《建筑几何》Chapter 19：形状自由形结构

2018-07-22 | 阅读：次

Discrete freeform structure

Triangle meshes 三角网格

Aesthetics is greatly enhanced if the mesh can be decomposed into three families of fair polygons (structure lines).

six beams meeting in a node means a significantly higher node complexity compared to other types of meshes.

the per-area cost of triangular glass panels is higher than that of quadrilateral panels.

one aims at less steel, more glass, and less weight, which also points to non-triangular faces.

triangle meshes in general torsion-free nodes do not exist.

triangle meshes do not possess offsets at constant face-face or edge-edge distance.

Quadrilateral meshes with palnar faces 具有平面面的四边网格

Rotational PQ mesh.

Conjugate curve networks.
Pick a curve c in the network and compute in each of its points the tangent to the curve from the other family. Then, these tangents must form a developable ruled surface.
A network of curves resulting by translation is another simple example of a conjugate curve network.
It carries infinitely many conjugate curve networks!

Hyperbolic paraboloid.

Principal curvature lines.
curve network that is conjugate and orthogonal.

Planar quad meshes are discrete versions of conjugate curve networks.

A planarization algorithm.
if the input mesh has been extracted from a conjugate curve network chances are high that the optimization will not get stuck in an undesirable solution and thus may yield a practically useful result

Combination of subdivision and planarization as a design tool.

Parallel meshes, offsets, and supporting beam layout 平行网格，等距面，支撑梁柱结构

We call two meshes M, M* parallel if there is a one-to-one correspondence among their vertices, edges, and faces (such meshes are called combinatorially equivalent) and if corresponding edges are parallel.

any mesh parallel to a PQ mesh is also a PQ mesh.

Requiring that the node axes be roughly orthogonal to the underlying design surface implies that the mesh M has planar or spherical shape. For a general freeform triangle mesh M, there is no chance to construct a practically useful support structure with torsion-free nodes.

Offset meshes 等距网格

Offset surfaces revisited.
Note that the normals of F are normals of F d as well
The tangent planes at corresponding points of F and F d are parallel and at constant distance d.

PQ meshes with exact offsets.
An offset mesh Md of a PQ mesh M is parallel to M and lies at constant distance d to M.

Vertex offsets: The distance of corresponding vertices of M and Md has a constant value d, which does not depend on the vertex.
Md is a vertex offset of M if and only if the vertices of the Gaussian image mesh S are contained in the unit sphere S*. In this case, M and Md are circular meshes (i.e., each face has a circum-circle).

Edge offsets: The distance of corresponding parallel edges of M and Md (actually, lines that carry these edges) does not depend on the edge and equals d.
Md is an edge offset of M exactly if the edges of the Gaussian image mesh S are tangent to the unit sphere S*.

Face offsets: The distance of corresponding faces of M and Md (actually, the parallel planes of corresponding faces) is independent of the face and equals d.
Md is a face offset of M if and only if the faces of the Gaussian image mesh S are tangent to the unit sphere S*. In this case, M and Md are conical meshes. cone condition at a vertex: The sum of opposite edge angles must be equal: $ω_1 + ω_3 = ω_2 +ω_4.$
This angle balance condition may also be seen as a discrete condition for orthogonality of the two mesh polygons passing through a vertex.
Because conical meshes are orthogonal and conjugate in a discrete sense, conical meshes are a discrete version of the network of principal curvature lines on a smooth surface.