaneously derived. Start with the equation, of a circle. Then write down three equations that force the collocation circle to go through the three points,, and on the curve. Solve the equations for and extract the formula for the radius of the collocation circle. Since it depends on we will store it as the function. The formula looks bewildering and one may wonder if it is of any value. Therefore, the limit in the numerator is derivative, hence the is.. The difference quotient in the denominator is recognized as the numerical approximation formula for the second The Osculating Circle file:///c /numerical_analysis/chap04/curvaturemod.html (2 of 6) :24:08
3 We now show that the limit of the collocation circle as is the osculating circle. Now we want to find the center (a, b) of the osculating circle. The abscissa for the center of the collocation circle is Take the limit as, to obtain the abscissa for the center of the circle of curvature. The numerators involve three difference quotients, all of which tend to when, and the difference quotient in the denominators tends to when. The Abscissa the Easy Way Subtract from the radius of curvature times. file:///c /numerical_analysis/chap04/curvaturemod.html (3 of 6) :24:08
4 The Ordinate for the Center of the Circle of Curvature The ordinate for the center of the collocation circle is Thus we have established the formula for the ordinate of center of the circle of curvature. The Ordinate of the Circle the Easy Way Add to the radius of curvature times The Osculating Circle At the point on the curve, the center and radius of the osculating circle are given by the limits calculated in the preceding discussion. file:///c /numerical_analysis/chap04/curvaturemod.html (4 of 6) :24:08
5 Example 5. Consider the parabola at the point. Draw the circle of curvature for various values Solution 5. Generalizations for 2D In two dimensions, a curve can be expressed with the parametric equations and. Similarly, the formulas for the radius of curvature and center of curvature can be derived using limits. At the point the center and radius of the circle of convergence is Remark. The absolute value is necessary, otherwise the formula would only work for a curve that is positively oriented. The Abscissa the Easy Way Subtract from the radius of curvature times. The abscissa of the circle of curvature is The Ordinate the Easy Way Add to the radius of curvature times. The ordinate of the circle of curvature is file:///c /numerical_analysis/chap04/curvaturemod.html (5 of 6) :24:08
6 Example 6. Consider the cardioid,. Draw the circle of curvature at. Solution 6. Example 7. Consider the cardioid,. Draw the circle of curvature at. Solution 7. file:///c /numerical_analysis/chap04/curvaturemod.html (6 of 6) :24:08
7 Example 1. Consider the parabola and the point {0,f[0]} = (0,0) on the curve. Find the radius of curvature and the circle of curvature. Solution 1. At x=0, we have We know the shape of this parabola and from symmetry we can conclude that the circle of curvature will have center and radius. file:///c /numerical_analysis/chap04/curvaturemod_lnk_1.html (1 of 3) :24:08
8 Discussion. What do other kinds of "tangent circles" look like? file:///c /numerical_analysis/chap04/curvaturemod_lnk_1.html (2 of 3) :24:08
9 It looks like the "circle of curvature" is the "best fitting" circle. file:///c /numerical_analysis/chap04/curvaturemod_lnk_1.html (3 of 3) :24:08
10 Example 2. Consider the parabola and the point (1,f(1)) = (1,1) on the curve. Find the unit tangent and unit normal at point (1,1). Solution 2. file:///c /numerical_analysis/chap04/curvaturemod_lnk_2.html :24:08
11 Example 3. Consider the parabola and the point (1,f(1)) = (1,1) on the curve. Find the radius of curvature and the circle of curvature. Solution 3. file:///c /numerical_analysis/chap04/curvaturemod_lnk_3.html (1 of 2) :24:08
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13 Example 4. Consider the parabola and the point (0,f(0)) = (0,0) on the curve. Find collocation circle to go through the three points,, and, and explore the situation for h = 1,.1,.01. Solution 4. Start with the equation of a circle. Then write down three equations that force the collocation circle to go through the three points,, and. Expand the equations and get Solve the equations for and extract the formula for the radius of the collocation circle. file:///c /numerical_analysis/chap04/curvaturemod_lnk_4.html (1 of 8) :24:08
14 Animation. Draw the circle of curvature for various values file:///c /numerical_analysis/chap04/curvaturemod_lnk_4.html (2 of 8) :24:08
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20 We can conjecture that the limit of the collocation polynomial is the circle of curvature. file:///c /numerical_analysis/chap04/curvaturemod_lnk_4.html (8 of 8) :24:08
21 Example 5. Consider the parabola at the point. Draw the circle of curvature for various values Solution 5. file:///c /numerical_analysis/chap04/curvaturemod_lnk_5.html (1 of 11) :24:08
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32 Example 6. Consider the cardioid,. Draw the circle of curvature at. Solution 6. file:///c /numerical_analysis/chap04/curvaturemod_lnk_6.html (1 of 3) :24:08
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35 Example 7. Consider the cardioid,. Draw the circle of curvature at. Solution 7. The radius of curvature formula can be used provided that the curve is positively oriented. Loosely speaking the curve must be oriented in the "counterclockwise direction". Let's investigate the situation at hand. file:///c /numerical_analysis/chap04/curvaturemod_lnk_7.html (1 of 2) :24:08
36 The formula for the radius of curvature computes a negative value. The correct the situation, change the sign and use. Caveat. The formula for the radius of curvature should include an absolute value. Be careful! In Mathematica it could be written as. file:///c /numerical_analysis/chap04/curvaturemod_lnk_7.html (2 of 2) :24:08
88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
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