Differential Geometry: Revisiting Curvatures
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1 Differenial Geomery: Reisiing Curaures
2 Curaure and Graphs Recall: hus, up o a roaion in he x-y plane, we hae: f 1 ( x, y) x y he alues 1 and are he principal curaures a p and he corresponding direcions of he cures a he poin p are he principal direcions.
3 Curaure and Graphs Recall: he produc of he principal curaures, 1, is he Gaussian Curaure. he sum of he principal curaures, 1 +, is he Mean Curaure.
4 Smooh Curaure (Cures) On a circle of radius r, an arc of uni-lengh will hae angle 1/r. 1/r r 1
5 Smooh Curaure (Cures) On a circle of radius r, an arc of uni-lengh will hae angle 1/r. Offseing he circle by a disance of in he normal direcion, we ge a circle wih radius (r+), and he lengh of he corresponding arc becomes 1+/r. 1/r r 1 1+/r
6 Smooh Curaure (Cures) We measure he curaure as he rae of change in lengh as a funcion of offse disance : 1 r 1/r r 1 1+/r
7 Smooh Curaure (Cures) In a similar fashion, we can define he curaure a a poin p on an arbirary cure by considering he rae of change in arc-lengh as we offse in he normal direcion by a disance of. p
8 Smooh Curaure (Cures) In a similar fashion, we can define he curaure a a poin p on an arbirary cure by considering he rae of change in arc-lengh as we offse in he normal direcion by a disance of. p
9 Discree Curaure (Cures) Gien a closed cure, consider he cure obained by offseing by in he normal direcion.
10 Discree Curaure (Cures) Gien a closed cure, consider he cure obained by offseing by in he normal direcion. he lengh of he offse cure is he lengh of he old cure l N 1 l i i0 l 1 l 0
11 Discree Curaure (Cures) Gien a closed cure, consider he cure obained by offseing by in he normal direcion. he lengh of he offse cure is he lengh of he old cure l N 1 l i i0 plus he lenghs of he arcs. i 1 l 1 l 0 0
12 Discree Curaure (Cures) Gien a closed cure, consider he cure obained by offseing by in he normal direcion. he lengh of he offse cure is he lengh of he old cure l N 1 l i i0 plus he lenghs of he arcs. hus, he rae of change in lengh hrough he erex i is i. i 1 l 1 l 0 0
13 Discree Curaure (Cures) Bu he angle of he arc is exacly he defici angle, so we ge he same definiion as before l 1 l 0 0
14 Discree Curaure (Cures) Bu he angle of he arc is exacly he defici angle, so we ge he same definiion as before. l l 0 0 0
15 Smooh Curaure (Surfaces) In a similar fashion, we can consider wha happens o he area of a surface as we offse i in he normal direcion by a disance of.
16 Smooh Curaure (Surfaces) In a similar fashion, we can consider wha happens o he area of a surface as we offse i in he normal direcion by a disance of. In his case we consider boh he rae of change and acceleraion in area, and we ge: A ( p) A ( p) H( p) K( p) where H is he mean curaure and K is he Gaussian curaure.
17 Smooh Curaure (Surfaces) In a similar fashion, we can consider wha happens o he area of a surface as we offse i in he normal direcion by a disance of.
18 Discree Curaure (Surfaces) Wha happens when we offse poins on a discree surface?
19 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) Wha happens when we offse poins on a discree surface? We can decompose he offse surface ino hree pars. = + +
20 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) he area of he offse surface is he sum of: he area of he original surface = + +
21 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) he area of he offse surface is he sum of: he area of he original surface he area of he cylindrical arcs defined by he edges = + +
22 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) he area of he offse surface is he sum of: he area of he original surface he area of he cylindrical arcs defined by he edges he area of he spherical caps defined by he erices = + +
23 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) he area of he offse surface is he sum of: A ris. A( ) eedges. e where: e is he angle a edge e is he solid angle a erex e ers. = + +
24 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) he area of he offse surface is he sum of: A ris. A( ) So he offse surface has e e as he 1 s -order erm of he area, and as he nd -order erm. eedges. e e ers. = + +
25 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) he area of he offse surface is he sum of: A ris. A( ) eedges. e e ers. We associae he discree mean curaure e e wih he edges of he polygon and discree Gaussian curaure wih he erices. = + +
26 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) If riangles 1 and mee a edge e, he angle e is defined as: cos e N N 1, N 1 e 1 N = + +
27 Face Offse Edge Offse erex Offse Discree Curaure (Surfaces) If riangles 1,, k mee a erex, he solid angle is he area of he spherical wedge going hrough N 1,,N k. N 1 N N 3 N 3 N N 1 = + +
28 Discree Curaure (Surfaces) If riangles 1,, k mee a erex, he solid angle is he area of he spherical wedge going hrough N 1,,N k. N 1 N N 3 N 3 N N 1 On a sphere, he area of a polygon wih angles 1,, k is: k A ( k) i1 i
29 Discree Curaure (Surfaces) Claim: he angle i a he inersecion of arcs N i-1 N i and N i N i+1 is minus he angle beween e i-1 and e i. N 3 N 1 - N 3 N e 1 e N N 1
30 Discree Curaure (Surfaces) Claim: he angle i a he inersecion of arcs N i-1 N i and N i N i+1 is minus he angle beween e i-1 and e i. Implicaions: If i is he angle (a ) beween e i-1 and e i, he Gaussian curaure is he angle of defici a : A ( k) k i1 i k i1 i N 1 - N 3 N e 1 e N N 3 N 1
31 Discree Curaure (Surfaces) Wha is he angles i? A (geodesic) arc beween poins p and q on he sphere is conained in he inersecion of he sphere wih he plane perpendicular o p and q. N 3 N 1 N 3 N N N 1
32 Discree Curaure (Surfaces) Wha is he angles i? A (geodesic) arc beween poins p and q on he sphere is conained in he inersecion of he sphere wih he plane perpendicular o p and q. he angle beween wo arcs is minus he angle beween he planes normals. N 3 N 1 N N 3 N N 1
33 Discree Curaure (Surfaces) Wha is he angles i? A (geodesic) arc beween poins p and q on he sphere is conained in he inersecion of he sphere wih he plane perpendicular o p and q. he angle beween wo arcs is minus he angle beween he planes normals. Bu he edge e i beween riangle i-1 and N i is perpendicular o boh he normals. N 1 N N N 3 N 3 N 1
34 Gauss-Bonne heorem (Smooh) Gien a (closed surface) S, he inegral of he Gaussian curaure oer he surface is: S K( p) dp where S is he Euler Characerisic of he surface S (an ineger ha is a opological inarian of he surface). S
35 Gauss-Bonne heorem (Smooh) Wha happens in he discree case?
36 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? Summing he Gaussian curaures we ge: where is a riangle conaining and is he inerior angle of a. K 0
37 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? Summing he Gaussian curaures we ge: E E K 0
38 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? Summing he Gaussian curaures we ge: E E K 0
39 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? Summing he Gaussian curaures we ge: E E K 0
40 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? Summing he Gaussian curaures we ge: E E K 0
41 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? Summing he Gaussian curaures we ge: E E K 0
42 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? In he discree case, he sum of he Gaussian curaure is equal o: K E
43 Gauss-Bonne heorem (Smooh) Wha happens in he discree case? In he discree case, he sum of he Gaussian curaure is equal o: Noe ha for a closed polyhedron: K is he Euler Characerisic, and saisfies: where g is he genus of he surface. E E g
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