The paper discusses the Triangle Collision Loss as a mechanism to avoid having triangles that penetrate each other in the simplified mesh. This loss is crucial to maintaining the geometric integrity of the mesh by ensuring that triangles do not intersect inappropriately.
Here’s a summary of what the paper says about Triangle Collision Loss:
Purpose: The Triangle Collision Loss directly penalizes triangles that collide with one another, meaning it penalizes situations where the surface of one triangle is penetrated by the edges of nearby triangles. This is essential to prevent overlapping or intersecting triangles that could result in a distorted or non-manifold mesh.
Method: The collision of a triangle is measured in terms of the number of line segments (edges of nearby triangles) that penetrate its surface. For each triangle, a plane is computed, and the loss measures how many other triangles intersect with this plane.
Penalty: The penalty applied to each irregular triangle is proportional to the number of planes it penetrates. The more planes a triangle penetrates, the higher the loss it incurs.
$p_t$ denotes the probability of the triangle $t$.
$m_c(t)$ is the number of faces penetrated by triangle $t$.
$\mathcal{F}_s$ represents the set of generated triangles.
This loss is a key component in ensuring that the resulting mesh remains valid and visually faithful to the original, by preventing the formation of triangles that would otherwise intersect in an unnatural or incorrect manner.
The paper discusses the Triangle Collision Loss as a mechanism to avoid having triangles that penetrate each other in the simplified mesh. This loss is crucial to maintaining the geometric integrity of the mesh by ensuring that triangles do not intersect inappropriately.
Here’s a summary of what the paper says about Triangle Collision Loss:
Purpose: The Triangle Collision Loss directly penalizes triangles that collide with one another, meaning it penalizes situations where the surface of one triangle is penetrated by the edges of nearby triangles. This is essential to prevent overlapping or intersecting triangles that could result in a distorted or non-manifold mesh.
Method: The collision of a triangle is measured in terms of the number of line segments (edges of nearby triangles) that penetrate its surface. For each triangle, a plane is computed, and the loss measures how many other triangles intersect with this plane.
Penalty: The penalty applied to each irregular triangle is proportional to the number of planes it penetrates. The more planes a triangle penetrates, the higher the loss it incurs.
Mathematical Formulation: The loss is defined as:
where:
p_t$ denotes the probability of the triangle $t$.m_c(t)$ is the number of faces penetrated by triangle $t$.\mathcal{F}_s$ represents the set of generated triangles.This loss is a key component in ensuring that the resulting mesh remains valid and visually faithful to the original, by preventing the formation of triangles that would otherwise intersect in an unnatural or incorrect manner.