Test e.ccw³ == e

We have shown that test e.ccw³ == 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.ccw³ depending upon whether or not there is an incoming edge f to vertex v -:

• (a) e.ccw³ has label i + 1
• (b) e.ccw³ has label i + 2

Therefore, e.ccw³ can never be e which has label i. The test always returns false for mouth triangle type.

Wedgepar 1

Let edge e have Schnyder label i. As shown in the picture below, there are four cases for e.ccw³ depending upon whether or not there are incoming edges between certain outgoing edges including -:

• (a) e.ccw³ has label i + 1
• (b) e.ccw³ has label i + 2
• (c) e.ccw³ has label i + 2

There is a fourth case where e.ccw³ 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.ccw², e.ccw³ will have label i. This edge e.ccw³ 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.ccw³ can never be e. The test always returns false for wedgepar triangle type.

Wedgerev 1

Let edge e have Schnyder label i. As shown in the picture below, there are four cases for e.ccw³ depending upon whether or not there are incoming edges between certain outgoing edges including -:

• (a) e.ccw³ has label i + 1
• (b) e.ccw³ has label i + 2
• (c) e.ccw³ has label i + 2

There is a fourth case where e.ccw³ 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.ccw², e.ccw³ will have label i. This edge e.ccw³ 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.ccw³ can never be e. The test always returns false for wedgepar triangle type.

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.

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.

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.

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.

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.

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.

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.

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.

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.

-- SisillaSookdeo - 01 Aug 2005
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