Curves - A lengthy storyReminder…Slide 3Slide 4Slide 5Slide 6In our previous episode: We encountered tangent vectorsNormal vectorsWho in the cast of characters might show up on the test?Let’s vote!Velocity vs. speedArc lengthSlide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Example: spiralExamplesExample: helixSlide 23Unit tangent vectorsSlide 25Slide 26Slide 27Tangent and normal vectors, and arc length.Slide 29Slide 30Slide 31Slide 32Slide 33Admittedly….Slide 35Tangent and normal lines:Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43The osculating planeSlide 45Slide 46Slide 47Slide 48Slide 49The binormal BSlide 51Slide 52Close-upSlide 54Slide 55Example: The helixSlide 57Slide 58The moving trihedronSlide 60Slide 61Slide 62Slide 63Slide 64Just what is curvature?Slide 66Slide 67Slide 68Slide 69How rapidly do T and N change?Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Slide 80Different expressions for Slide 82Slide 83Slide 84ExampleSlide 86Slide 87Slide 88Slide 89Slide 90In our previous episode:Dimensional analysisSlide 93Slide 94Slide 95Slide 96Slide 97Curves - A lengthy storyLecture 4MATH 2401 - HarrellCopyright 2007 by Evans M. Harrell II.Reminder…No class on Monday, but ….No class on Monday, but ….Reminder…There’s a test on Thursday!There’s a test on Thursday!No class on Monday, but ….No class on Monday, but ….What a lonely archive!In our previous episode: We encountered tangent vectorsThe velocity vector is tangent to the curve.All tangents (velocities) are proportional. They may have different lengths but their directions are the same. Or exactly opposite (negative multiple).Exceptional case: v = 0.Normal vectors N = T/|T|.Unless the curve is straight at position P, N is defined as a unit vector perpendicular to T.Who in the cast of characters might show up on the test?Curves r(t), velocity v(t).Tangent and normal lines.Angles at which curves cross.T,N, B, and the curvature and torsion .The arc length s.The osculating plane.Let’s vote!Choice #1…. Charge ahead and learn about curvature and osculating planes.Choice #2…. Spend more time on curves.Velocity vs. speedThe velocity v(t) = dr/dt is a vector function.The speed |v(t)| is a scalar function. |v(t)| ≥ 0.Arc lengthIf an ant crawls at 1 cm/sec along a curve, the time it takes from a to b is the arc length from a to b.Arc lengthIf an ant crawls at 1 cm/sec along a curve, the time it takes from a to b is the arc length from a to b.More generally, ds = |v(t)| dtArc lengthIf an ant crawls at 1 cm/sec along a curve, the time it takes from a to b is the arc length from a to b.More generally, ds = |v(t)| dtIn 2-D ds = (1 + y2)1/2 dx, or ds2 = dx2 + dy2 or…Arc lengthExample: spiralExamplesExample: helixExamplesMiraculously - don’t expect this in other examples - the speed does not depend on t. The arclength in 2 coils, t from 0 to 1, is the integral of |r| over this integral, i.e., (1+ 16 π2)1/2.dtUnit tangent vectorsNot only useful for arc length, also for understanding the curve ‘from the inside.’Move on curve with speed 1. T(t) = r(t)/ |r(t)|TNNormal vectors N = T /|T|.Unless the curve is straight at position P, N is defined as a unit vector perpendicular to T.Tangent and normal vectors, and arc length.If you “parametrize with arc length,” what does that mean for T and N?Tangent and normal vectors, and arc length.If you “parametrize with arc length, what does that mean for T and N? T = dr(s)/ds - No denominator!Tangent and normal vectors, and arc length.If you “parametrize with arc length, what does that mean for T and N? T = dr(s)/ds - No denominator! N = T/|T| - You still have to “normalize”Tangent and normal vectors, and arc length.If you “parametrize with arc length, what does that mean for T and N? T = dr(s)/ds - No denominator! N = T/|T| - You still have to “normalize”Next week we’ll use |T| to quantify curvature.sAdmittedly….You can really get tangled up in these calculations!Tangent and normal lines:Recall the helix:Tangent and normal lines:Recall the helix:Tangent and normal lines:Recall the helix:Tangent and normal lines:Ways to describe a line: slope-intercept y = m x + b 2 points, point-slopeThese are not so useful in 3-D. Better: parametric form: r(t) = r0 + u v (call parameter something other than t)Tangent and normal lines:The essential facts about the helix:N(t) = - cos(4πt) i - sin(4πt) jT(t) = (1 /(1+16π2)1/2) (-4πsin(4πt) i + 4π cos(4πt) j + k)r(t) = cos(4πt) i + sin(4πt) j + t kExample: Tangent and normal lines at (1,0,1)Tangent and normal lines:Example: Tangent and normal lines at (1,0,1)r(t) = (cos(4πt),sin(4πt),t)= (1,0,1) when t = 1.T(1) = (1 /(1+16π2)1/2) (-4πsin(4π) i + 4π cos(4π) j + k) = (1 /(1+16π2)1/2) (4π j + k) .Line: (1,0,1) + u (4π j + k)Tangent and normal lines:Example: Tangent and normal lines at (1,0,1)r(t) = (cos(4πt),sin(4πt),t)= (1,0,1) when t = 1.T(1) = (1/(1+16π2)1/2) (-4πsin(4π) i + 4π cos(4π) j + k) = (1/(1+16π2)1/2) (4π j + k) .Line: (1,0,1) + u (4π j + k)Hey! What in &#*$ happened to the (1/(1+16π2)1/2) ?Tangent and normal lines:Example: Tangent and normal lines at (1,0,1)r(t) = (cos(4πt),sin(4πt),t)= (1,0,1) when t = 1.N(1) = - cos(4π) i - sin(4π) j = - i .Line: (1,0,1) + u i .Wait a minute! What about the sign ?The osculating planeBits of curve have a “best plane.”stickies on wire.The osculating planeBits of curve have a “best plane.”stickies on wire.Each stickie contains T and N.The osculating planeBits of curve have a “best plane.”One exception - a straight line lies in infinitely many planes.The osculating planeWhat’s the formula, for example for the helix?1. Parametric form2. Single equationThe binormal BThe normal vector to a plane is not the same as the normal to a curve in the plane. It has to be to all the curves and vectors that lie within the plane.The binormal BThe normal vector to a plane is not the same as the normal to a curve in the plane. It has to be to all the curves and vectors that lie within the plane.Since the osculating plane contains T and N, a normal to the plane is
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