assignment discription

Divide and conquer the problem into four phases as suggested:
1. Phase 1—tangent planes
Given an unorganized set of points in 3-space (point cloud), compute an estimate of a tangent plane at each point.

2. Phase 2—consistent normals
Given a set of tangent planes at each input point, Tp(xi), each consisting of:

* a neighbourhood of points defining the tangent plane's neighborhood, Nbhd(xi),
* the tangent plane origin, oi, and
* the tangent plane's orientation represented by its normal, ni,

find a consistent global orientation for each connected component of the input data.

3. Phase 3—signed distance function
Given a set of consistently oriented tangent planes at each input point, Tp(xi), each consisting of:

* a neighbourhood of points defining the tangent plane's neighborhood, Nbhd(xi),
* the tangent plane origin, oi, and
* the tangent plane's orientation represented by its normal, ni,

calculate the signed distance function f(p) at each point p situated at the vertex of a cube in a 3D cube lattice, where f(p) is defined as the signed distance between point p and its projection z onto Tp(xi): f(p) = (p - oi) ⋅ ni.

4. Phase 4—contour tracing (marching cubes)

Given the signed distance function f(p) at each point p situated at the vertex of a cube in a 3D cube lattice, where f(p) is defined as the signed distance between point p and its projection z onto Tp(xi): f(p) = (p - oi) . ni, extract the isosurface from the signed distance function (field data). This is accomplished via the marching cubes algorithm. The resulting simplical surface contains triangles oriented at the surface of the point-sampled object.