Introduction to Meshes

1. Introduction to Meshes Lecture 22 Mon, Oct 20, 2003

2. Chapter 6 Modeling Shapes with Polygonal Meshes

3. Introduction to Solid Modeling with Polygonal Meshes • In computer graphics, we do not draw surfaces that are truly curved. • Instead, each surface consists of many small polygons connected in a mesh. • This is also called a wireframe. • If we fill the polygons in the mesh, then the surface looks solid.

4. Solid Model

5. Wireframe Model

6. Solid Model (69451 faces)

7. Wireframe Model

8. Close-up of Wireframe Model

9. Polygonal Faces • The simplest of all polygonal faces is the triangle. • Triangles have two major advantages. • All triangles are planar. • All triangles are convex.

10. R n P Q One Normal for the Face • The normal may be calculated from the vertices. • n = (Q – P)  (R – P).

11. n2 n1 n3 One Normal for each Vertex • The normal at each vertex may be perpendicular to the face. • n1 = n2 = n3.

12. n2 n1 n3 One Normal for each Vertex • The normal at each vertex may point in a unique direction. • n1n2n3.

13. Lighting and Normals • When using a lighting model, the brightness and color of the surface are determined, in part, by the normals. • If there is one normal, the surface is of uniform brightness. • This is called flat shading.

14. Lighting and Normals • If each vertex has a distinct normal, then the brightness may vary over the surface. • This is called smooth shading.

15. One Normal, Uniform Shading

16. Distinct Normals, Varying Shading

17. Shading a Triangle • We will study lighting models later. • Lighting models determine how to color, or shade, a vertex. • For now, we will assume that each vertex has been assigned a shade. • Note: “shading” has nothing to do with shadows. It refers to the shade of color.

18. Shading a Triangle • Each point in the interior and on the boundary of a triangle can be expressed as a linear combination aP + bQ + cR of the three vertices of the triangle, where a + b + c = 1. • These are the barycentric coordinates.

19. Shading a Triangle • The coefficients a, b, c are used to determine the color of the point in the triangle. • The color of the point is the same linear combination of the colors of the vertices. • This is done for each pixel that is part of the triangle.

20. R(4, 8), color = (1, 0, 0) Q(12, 0) color = (0, 1, 1) P(0, 0) color = (1, 1, 0) Shading a Triangle • Consider this triangle.

21. R(4, 8), color = (1, 0, 0) Q(12, 0) color = (0, 1, 1) P(0, 0) color = (1, 1, 0) Shading a Triangle • What is the color of the outlined pixel?

22. Shading a Triangle • The coordinates are (3, 3). • Solve the system a(0, 0) + b(12 ,0) + c(4, 8) = (3, 3), a + b + c = 1. • The equations are 12b + 4c = 3, 8c = 3, a + b + c = 1.

23. Shading a Triangle • The solution is • a = 1/2. • b = 1/8. • c = 3/8. • Therefore, the color of the pixel is (1/2)(1, 1, 0) + (1/8)(0, 1, 1) + (3/8)(1, 0, 0) = (7/8, 5/8, 1/8).

24. R(4, 8), color = (1, 0, 0) color = (7/8, 5/8, 1/8) Q(12, 0) color = (0, 1, 1) P(0, 0) color = (1, 1, 0) Shading a Triangle

25. R(4, 8) Q(12, 0) P(0, 0) Shading a Triangle • The entire triangle