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TModeling.TriangleTests
r1.1 - 01 Sep 2005 - 19:13 - Main.guest
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%TOC% ---++ Test e.ccw<sup>3</sup> == e We have shown that test e.ccw<sup>3</sup> == e returns true for triangle type cycle left of an edge e. It is important, however, to show that this test returns false for the other three triangle types. ---+++ Mouth 1 Let edge _e_ have Schnyder label _i_. As shown in the picture below, there are two cases for e.ccw<sup>3</sup> depending upon whether or not there is an incoming edge _f_ to vertex _v_ -: * (a) e.ccw<sup>3</sup> has label _i_ + 1 * (b) e.ccw<sup>3</sup> has label _i_ + 2 Therefore, e.ccw<sup>3</sup> can never be _e_ which has label _i_. The test always returns false for mouth triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-4.40.22PM.jpg" alt="e.ccw^3 for mouth" /> ---+++ Wedgepar 1 Let edge _e_ have Schnyder label _i_. As shown in the picture below, there are four cases for e.ccw<sup>3</sup> depending upon whether or not there are incoming edges between certain outgoing edges including -: * (a) e.ccw<sup>3</sup> has label _i_ + 1 * (b) e.ccw<sup>3</sup> has label _i_ + 2 * (c) e.ccw<sup>3</sup> has label _i_ + 2 There is a fourth case where e.ccw<sup>3</sup> could have label _i_. In the picture below, edges _j_ and _k_ are outgoing from the same vertex _x_. If there are no incoming edges between edges _k_ and _j_ and _k_ is e.ccw<sup>2</sup>, e.ccw<sup>3</sup> will have label _i_. This edge e.ccw<sup>3</sup> can be edge _e_ iff there exists a triangle T so that edge _j_ is incoming to vertex _w_. This triangle cannot exist for the following reasons-: * Vertex _v_ must have an outgoing edge between edges _g_ and _h_ * Incoming edges between edges f and g should have label _i_ Therefore, e.ccw<sup>3</sup> can never be _e_. The test always returns false for wedgepar triangle type. <img src="%ATTACHURLPATH%/Capture8-4-2005-6.10.09PM.jpg" alt="e.ccw^3 for wedgepar" /> ---+++ Wedgerev 1 Let edge _e_ have Schnyder label _i_. As shown in the picture below, there are four cases for e.ccw<sup>3</sup> depending upon whether or not there are incoming edges between certain outgoing edges including -: * (a) e.ccw<sup>3</sup> has label _i_ + 1 * (b) e.ccw<sup>3</sup> has label _i_ + 2 * (c) e.ccw<sup>3</sup> has label _i_ + 2 There is a fourth case where e.ccw<sup>3</sup> could have label _i_. In the picture below, edges _j_ and _k_ are outgoing from the same vertex _x_. If there are no incoming edges between edges _k_ and _j_ and _k_ is e.ccw<sup>2</sup>, e.ccw<sup>3</sup> will have label _i_. This edge e.ccw<sup>3</sup> can be edge _e_ iff there exists a triangle T so that edge _j_ is incoming to vertex _w_. This triangle cannot exist for the following reasons-: * Vertex _v_ must have an outgoing edge between edges _g_ and _h_ * Incoming edges between edges f and g should have label _i_ Therefore, e.ccw<sup>3</sup> can never be _e_. The test always returns false for wedgepar triangle type. <img src="%ATTACHURLPATH%/Capture8-4-2005-6.09.17PM.jpg" alt="e.ccw^3 for wedgerev" /> ---++ Test e.ccw.inc.cw == e We have shown that test e.ccw.inc.cw == e returns true for triangle type mouth left of an edge e. We show below that this test returns false for the other three triangle types. ---+++ Cycle 1 Let edge _e_ have Schnyder label _i_. e.ccw.inc.cw has label _i_ + 1 and is an edge on the triangle as shown in the picture below. Therefore, edge e.ccw.inc.cw cannot be e. The test always returns false for cycle triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.38.31PM.jpg" alt="e.ccw.inc.cw for cycle" /> ---+++ Wedgepar 2 Let edge _e_ have Schnyder label _i_. As shown in the picture below, there are two cases for e.ccw.inc.cw depending upon whether or not there is an incoming edge _f_ to vertex _v_ including -: * (a) e.ccw.inc.cw has label _i_ + 2 * (b) e.ccw.inc.cw has label _i_ + 1 Therefore, e.ccw.inc.cw can never be _e_. The test always returns false for wedgepar triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.40.43PM.jpg" alt="e.ccw.inc.cw for wedgepar" /> ---+++ Wedgerev 2 Let edge _e_ have Schnyder label _i_. As shown in the picture below, there are two cases for e.ccw.inc.cw depending upon whether or not there is an incoming edge _f_ to vertex _v_ -: * (a) e.ccw.inc.cw has label _i_ + 2 * (b) e.ccw.inc.cw has label _i_ + 1 Therefore, e.ccw.inc.cw can never be _e_. The test always returns false for wedgerev triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.41.53PM.jpg" alt="e.ccw.inc.cw for wedgerev" /> ---++ Test e.ccw.cw.dec == e We have shown that test e.ccw.cw.dec == e returns true for triangle type wedgepar left of an edge e. We show below that this test returns false for the other three triangle types. ---+++ Cycle 2 Let edge _e_ have Schnyder label _i_. As shown in the picture below, there are two cases for e.ccw.inc.cw depending upon whether or not there is an incoming edge _f_ to vertex v -: * (a) e.ccw.inc.cw has label _i_ + 2 * (b) e.ccw.inc.cw has label _i_ + 1 Therefore, e.ccw.cw.dec can never be _e_. The test always returns false for cycle triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.44.30PM.jpg" alt="e.ccw.cw.dec for cycle" /> ---+++ Mouth 2 Let edge _e_ have Schnyder label _i_. e.ccw.inc.cw has label _i_ + 1 as shown in the picture below. Therefore, e.ccw.inc.cw can never be _e_. The test always returns false for mouth triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.46.14PM.jpg" alt="e.ccw.cw.dec for mouth" /> ---+++ Wedgerev 3 Let edge _e_ have Schnyder label _i_ and be outgoing from vertex _v_ and incoming to vertex _w_. e.ccw.cw.dec also has label _i_. e.ccw.cw.dec is outgoing from w while _e_ is outgoing from v. Therefore, e.ccw.cw.dec can never be _e_. The test always returns false for wedgerev triangle type. <img src="%ATTACHURLPATH%/Capture8-4-2005-6.07.36PM.jpg" alt="e.ccw.cw.dec for wedgerev" /> ---++ Test e.inc.cw.ccw == e We have shown that test e.ccw.cw.dec == e returns true for triangle type wedgerev left of an edge e. We show below that this test returns false for the other three triangle types. ---+++ Cycle 3 Let edge _e_ have Schnyder label _i_. As shown in the picture below, e.inc.cw.ccw has label _i_ + 1 and is an edge on the triangle. Therefore, e.inc.cw.ccw can never be _e_. The test always returns false for cycle triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.49.54PM.jpg" alt="e.inc.cw.ccw for cycle" /> ---+++ Mouth 3 Let edge _e_ have Schnyder label _i_. As shown in the picture below, e.inc.cw.ccw has label _i_ + 2. Therefore, e.inc.cw.ccw can never be _e_. The test always returns false for mouth triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.50.46PM.jpg" alt="e.inc.cw.ccw for mouth" /> ---+++ Wedgepar 3 Let edge _e_ have Schnyder label _i_. e.inc.cw.ccw has label _i_ but is an edge _f_ on the triangle. Since the edges on the triangle must be unique, _f_ or e.inc.cw.ccw can never be _e_. The test always returns false for cycle triangle type. <img src="%ATTACHURLPATH%/Capture8-1-2005-7.51.32PM.jpg" alt="e.inc.cw.ccw for mouth" /> -- Main.SisillaSookdeo - 01 Aug 2005
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