Initial value problem of discrete geodesics and its application
The commonly used shortest geodesic paths neither simulate properties of geodesics on smooth surface nor provide a unique solution on triangle meshes. We focus on the initial value problem, i.e., finding a uniquely determined geodesic path from a given point in any direction. Firstly, we propose...
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
2016
|
| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/67025 |
| _version_ | 1826114335588483072 |
|---|---|
| author | Cheng, Peng |
| author2 | Nadia Magnenat-Thalmann |
| author_facet | Nadia Magnenat-Thalmann Cheng, Peng |
| author_sort | Cheng, Peng |
| collection | NTU |
| description | The commonly used shortest geodesic paths neither simulate properties of geodesics on smooth surface nor provide a unique solution on triangle meshes. We focus on the initial value problem, i.e., finding a uniquely determined geodesic path from a given point in any direction.
Firstly, we propose a first-order tangent ODE method. Our method is different from the conventional methods of directly solving the geodesic equation (i.e., a second-order ODE of the position) on piecewise smooth surfaces, which is difficult to implement due to complicated representation of the geodesic equation involving Christoffel symbols. Our method is particularly useful for computing geodesic paths on low-resolution meshes which may have large and/or skinny triangles.
Moreover, we employ the initial value problem geodesic to solve the constrained texture mapping problem. The proposed method provides a valid one-to-one mapping, which not only satisfies user-defined constraints but also preserves the metric structure of the original mesh. |
| first_indexed | 2024-10-01T03:37:49Z |
| format | Thesis |
| id | ntu-10356/67025 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:37:49Z |
| publishDate | 2016 |
| record_format | dspace |
| spelling | ntu-10356/670252023-03-04T00:51:16Z Initial value problem of discrete geodesics and its application Cheng, Peng Nadia Magnenat-Thalmann Miao Chun Yan School of Computer Engineering DRNTU::Engineering::Computer science and engineering The commonly used shortest geodesic paths neither simulate properties of geodesics on smooth surface nor provide a unique solution on triangle meshes. We focus on the initial value problem, i.e., finding a uniquely determined geodesic path from a given point in any direction. Firstly, we propose a first-order tangent ODE method. Our method is different from the conventional methods of directly solving the geodesic equation (i.e., a second-order ODE of the position) on piecewise smooth surfaces, which is difficult to implement due to complicated representation of the geodesic equation involving Christoffel symbols. Our method is particularly useful for computing geodesic paths on low-resolution meshes which may have large and/or skinny triangles. Moreover, we employ the initial value problem geodesic to solve the constrained texture mapping problem. The proposed method provides a valid one-to-one mapping, which not only satisfies user-defined constraints but also preserves the metric structure of the original mesh. DOCTOR OF PHILOSOPHY (SCE) 2016-05-11T01:19:27Z 2016-05-11T01:19:27Z 2016 Thesis Cheng, P. (2016). Initial value problem of discrete geodesics and its application. Doctoral thesis, Nanyang Technological University, Singapore. https://hdl.handle.net/10356/67025 10.32657/10356/67025 en 110 p. application/pdf |
| spellingShingle | DRNTU::Engineering::Computer science and engineering Cheng, Peng Initial value problem of discrete geodesics and its application |
| title | Initial value problem of discrete geodesics and its application |
| title_full | Initial value problem of discrete geodesics and its application |
| title_fullStr | Initial value problem of discrete geodesics and its application |
| title_full_unstemmed | Initial value problem of discrete geodesics and its application |
| title_short | Initial value problem of discrete geodesics and its application |
| title_sort | initial value problem of discrete geodesics and its application |
| topic | DRNTU::Engineering::Computer science and engineering |
| url | https://hdl.handle.net/10356/67025 |
| work_keys_str_mv | AT chengpeng initialvalueproblemofdiscretegeodesicsanditsapplication |