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GT MATH 2401 - Curves - A lengthy story
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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 vectorsThe 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. speedThe velocity v(t) = dr/dt is a vector function.The speed |v(t)| is a scalar function. |v(t)| ≥ 0.Arc lengthIf 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 lengthIf 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 lengthIf 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)| dtIn 2-D ds = (1 + y2)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 vectorsNot 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 planeBits of curve have a “best plane.”stickies on wire.The osculating planeBits of curve have a “best plane.”stickies on wire.Each stickie contains T and N.The osculating planeBits of curve have a “best plane.”One exception - a straight line lies in infinitely many planes.The osculating planeWhat’s the formula, for example for the helix?1. Parametric form2. Single equationThe binormal BThe 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 BThe 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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