2.3 Geometry Of Curves: Arclength, Curvature, Torsion .
82.3Chapter 2 draftGeometry of curves: arclength, curvature, torsionOverview: The geometry of curves in space is described independently ofhow the curve is parameterized. The key notion of curvature measures howrapidly the curve is bending in space. In 3-D, an additional quantity, torsion, describes how much the curve is wobbling out of a plane. Alternativemethods of calculation for curvature and torsion are developed.2.3.1Idea of curved vs. straight as independent of parametrizationThe goal of this subsection is to create some measurement of curvedness that is aproperty of the road, not the driver. A first question: which of the two circles is“more curved”?Figure 1: The two circles have different “curviness”You most likely decided that the one with the smaller radius is more curved. Howcan we quantify that? How should we relate more general curves to circles? Whatcould play the role of the radius? How curved is a line?2.3.2Circular model: tied to radius of circleIf a smaller radius leads to a more curved circle, it follows that the measurement ofcurvature should increase as the radius of a circle decreases. Curvature has a longhistory including work in ancient Greece, but the curvature of a general curve inthe plane was solved by Newton, building on earlier work by Oresme, Huyghens,and others (see Lodder[TBA REF] and references therein). Newton defined thecurvature for a circle as the reciprocal of the radius. Newton then went deeperand developed the general case in terms of the radius of the best circular approximation, made instantaneously at each point. For each such approximating circle,known now as the osculating circle, the radius of curvature is defined as the radius of the osculating circle and curvature is then the reciprocal of the radius of
2.3Geometry of curves: arclength, curvature, torsion9curvature. From our recent analysis of circular motion, the acceleration will play arole in this definition, particularly its normal component.Newton’s definition is not particularly easy to use for calculation, but it will beillustrated in a simple example and the extension to general planar curves is givenin a sequence of problems at the end of the section, problems TBN.Example 2.10 Curvature at the vertex of a parabola: Let y a x2 for a 0define a parabola. Find the best instantaneous circle approximation at the vertex(0, 0) and use it to calculate the radius of curvature and the curvature at the vertex.By symmetry, we can suppose the circle to have center along the y-axis. Since theparabolas bend up, the circles that vie for best approximation should lie above thex axis. The circles of radius R of that form pass through (0, 0) with center at(0, R) so they have equations: x2 (y R)2 R2 . Now we can look for secondderivatives to match up by choice of radius R. The circle splits into two semicircleswhen we express y as a function of x and we are focusing on the lower half, whichgoes through our origin. Thus y R R2 x2 near (0, 0). Using a Taylorexpansion in powers of x to second order is best done by algebraic substitutiont x2 from R2 t R2 12 t/R, which now becomes: y R R 12 x2 /R,which should equal y a x2 locally. This means we choose 2R 1/a. The radiusof curvature at the vertex of the family of parabolas is R 1/2a and the curvatureis 1/R 2a. Note that this is also the value of the second derivative at the vertex.A graphical illustration of the approximation to a parabola by circles is given in thefigure below, where the value of a is 5, so the radius of curvature at the vertex isR 0.1.Competing circles for best approximation to y 5x 20.050.040.030.020.01-0.10-0.050.050.10Figure 2: The parabola is the blue curve, while the red circles have radii:0.05, 0.075, 0.1, 0.15A question to ponder: The parabola has a constant second derivative. Do you thinkthe parabola has constant curvature? Why or why not? What does this suggestabout the relation between curvature an n the tangent and the change perpendicular to it. It will turn out that in thosenew coordinates you are back to the previous problem, so the curvature willbe the reciprocal of the second derivative in the new variables. Then show(by Taylor approximation or otherwise) that the second derivative in thesenew variables is the expression f 00 (x0 )(cos φ0 )3 where f 00 (x0 ) is the secondderivative without rotation (meaning in (x, y) world) and the cosine termuses the angle of rotation, which means tan φ0 f 0 (x0 ).13. (circular matching property in 2-D general case directly) Consider a curvein the plane given as x(t)ı y(t) . Remind yourself what the tangent vectorof such a curve is and therefore find a convenient vector perpendicular to
2.3Geometry of curves: arclength, curvature, torsion15the tangent direction. The best fit circle has its center, say (a, b), on thenormal line (the perpendicular to the tangent line at the given point), so thedistance from the center to the curve at t t0 will determine the circle. Youcan parameterize the family of circles tangent to the curve at some locationas (x(t) a)2 (y(t) b)2 R2 (perhaps an overuse of t) and thenseek to match the circle with the graph as closely as possible. This requires(a, b) to lie on the normal line and the square of the radius to be R2 (x(t0 ) a)2 (y(t0 ) b)2 . Show that taking first and second derivatives int at t t0 yields the center and radius and that the radius as determined inthis way is the radius of curvature.14. Challenging (limit version like tangent as limit of secant lines): Considera curve in the plane (for simplicity). Pick three distinct points on it, say(x0 , y0 ), (x1 , y1 ), and (x2 , y2 ). These points determine a circle through allthree. Take the limit as the points (x1 , y1 ) and (x2 , y2 ) approach the fixedlocation (x0 , y0 ) to find the equation of the osculating circle and the radiusof curvature.
2.3 Geometry of curves: arclength, curvature, torsion Overview: The geometry of curves in space is described independently of how the curve is parameterized. The key notion of curvature measures how rapidly the curve is bending in space. In 3-D, an additional quantity, tor-sion, describes h
Sebastian Montiel, Antonio Ros, \Curves and surfaces", American Mathematical Society 1998 Alfred Gray, \Modern di erential geometry of curves and surfaces", CRC Press 1993 5. 6 CHAPTER 1. CURVES . Contents: This course is about the analysis of curves and surfaces in 2-and 3-space using the tools of calculus and linear algebra. Emphasis will
Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces . Hermite geometry matrix M H satisfying. 02/11/2003 15-462 Graphics I 25 Blending FunctionsBlending Functions Explicit Hermite geometry matrix
SOLIDWORKS creates surface bodies by forming a mesh of curves called U-V curves. The U curves run along a 4-sided surface and are shown in the magenta color. The curves that run perpendicular to the U curves are called the V Curves, and they are shown in the green color. Certain commands may offer the preview mesh where these
Differential Geometry of Curves 1 Mirela Ben‐Chen. Motivation Applications From “Discrete Elastic Rods” by Bergou et al. Good intro to dff ldifferential geometry on surfaces 2 Nice theorems. Parameterized Curves Intuition A particle is moving in space At .File Size: 658KB
Curves Di erential Geometry Introduction. Di erential geometry is a mathematical discipline that uses methods of multivariable calculus and linear algebra to study problems in geometry. In this course, we will study curves and surfaces and, later in the course, their generalizations manifolds.
Figure 10. Centrifugal Pump Performance Curves 37 Figure 11. Family of Pump Performance Curves 38 Figure 12. Performance Curves for Different Impeller Sizes 38 Figure 13. Performance Curves for a 4x1.5-6 Pump Used for Water Service 39 Figure 14. Multiple Pump Operation 44 Figure 15. Multiple-Speed Pump Performance Curves 45 Figure 16.
applications. Smooth degree-3 curves, known as elliptic curves, were used in Andrew Wiles’s proof of Fermat’s Last Theorem [11]. The points on elliptic curves form a group with a nice geometric description. Hendrick Lenstra [5] exploited this group structure to show that elliptic curves can be used to factor large numbers with a relatively .
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