Download presentation

Presentation is loading. Please wait.

Published byWilliam Somers Modified over 2 years ago

1
Gopi -ICS280F02 - Slide 1 Characteristics of an Object

2
Gopi -ICS280F02 - Slide 2 Object Characteristics Topological propertiesTopological properties –Manifolds (w/ boundaries) /Non-manifold –Euler characteristic/Genus/Betti numbers/ –Orientability –Homeomorphism/ Homology groups etc. Geometric propertiesGeometric properties –Curvature, continuity –Surface parameterization –Convexity and concavity –Silhouette visibility

3
Gopi -ICS280F02 - Slide 3 (Layman) Manifold Definitions ManifoldsManifolds –2D: Every edge has exactly two incident triangles. –3D: Every triangle has exactly two incident tetrahedrons. Manifolds with boundariesManifolds with boundaries –2D: Every edge has either one or two incident triangles. –3D: Every triangle has either one or two incident tetrahedrons. Non-manifoldsNon-manifolds –That does not have the above restrictions.

4
Gopi -ICS280F02 - Slide 4 (Expert) Manifold Definitions ManifoldsManifolds –2D: Neighborhood of every point belonging to the object is homeomorphic to an open disc. –3D: Neighborhood of every point belonging to the object is homeomorphic to an open ball. Manifolds with boundariesManifolds with boundaries –2D: …(as above) or a half-disk. –3D:…(as above) or a half-ball Non-manifoldsNon-manifolds –That does not have the above restrictions.

5
Gopi -ICS280F02 - Slide 5 Genus (g) of a manifold Applicable only for manifoldsApplicable only for manifolds (Naïve) Number of “handles”.(Naïve) Number of “handles”. Sphere has g=0; cube has g=0; torus has g=1; coffee cup has g=1.Sphere has g=0; cube has g=0; torus has g=1; coffee cup has g=1.

6
Gopi -ICS280F02 - Slide 6 Euler Characteristic (e) of a manifold e = V-E+F (V: Vertices, E: Edges, F: Faces).e = V-E+F (V: Vertices, E: Edges, F: Faces). Applicable only for manifoldsApplicable only for manifolds In generalIn general –e=(0 dim)-(1 dim)+(2 dim)-(3 dim)+(4 dim)… Relationship between e and g: e=2-2gRelationship between e and g: e=2-2g –Sphere or Cube: e=2-2(0)=2 –Torus: e=2-2(1)=0 Verify: Cube has 8 vertices, 12 edges, 6 facesVerify: Cube has 8 vertices, 12 edges, 6 faces –e = V-E+F = 8-12+6 = 2

7
Gopi -ICS280F02 - Slide 7 Orientability of an object If you have consistent normal direction for a point then the object is orientable. Otherwise, non- orientable.If you have consistent normal direction for a point then the object is orientable. Otherwise, non- orientable. Mobius Strip

8
Gopi -ICS280F02 - Slide 8 In this course.. 2D orientable manifolds with boundaries.2D orientable manifolds with boundaries. Remember: manifolds with boundaries is a superset of manifolds, and non-manifold is a superset of manifolds with boundaries. Non-manifold actually means that “need not” be a manifold; not “is not” a manifold.

Similar presentations

OK

Lecture 1: The Euler characteristic of a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific.

Lecture 1: The Euler characteristic of a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Fiber post ppt online Ppt on catenation of carbon Ppt on lhasa Free ppt on mobile number portability faq Ppt on types of parallelograms quadrilateral Ppt on social contract theory hobbes Ppt on refraction and reflection of light Ppt on environmental protection act 1986 Ppt on travel and tourism industry Ppt on power grid control through pc