Surfaces

#comp37111

Various ways to represent surfaces:

• Explicit: Mesh (analogous to polylines), Point Clouds.
  • Can be triangle or polylines meshes. Each polygon is called a face and each point is called a vertex.
• Tensor Product Splines: Analogous to Spline Curves.
• Implicit: Formulaic
• Procedural

1. Point Set / Clouds

• When scanning some object with a laser the output is a point cloud.
• Being favoured recently in combination with implicit surfaces.

2. Tensor Product Splines

• Cubic Bezier Surface $$P(u, v) = (1-u)^3 P_1 (v) + 3u (1-u)^2 P_2(v) + 3u^2 (1-u) P_3 (v) + u^3 P_4(v)$$

• A small part of a surface is known as a patch.

• Matrix Notation: $$P^x(u ,v) = \begin{pmatrix} B_1 (u) \ldots B_4(u) \end{pmatrix} \begin{pmatrix} P^x_{1, 1} \ldots P^x_{1, 4} \ . \ . \ P^x_{4, 1} \ldots P^x_{4, 4}\end{pmatrix} \begin{pmatrix} B_1 (v) \ . \ . \ B_4(v) \end{pmatrix}$$

• Tensor Product Splines:

  • Pros: resulting surface is quite smooth + can be described using small set of points
  • Cons: harder to render (converted to triangle meshes) + tricky to ensure continuity at patch boundaries

• Displacement is a way to add discontinuity to a surface. Some noise (displacement map) is added to the surface and points are displaced by the corresponding value in the direction of the surface normal. We can then use Tessellation to render the curve.

• Corner Cutting Subdivision Technique (Chaikin's Algorithm):

  • Starting from a polygon mesh (control polygon), connect the mid points of the line segment joining the control points
  • Results in a smoother surface
  • This produces a quadratic B-Spline curve

3. Implicit Surfaces (Formulaic)

• Pros:
  • Efficient to check if a point is in the surface or not (just plug into the formula)
  • Efficient for boolean operations
  • Can handle weird topology (holes in objects)
• Cons:
  • Hard to generate points on the surface or add points to the surface.

4. Procedural

• Ex: Surface of Revolution, revolve a base shape around the basis $s(u,v)=R(v)q(u)$.
• Ex: Generalised Cylinder, move a 2D curve along a trajectory $s(u,v)=M(c(v))q(u)$. Surface of a revolution can be seen as a special case of this.

Differential Properties of Surfaces

• Frenet Frame:
  • derivatives of a surface.
  • problem with using the 2nd derivative because of inflection points, points where the direction of normal changes its sign.