Curves - A lengthy story Lecture 4 MATH 2401 - Harrell Copyright 2007 by Evans M. Harrell II.Reminder…What a lonely archive!!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 κ.  The arc length s.  The osculating plane.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.  More generally, ds = |v(t)| dt  In 2-D ds = (1 + y′2)1/2 dx, or ds2 = dx2 + dy2 or…Arc lengthExample: spiralExamplesExample: helixExamples Miraculously - 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)|T NNormal 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?  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: Ways to describe a line: slope-intercept y = m x + b 2 points, point-slope These 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) j T(t) = (1 /(1+16π2)1/2)(-4πsin(4πt) i + 4π cos(4πt) j + k)r(t) = cos(4πt) i + sin(4πt) j + t k Example: 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)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. 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 form 2. 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.  Since the osculating plane contains T and N, a normal to the plane is B = T × NClose-upThe osculating plane  What’s the formula, for example for the helix? 1. Parametric form 2. Single equationExample: The helix T(t) = (- 4 π sin(4 π t) i + 4 π cos(4 π t) j + k )/(1+16π2)1/2 N(t) = - cos(4 π t) i - sin(4 π t) j r(t) = cos(4 π t) i + sin(4 π t) j + t kExample: The helix r(t) = cos(4 π t) i + sin(4 π t) j + t k T(t) × N(t) = (- sin(4 π t) i + cos(4 π t) j - 4 π k )/(1+16π2)1/2 Osculating plane at (1,0,1): Calculate at t=1. (rosc - (i + k)) ·(1 j - 4π k) = 0 (The factor (1+16π2)1/2 can be dropped.)The moving trihedron  The curve’s preferred coordinate system is oriented along (T,N,B), not some Cartesian system (i,j,k) in the sky.The moving trihedron  A vehicle can rotate around any of these axes. A rotation around T is known as roll. If the vehicle has wings (or a hull) it may prefer a second direction over N. For example, the wing direction may correlate with N when the airplane turns without raising or lowering the nose. Such an acceleration is called yaw.Figure from JPL/NASAThe moving trihedron  However, when the aircraft soars or dives (this kind of acceleration is called pitch), the normal vector N is perpendicular to the wing axis, which in this case correlates with the binormal B.  An aircraft can accelerate, roll, yaw, and pitch all at once. Fasten your seatbelt!Watercraft have the same kinds of accelerations as aircraft. The rudder controls yaw. The boat is usually designed to minimize pitch and roll. pitch roll yawJust what is curvature?  How do you know a curve is curving? And how much?  The answer should depend just on the shape of the curve, not on the speed at which it is drawn. So it connects with arclength s, not with a time-parameter t.Just what is curvature? WHICH CURVES MORE? HOW ABOUT HERE?How rapidly do T and N change? T T T T T T N N N N N N N TJust what is curvature?  And let’s be quantitative about it!  2D: How about |dφ/ds|, where φ is the direction of T with respect to the x-axis?  To get started, notice that the direction of T is the same as that of the tangent line. That is, tan φ = dy/dx = (dy/ds)/(dx/ds) (fasten seatbelts for the next slide!)It’s our old friend the chain rule, used in a creative way!Different expressions for κ  κ = |dφ/ds|  κ = |(dφ/dt)/(ds/dt)|  κ = |x′(s) y′′(s) - y′(s) x′′(s)|  κ = |x′(t) y′′(t) - y′(t) x′′(t)| |(x′(t))2 + (y′(t))2|3/2 Huh??Example  Circle of radius 5.  No calculus needed!  If you move distance Δs along the perimeter, the change in angle is Δs/5. So Δφ/Δs = 1/5. The general rule for a circle is that the curvature is the reciprocal of the radius.Example  Spiral: The formula for curvature is complicated, but the spiral is simple, so the curvature should be simple.  Still, we’ll be