Parametric Representation and Tessellation Lecture Notes

Parametric Representation (Continued)

• Focus on subdivision algorithms, curves, and surfaces.

• Subdivision is favored for being small, affordable, and efficient.

• Integrated into many GPUs and software.

Subdivision Algorithms

• Basic idea: Divide curves into segments and generate new control points on those segments iteratively.

Process:

1. Start with a curve defined by control points.

2. Generate new control vertex points along each segment.

3. Connect the new control points to form a new curve.

4. Repeat the process indefinitely. The definition of the rules is customizable.

• Quadratic B-spline example: Uses a specific rule set, resulting in a C1 continuous curve.

  • C1 Continuity: Continuity at the joint control points, meaning the segments are the same.

• Stopping condition in OpenGL: When the line segment is approximately one pixel.

• C2 Continuity: Derivative is the same at the joint control points.

  • Circle example: The derivative is always the same.

• Midpoint insertion: Another way to achieve C2 continuity, resulting in a smoother curve.

Subdivision for Smooth Surfaces

• Goal: To make the overall structure (curves or surfaces) as smooth as possible in 2D or 3D.

• Curves are represented using a single parameter $t$ for $x$ and $y$, and surfaces use $u$ and $v$ to represent $x$, $y$, and $z$ in 3D.

Loop Subdivision (Surfaces)

• A generalization of curve subdivision to 2D surfaces.

• Iteratively refines a mesh to create a smoother and more detailed surface.

• Process: Add new surface points, creating smaller polygons, gradually approximating a smoother surface.

Weights:

• Different weights are assigned to different control points (similar to using a fraction $t$ from 0 to 1).

• Control points contribute to the new polygon vertex based on assigned weights.

• Points closer to a vertex have a larger weight.

• The process is stopped in iterations.

Extraordinary Polygons Subdivision

• Addresses cases with irregular polygon shapes and sizes.

• Involves generating new control points based on the polygons.

• Equations for weight calculation:

  • $\alpha_A$ = weight for different polygons

  • $k$ = number of vertices in the polygon

  • $\alpha_i$ = calculated weight

  • $r_{part \, 0}$ = Weight calculated nearest using this equation

  • $r_{part \, i}$ = Weight calculation far away from point using this polygon.

  • $R_i = \frac{4 - (2cos(2\pi i / k) + 2)}{k}$

• The goal is uniformity when dealing with surfaces.

Coupler Clock

Key Terms:

• Face Point: A new point defined on a face.

• Edge Point: A new point defined on an edge.

• Original Point: The original control point.

• Process involves defining new face points and edge points, reconnecting them to form new segments and structures.

• New vertices may occur on faces or edges of the original mesh.

Bump Mapping vs. Displacement Mapping

• Bump Mapping: Simulates surface detail without changing the actual shape.

• Displacement Mapping: Modifies the actual surface by giving different heights.

• Limitation of Bump Mapping: It doesn't change the actual shape or surface size, leading to smooth shading even on edges.

• Advantage of Bump Mapping: More efficient as it simulates detail without altering the geometry.

Tessellation

• A built-in technique in modern GPUs for rendering.

• Offloads the refinement of triangles from the CPU to the GPU.

• Refines coarse geometry into finer, more detailed geometry.

• Involves subdividing polygons into smaller tessellations or triangles.

Tessellation Stages:

• Control Shader (Tessellation Control Shader):

  • Decides the amount of tessellation applied to a patch (a coarse polynomial surface).

  • Outputs tessellation levels for edges and the interior of patches.

  • Ensures continuity between neighboring patches (C1, C2 continuity).

• Primitive Generator:

  • Subdivides input patches into smaller geometric primitives (usually triangles).

• Evaluation Shader (Tessellation Evaluation Shader):

  • Processes the subdivided geometry from the generator.

  • Calculates attributes for the new vertices.

Tessellation Levels:

• Outer Tessellation Level: Defines tessellation for the edges.

• Inner Tessellation Level: Defines tessellation inside the patch.

• Outer tessellation levels (four values) determine how edges are subdivided (e.g., a level of 4 means an edge is divided into four).

• Inner levels define the division of the interior of the object.

Recap of Parametric Representations

• Using a single parameter $t$ to represent curves where a single x corresponds to two y values.

• Extending from 2D curves to 3D surfaces using parameters $u$ and $v$.

• Generating new curves based on control points by assigning different weights (fractions between 0 and 1).

• Subdivision methods for curves and surfaces involve defining new control points based on midpoints or other rules.

• C1 continuity (joint continuity) vs. C2 continuity (derivative continuity).

• Tessellation is now implemented in OpenGL, offloading computations from CPU to GPU for faster processing.