Lecture 06

CSCI 461: Computer Graphics

Middlebury College, Fall 2023

Lecture 06: Meshes

By the end of today's lecture, you will be able to:

represent meshes using two flattened (1d) arrays for vertices and elements,
access vertices and elements in a triangle mesh,
extract the edges of a triangle mesh,
extract vertex-to-vertex adjacency information in a triangle mesh,
subdivide a mesh using the Loop subdivision scheme.

Previously, models were represented as a "soup" of triangles.

What if we want to modify our models?

We need a concept of "connectedness."

What about memory consumption?

Assume floating-point values (each component of a vec3) are stored as Float32 (4 bytes). How much memory is used if every triangle directly stores 3 vec3s?

Meshes have two ingredients: geometry & topology.

Extracting the edges of a triangle mesh.

Getting the "opposite" vertex of an edge.

Subdividing a triangle mesh quadruples # triangles.

What are the indices of the four new triangles resulting from the subdivision?

Exercise: subdividing a mesh to approximate a sphere.

We'll get started with keeping track of new edge vertex indices.
First assign edge midpoint as coordinates for new edge vertex.
Then normalize to place this point on the unit sphere.

Thursday's Lab: Loop Subdivision Surfaces.

$k$: # vertices adjacent to vertex, $\rightarrow$ we need vertex-to-vertex information!
$\beta$: (k === 3) ? 0.1875 : 0.375 / k

Creating the vertex-to-vertex information.

// initialize an array for each vertex to place the v2v information let v2v = new Array(nVertices); for (let i = 0; i < nVertices; i++) v2v[i] = []; for (let i = 0; i < nTriangles; i++) { for (let j = 0; j < 3; j++) { let p = this.triangles[3 * i + j]; let q = this.triangles[3 * i + ((j + 1) % 3)]; // add q as a neighbor to p // (p will be added as a neighbor to q when looping through the adjacent triangle) v2v[p].push(q); } }