《建筑几何》Chapter 11：自由形曲面

2018-06-05 | 阅读：次

Freeform surfaces 自由形曲面

Bezier Surfaces

Translational Bézier surfaces：

Each boundary polygon of the control mesh defines a Bézier curve that is a boundary curve of the designed Bézier surface patch. Bezier曲面的边界是Bezier曲线.

The boundary polygons are the only row and column polygons that define curves on the surface. 内部控制多边形生成的Bezier曲线，不一定在曲面上，除非是translational bezier 曲面:

Bézier surfaces of degree (1,1)：

$r_0 = (1 – v)b_{00} + v b_{01} $

$r_1 = (1 – v)b_{10} + vb_{11}$

$b(u,v) = (1 – u)(1 – v)b_{00} + (1 – u)vb_{01} + u(1 – v)b_{10} + uvb_{11}$

This shows that the surface is (part of ) a hyperbolic paraboloid (or part of a plane if the control quad is planar).

Bézier surfaces that are also ruled surfaces：

Consider a Bézier surface of degree $(1,n)$. Its u-curves are Bézier curves of degree 1 and therefore straight line segments. Therefore, this surface is a ruled surface—spanning two Bézier curves of degree $n$.

Bézier surfaces joined smoothly：

B-spline surfaces and NURBS Surfaces

Open mode for u- and v-curves: the surface is a four-sided patch.

Closed mode in one direction (u or v), open mode in the other direction: such a surface looks like a deformed piece of a pipe.

Closed mode in both directions: the surface looks like a deformed torus.

Interpolating spline surfaces:

This interpolation problem in its most general form is difficult to solve. However, the following solution is easy and is implemented in most 3D modelers: Given a set of points, arranged like the control points in a quadrilateral mesh, pass a B-spline surface through it.

Meshes

Roughly speaking, a mesh is a collection of points (vertices) arranged into basic elements called faces.

The faces are bounded by polygons. Typically, one type of polygon dominates (e.g., triangle, quadrilateral, or even hexagon).

Geometry and connectivity:

first discuss their connectivity—also refered to as their “mesh topology.”
Quadrilateral meshes:

In a quad mesh, an interior vertex of valence 4 is called a regular vertex. If the valence is different from four, we talk about an irregular vertex.

Triangle meshes:

Triangle meshes are well suited for architecture because their faces are planar.

we may roughly need twice as many triangles as quads to represent the same shape

Hexagonal meshes:

It is useful to think about refinement as a two-step procedure:

change the connectivity (number of vertices and the way they are connected)

and then change the geometry (the position of the vertices).

Mesh decimation

reverse of mesh refinement (mesh decimation)

Aesthetics of meshes and relaxation:

One basic idea is to adjust all mesh vertices so that the resulting mesh consists of nearly regular faces only.

A related idea is a physical interpretation using a so-called mass-spring simulation system.

one implements the vertices as mass points and the edges as springs.

Then, one fixes some points (e.g., all boundary vertices) and lets the other vertices move freely until the entire system achieves an equilibrium . -This technique is called relaxation.