TD 5 - Subdivision surfaces

(Luca Castelli Aleardi)

The goal of this TD session is to implement subdivision schemes for surface meshes

 The original mesh (genus 1 torus), and the mesh after one subdivision (Loop scheme) The mesh after some subdivision rounds

Example of surface subdivision using the Loop scheme.

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1. Getting started (a Processing frame for 2D and 3D rendering)

• meshes.zip: examples of triangle meshes (OFF format).

• Jcg.jar: library for the manipulation of triangle meshes (here are the sources)
• TC.jar: input/output from files.

For the simplification and rendering of meshes:

TD5 documentation: instructions

Important: here are a few slides illustrating how to implement subdivision schemes for TD5.

class MeshViewer

Class MeshViewer provides simple methods for drawing a surface mesh in a 3D frame; it also provides a simple trackball (allowing to rotate the 3D scene with mouse events).
You can choose three distinct rendering modes (as illustrated below): press 'r' key (on the keyboard) in order to change rendering mode.
Method setup() allows to select the preferred subdivision scheme: press 's' key (on the keyboard) in order to run the subdivision procedure.
`Three rendering modes are available`

1. Loop subdivision

Goal

Given a triangle mesh (class Polyhedron_3<Point_3>), we want to subdivide it, by computing the Loop subdivision (see slides for TD5).
• complete method subdivide() of class LoopSubdivision (which extends class  MeshSubdivision).

Suggestion: in order to implement the Loop subdivision scheme, we suggest to you to follow the steps below (feel you free in choosing and implementing a different solution)

1.1 compute edge point coordinates

Add/complete method computeEdgePoints() to class LoopSubdivision, which computes the list of edge points (just geometric locations).

1.2 split edges by inserting edge points

Now we want to insert edge vertices: just use splitEdge(Halfedge<X> h) operator (already implemented in Jcg).
• add method splitEdges(...) to class LoopSubdivision which perform the splitting of all original edges.
• complete method subdivide() which calls previous method.

Splitting edges

1.3 subdivide faces

In order to subdivide each face (of degree 3+3) into 4 triangle faces, you can perform three splitFace() operations (see slides for TD5).

• add method subdivideFace(f), which perform the subdivision of a face.
• complete method subdivide() in order to subdivide all faces (recall that only original faces must to be subdivided).

Subdividing faces

1.4 compute new locations for original vertices

To conclude, it only remains to compute the final new locations of vertices of the original mesh.
• add method computeNewVertexLocation(Vertex<Point_3> v) in order to compute the new coordinates of original vertices.

One round subdivision

2. Catmull-Clark subdivision

2.0 A subdivision scheme for polygonal meshes

Given a triangle mesh (class Polyhedron_3<Point_3>), we want to subdivide it, by computing the Catmull-Clark subdivision (see slides for TD5).
• write a new class CatmullClarkSubdivision (which extends class  MeshSubdivision) and complete its method subdivide().

Suggestion: you can follow the same steps as above, taking a look to the slides for TD5 (feel you free in choosing and implementing a different solution). For testing, uncomment the corresponding line in the class MeshViewer (in the setup function).