Bilinear interpolation on a quadrilateral using Barycentric coordinates

Summary

In computer graphics, we rarely encounter continuous data. We often work with digital data, and in the context of geometric modeling, this means we typically work with polygon meshes rather than procedural surfaces like Bézier patches. The most popular technique for constructing digital three-dimensional objects in dedicated modeling software is polygon modeling. The result of the creation phase is a set of polygons (mesh), where the polygons in the mesh can share vertices and edges with other polygons. Although users can create various types of surfaces (e.g., non-manifold), the most common surface is the topological 2-manifold. In short, a 2-manifold is a mathematical concept in topology, where the space locally resembles the Euclidean plane in \mathbb{R}^2. Essentially, every point on a 2-manifold has a neighborhood that looks like a piece of the plane. Examples of 2-manifold mesh triangle-based Examples of 2-manifold mesh quad-based At the vertices of a polygon, users can store additional data (per-vertex attributes), such as vertex normals (for simulating curved surfaces), texture coordinates (for texture mapping), or RGBA color. In theory, all types of polygons can be used. In practice, however, 3D graphics artists most commonly use triangles and quadrilaterals. These polygons are typically referred to as topology primitives in computer graphics APIs (such as Microsoft DirectX® or Vulkan®). From an artist’s point of view, quadrilaterals are more advantageous because they are easier to work with. The properties of modeling based on quadrilaterals include: The ease of generating grid-like surfaces. Creating a topology that provides an edge flow that can be easily adjusted (when adding or deleting edge loops). Producing a predictable result for subdivision algorithms. not clean looking mesh clean looking mesh These arguments make the quadrilateral-based topology preferred by artists when modeling 3D objects. Long ago, GPUs abandoned support for hardware accelerat...