How to use Geometric Software in Courses of Differential Geometry
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1 How to use Geometric Software in Courses of Differential Geometry TOMICZKOVÁ Světlana, JEŽEK František KMA FAV ZČU Plzeň 2018 Coimbra 2018 How to use Geometric Software in Courses of Differential Geometry / 26
2 Content Course of Differential Geometry. How we use software Practical lessons of differential geometry How to use Geometric Software in Courses of Differential Geometry / 26
3 Course of Differential Geometry Curve equations, parameterization. Curvature, torsion, Frenet equations. Surface equations, parameterization. The first fundamental form of surfaces: the length of the curve on the surface, projection and developing surfaces, conformal mapping. The second fundamental form of surfaces: normal surface curvature, Meusnier proposition, Dupin indikatrix. Mean and Gaussian curvature. Developable and minimal surfaces. The second fundamental form of surfaces: normal surface curvature, Geodetic curvature, geodesic. How to use Geometric Software in Courses of Differential Geometry / 26
4 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros ow to use Geometric Software in Courses of Differential Geometry / 26
5 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
6 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
7 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
8 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
9 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
10 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
11 How to use software in courses Vizualizations of curves Vizualizations of surfaces Theoretical lectures - ilustration of definitions Theoretical lectures - theorems Ilustration of exercise or examples Ilustration of properties Practical lesson with Rhinoceros ow to use Geometric Software in Courses of Differential Geometry / 26
12 Geometric software Geogebra How to use Geometric Software in Courses of Differential Geometry / 26
13 Geometric software Wolfram Mathematica How to use Geometric Software in Courses of Differential Geometry / 26
14 Geometric software Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
15 Why Rhinoceros How to use Geometric Software in Courses of Differential Geometry / 26
16 Curvature Graph How to use Geometric Software in Courses of Differential Geometry / 26
17 Helix Example 1: Compute the curvature for the helix. Plot the helix and display the curvature graph. P(s) = (r cos (, ) Ṗ(s) = ( r 1 r2 +c 2 sin s r 2 +c 2, r sin 1 P(s) = ( r r 2 +c cos 2 1 k = P = s r 2 +c 2, s, r 1 cos r2 +c 2 r2 +c 2 (r 1 r 2 +c 2 ) 2 = cs r 2 +c 2 ), s s r2 +c 2, s, r 1 r 2 +c 2 r 2 +c sin 2 r r 2 +c 2 c r2 +c 2 ) s r 2 +c 2, 0) ow to use Geometric Software in Courses of Differential Geometry / 26
18 Helix Example 1: Compute the curvature for the helix. Plot the helix and display the curvature graph. ow to use Geometric Software in Courses of Differential Geometry / 26
19 Helix Example 1: Compute the curvature for the helix. Plot the helix and display the curvature graph. ow to use Geometric Software in Courses of Differential Geometry / 26
20 Curvature Analysis false-color mapping How to use Geometric Software in Courses of Differential Geometry / 26
21 Developable surfaces Ruled surfaces which are characterized by property that they can be mapped isometrically into the plane. Three basic types of developable surfaces: cylinders cones tangent surfaces of space curves How to use Geometric Software in Courses of Differential Geometry / 26
22 Gaussian curvature Example 2: Find the Gauss curvature for the cylinder. Find a patch for some cylinders compare your calculation and visualisation of Gauss curvature in Rhinoceros. X(s, t) = P (s) + tu, s, t R g 11 = P P g 12 = P u g 22 = u u h 11 = P (P 1 u) P u h 12 = 0 h 22 = 0 K = h 11h 22 (h 12 ) 2 g 11 g 22 (g 12 ) 2 = 0 How to use Geometric Software in Courses of Differential Geometry / 26
23 Gauss curvature How to use Geometric Software in Courses of Differential Geometry / 26
![Exercise Find the the Gauss and mean curvature for the oblique circular cylinder (the circle in the plane xy, centre S[0, 0, 0], radius r = 2, direction of straight lines a = (0, 1, 1) ) in the point](/docs-images/87/96526709/images/24-0.jpg "U[0, 2, 0].")
24 Exercise Find the the Gauss and mean curvature for the oblique circular cylinder (the circle in the plane xy, centre S[0, 0, 0], radius r = 2, direction of straight lines a = (0, 1, 1) ) in the point U[0, 2, 0]. X(u, v) = (2 cos v, u + 2 sin v, u) point U for v = 3π 2, u = 0 g 11 = 2 g 12 = 2 cos v g 22 = 4 h 11 = 0 h 12 = 0 2 h 22 = 1+sin 2 v dx du dx dv = (0, 1, 1) = ( 2 sin v, 2 cos v, 0) d 2 X du = (0, 0, 0) 2 d 2 X dv = ( 2 cos v, 2 sin v, 0) 2 d 2 X dudv = (0, 0, 0) 1 n = ( 2 cos v, 2 sin v, 2 sin v) 2 1+sin 2 v H = 1 2 g11 h22 2g12 h12+g22 h11 g 11g 22 (g 12) = 2 1 = 2 1+sin, for v = 3π 2 v(2 cos 2 v) 2 H = 2 8 K = h11h22 (h12)2 g 11g 22 (g 12) 2 = 0 How to use Geometric Software in Courses of Differential Geometry / 26
25 Exercise Find the the Gauss and mean curvature for the oblique circular cylinder (the circle in the plane xy, centre S[0, 0, 0], radius r = 2, direction of straight lines a = (0, 1, 1) ) in the point U[0, 2, 0]. Ussing Meusnier theorem: H = 1 2 (n k min + n k max ) = = 1 2 ( ) = 2 8 K = n k min n k max = = 0 How to use Geometric Software in Courses of Differential Geometry / 26
26 Gauss curvature How to use Geometric Software in Courses of Differential Geometry / 26
27 Gauss curvature How to use Geometric Software in Courses of Differential Geometry / 26
28 Gauss curvature How to use Geometric Software in Courses of Differential Geometry / 26
29 Minimal surfaces Catenoid Helicoid Eneper surface Catalan surface ow to use Geometric Software in Courses of Differential Geometry / 26
30 Mean curvature How to use Geometric Software in Courses of Differential Geometry / 26
31 Mean curvature How to use Geometric Software in Courses of Differential Geometry / 26
32 Mean curvature How to use Geometric Software in Courses of Differential Geometry / 26
33 Mean curvature How to use Geometric Software in Courses of Differential Geometry / 26
34 Conclusion Terms, definition, theorems, properties Visualization Discussion problems of visualization of properties Connection between different parts of geometry Motivation for course Geometric and Computer modelling How to use Geometric Software in Courses of Differential Geometry / 26
35 Thank you for your attention How to use Geometric Software in Courses of Differential Geometry / 26
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