By Elsa Abbena, Alfred Gray, Simon Salamon
Featuring idea whereas utilizing Mathematica in a complementary manner, Modern Differential Geometry of Curves and Surfaces with Mathematica, the 3rd version of Alfred Gray’s well-known textbook, covers how to find and compute common geometric features utilizing Mathematica for developing new curves and surfaces from present ones. in view that Gray’s loss of life, authors Abbena and Salamon have stepped in to convey the booklet brand new. whereas retaining Gray's intuitive method, they reorganized the fabric to supply a clearer department among the textual content and the Mathematica code and extra a Mathematica workstation as an appendix to every bankruptcy. in addition they deal with very important new subject matters, reminiscent of quaternions.
The process of this publication is from time to time extra computational than is common for a e-book at the topic. for instance, Brioshi’s formulation for the Gaussian curvature when it comes to the 1st basic shape should be too complex to be used in hand calculations, yet Mathematica handles it simply, both via computations or via graphing curvature. one other a part of Mathematica that may be used successfully in differential geometry is its targeted functionality library, the place nonstandard areas of continuous curvature will be outlined by way of elliptic services after which plotted.
Using the strategies defined during this publication, readers will comprehend options geometrically, plotting curves and surfaces on a display screen after which printing them. Containing greater than three hundred illustrations, the ebook demonstrates the right way to use Mathematica to devise many attention-grabbing curves and surfaces. together with as many issues of the classical differential geometry and surfaces as attainable, it highlights vital theorems with many examples. It comprises three hundred miniprograms for computing and plotting a variety of geometric items, assuaging the drudgery of computing issues similar to the curvature and torsion of a curve in area.
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Extra resources for Modern Differential Geometry of Curves and Surfaces with Mathematica
16) Thus κ2[β] = = (α ◦ h)h 2 + (α ◦ h)h (α ◦ h)h h3 |h |3 3 · J(α ◦ h) h (α ◦ h) · J (α ◦ h) (α ◦ h) 3 . 15). 21. Let β be a unit-speed curve in the plane. 17) β = κ2[β]J β . Proof. Differentiating β · β = 1, we obtain β · β = 0. Thus β must be a multiple of J β . 17) . Finally, we give simple characterizations of straight lines and circles by means of curvature. 22. Let α: (a, b) → R2 be a regular curve. (i) α is part of a straight line if and only if κ2[α](t) ≡ 0. (ii) α is part of a circle of radius r > 0 if and only if κ2[α] (t) ≡ 1/r.
46 CHAPTER 2. 5), we consider a portion pq of the cable between the lowest point p and an arbitrary point q. Three forces act on the cable: the weight of the portion pq, as well as the tensions T and U at p and q. If w is the linear density and s is the length of pq, then the weight of the portion pq is ws. 11. 6) |T| = |U| cos θ and w s = |U| sin θ. Let q = (x, y), where x and y are functions of s. 6) we obtain dy ws = tan θ = . 8) ds = dx 1+ dy dx 2 . 5) with a = ω/|T |. 5. CISSOID OF DIOCLES 47 Although at first glance the catenary looks like a parabola, it is in fact the graph of the hyperbolic cosine.
Ellipses These are perhaps the next simplest curves after the circle. The name ‘ellipse’ (which means ‘falling short’) is due to Apollonius4 . 26) ellipse[a, b](t) = a cos t, b sin t , 0 t < 2π. 8: An ellipse and its foci The curvature of this ellipse is κ2[ellipse[a, b]](t) = (b2 cos2 ab . t + a2 sin2 t)3/2 4 Apollonius of Perga (262–180 BC). His eight volume treatise on conic sections is the standard ancient source of information about ellipses, hyperbolas and parabolas. 22 CHAPTER 1. 8) is illustrated below with an exaggerated vertical scale.